TPTP Problem File: SLH0844^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Query_Optimization/0013_IKKBZ_Optimality/prob_02764_123151__15795878_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1706 ( 523 unt; 439 typ;   0 def)
%            Number of atoms       : 3802 (1653 equ;   0 cnn)
%            Maximal formula atoms :   19 (   3 avg)
%            Number of connectives : 12938 ( 598   ~;  46   |; 366   &;10253   @)
%                                         (   0 <=>;1675  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   7 avg)
%            Number of types       :   84 (  83 usr)
%            Number of type conns  : 1589 (1589   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  359 ( 356 usr;  23 con; 0-4 aty)
%            Number of variables   : 4500 ( 379   ^;3856   !; 265   ?;4500   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:07:55.448
%------------------------------------------------------------------------------
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    accp_P4272997315293813680st_a_b: ( produc2725289747333967015st_a_b > produc2725289747333967015st_a_b > $o ) > produc2725289747333967015st_a_b > $o ).

thf(sy_c_member_001_062_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_M_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_Mt__Real__Oreal_J_J,type,
    member2196815172658074515b_real: ( list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real ) > set_li8863668716876874418b_real > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_M_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J_J,type,
    member3358527249890354327_a_b_o: ( produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o ) > set_Pr750648071918454478_a_b_o > $o ).

thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
    member_a_o: ( a > $o ) > set_a_o > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member1308482598881763409st_a_b: list_P2281458131032066856st_a_b > set_li7460651515946730888st_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_I_062_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_M_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_Mt__Real__Oreal_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    member2006299769585415352st_a_b: produc6292069877431205281st_a_b > set_Pr3822622692308018519st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_M_062_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    member3277860331849966602st_a_b: produc8664631368769065441st_a_b > set_Pr7060751113198729409st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
    member6028662021613063888st_a_b: produc2725289747333967015st_a_b > set_Pr3892374080263479431st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J,type,
    member8006451236149131979st_a_b: produc2579390649552321826st_a_b > set_Pr9033091269622839554st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member7191766420993298128st_a_b: produc2313410840657238951st_a_b > set_Pr4988534092439063687st_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member2248238449149680884st_a_b: produc5365630598061846603st_a_b > set_Pr4431676530321648299st_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__b_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member5954712591949503667st_a_b: produc9072104740861669386st_a_b > set_Pr5427516116882750186st_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    member_set_list_a: set_list_a > set_set_list_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
    member358942219636986539st_a_b: set_Pr9033091269622839554st_a_b > set_se1825328058903261410st_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_a,type,
    a2: a ).

thf(sy_v_cmp,type,
    cmp: compar7542523840845723048st_a_b ).

thf(sy_v_e1____,type,
    e1: b ).

thf(sy_v_e2____,type,
    e2: b ).

thf(sy_v_v1____,type,
    v1: list_a ).

thf(sy_v_v2____,type,
    v2: list_a ).

thf(sy_v_x____,type,
    x: produc2579390649552321826st_a_b ).

thf(sy_v_xsa____,type,
    xsa: list_P2281458131032066856st_a_b ).

thf(sy_v_y____,type,
    y: produc2579390649552321826st_a_b ).

thf(sy_v_ysa____,type,
    ysa: list_P2281458131032066856st_a_b ).

% Relevant facts (1261)
thf(fact_0__092_060open_062y_A_061_A_Iv2_M_Ae2_J_092_060close_062,axiom,
    ( y
    = ( produc4781227316648555538st_a_b @ v2 @ e2 ) ) ).

% \<open>y = (v2, e2)\<close>
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v1_Ae1_O_Ax_A_061_A_Iv1_M_Ae1_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [V1: list_a,E1: b] :
        ( x
       != ( produc4781227316648555538st_a_b @ V1 @ E1 ) ) ).

% \<open>\<And>thesis. (\<And>v1 e1. x = (v1, e1) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v2_Ae2_O_Ay_A_061_A_Iv2_M_Ae2_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [V2: list_a,E2: b] :
        ( y
       != ( produc4781227316648555538st_a_b @ V2 @ E2 ) ) ).

% \<open>\<And>thesis. (\<And>v2 e2. y = (v2, e2) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3__C3_OIH_C_I2_J,axiom,
    ( ( ( compar8165832444956594340st_a_b @ cmp @ x @ y )
     != greater )
   => ( ! [X: list_a] :
          ( ( member_list_a @ X @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ xsa ) ) )
         => ~ ( member_a @ a2 @ ( set_a2 @ X ) ) )
     => ( ord_less_eq_set_a @ ( iKKBZ_8930711501659780123st_a_b @ ( cons_P2018802354021969880st_a_b @ y @ ysa ) @ a2 ) @ ( iKKBZ_8930711501659780123st_a_b @ ( sortin5994461831780029029st_a_b @ cmp @ xsa @ ( cons_P2018802354021969880st_a_b @ y @ ysa ) ) @ a2 ) ) ) ) ).

% "3.IH"(2)
thf(fact_4__C3_OIH_C_I1_J,axiom,
    ( ( ( compar8165832444956594340st_a_b @ cmp @ x @ y )
      = greater )
   => ( ! [X: list_a] :
          ( ( member_list_a @ X @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ x @ xsa ) ) ) )
         => ~ ( member_a @ a2 @ ( set_a2 @ X ) ) )
     => ( ord_less_eq_set_a @ ( iKKBZ_8930711501659780123st_a_b @ ysa @ a2 ) @ ( iKKBZ_8930711501659780123st_a_b @ ( sortin5994461831780029029st_a_b @ cmp @ ( cons_P2018802354021969880st_a_b @ x @ xsa ) @ ysa ) @ a2 ) ) ) ) ).

% "3.IH"(1)
thf(fact_5_subsetI,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A )
         => ( member1426531477525435216od_a_a @ X @ B ) )
     => ( ord_le746702958409616551od_a_a @ A @ B ) ) ).

% subsetI
thf(fact_6_subsetI,axiom,
    ! [A: set_li7460651515946730888st_a_b,B: set_li7460651515946730888st_a_b] :
      ( ! [X: list_P2281458131032066856st_a_b] :
          ( ( member1308482598881763409st_a_b @ X @ A )
         => ( member1308482598881763409st_a_b @ X @ B ) )
     => ( ord_le2045451444129209640st_a_b @ A @ B ) ) ).

% subsetI
thf(fact_7_subsetI,axiom,
    ! [A: set_se1825328058903261410st_a_b,B: set_se1825328058903261410st_a_b] :
      ( ! [X: set_Pr9033091269622839554st_a_b] :
          ( ( member358942219636986539st_a_b @ X @ A )
         => ( member358942219636986539st_a_b @ X @ B ) )
     => ( ord_le4557451523740326018st_a_b @ A @ B ) ) ).

% subsetI
thf(fact_8_subsetI,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ! [X: list_a] :
          ( ( member_list_a @ X @ A )
         => ( member_list_a @ X @ B ) )
     => ( ord_le8861187494160871172list_a @ A @ B ) ) ).

% subsetI
thf(fact_9_subsetI,axiom,
    ! [A: set_set_list_a,B: set_set_list_a] :
      ( ! [X: set_list_a] :
          ( ( member_set_list_a @ X @ A )
         => ( member_set_list_a @ X @ B ) )
     => ( ord_le8877086941679407844list_a @ A @ B ) ) ).

% subsetI
thf(fact_10_subsetI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ A )
         => ( member_set_a @ X @ B ) )
     => ( ord_le3724670747650509150_set_a @ A @ B ) ) ).

% subsetI
thf(fact_11_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ X @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_12_subset__antisym,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_13_subset__antisym,axiom,
    ! [A: set_set_list_a,B: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B )
     => ( ( ord_le8877086941679407844list_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_14_subset__antisym,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_15_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_16_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_17_list_Oinject,axiom,
    ! [X21: list_P2281458131032066856st_a_b,X22: list_l3733004638376709294st_a_b,Y21: list_P2281458131032066856st_a_b,Y22: list_l3733004638376709294st_a_b] :
      ( ( ( cons_l7524743740370447070st_a_b @ X21 @ X22 )
        = ( cons_l7524743740370447070st_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_18_list_Oinject,axiom,
    ! [X21: produc2313410840657238951st_a_b,X22: list_P1351356133043840941st_a_b,Y21: produc2313410840657238951st_a_b,Y22: list_P1351356133043840941st_a_b] :
      ( ( ( cons_P5669161040283791197st_a_b @ X21 @ X22 )
        = ( cons_P5669161040283791197st_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_19_list_Oinject,axiom,
    ! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
      ( ( ( cons_list_a @ X21 @ X22 )
        = ( cons_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_20_list_Oinject,axiom,
    ! [X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b,Y21: produc2579390649552321826st_a_b,Y22: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X21 @ X22 )
        = ( cons_P2018802354021969880st_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_21_order__refl,axiom,
    ! [X2: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_22_order__refl,axiom,
    ! [X2: set_set_list_a] : ( ord_le8877086941679407844list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_23_order__refl,axiom,
    ! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_24_order__refl,axiom,
    ! [X2: list_a > b > $o] : ( ord_le8305133717419367763_a_b_o @ X2 @ X2 ) ).

% order_refl
thf(fact_25_order__refl,axiom,
    ! [X2: a > $o] : ( ord_less_eq_a_o @ X2 @ X2 ) ).

% order_refl
thf(fact_26_order__refl,axiom,
    ! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_27_dual__order_Orefl,axiom,
    ! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
    ! [A2: set_set_list_a] : ( ord_le8877086941679407844list_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
    ! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
    ! [A2: list_a > b > $o] : ( ord_le8305133717419367763_a_b_o @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_31_dual__order_Orefl,axiom,
    ! [A2: a > $o] : ( ord_less_eq_a_o @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_32_dual__order_Orefl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_33__C3_Oprems_C,axiom,
    ! [X3: list_a] :
      ( ( member_list_a @ X3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ x @ xsa ) ) ) )
     => ~ ( member_a @ a2 @ ( set_a2 @ X3 ) ) ) ).

% "3.prems"
thf(fact_34_v1__def,axiom,
    ( x
    = ( produc4781227316648555538st_a_b @ v1 @ e1 ) ) ).

% v1_def
thf(fact_35_merge__assoc,axiom,
    ! [Cmp: comparator_a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( sorting_merge_a @ Cmp @ Xs @ ( sorting_merge_a @ Cmp @ Ys @ Zs ) )
      = ( sorting_merge_a @ Cmp @ ( sorting_merge_a @ Cmp @ Xs @ Ys ) @ Zs ) ) ).

% merge_assoc
thf(fact_36_merge__assoc,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ ( sortin5994461831780029029st_a_b @ Cmp @ Ys @ Zs ) )
      = ( sortin5994461831780029029st_a_b @ Cmp @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) @ Zs ) ) ).

% merge_assoc
thf(fact_37_path__lverts__list__merge__supset__xs__notin,axiom,
    ! [Ys: list_P2281458131032066856st_a_b,A2: a,Xs: list_P2281458131032066856st_a_b,Cmp: compar7542523840845723048st_a_b] :
      ( ! [X: list_a] :
          ( ( member_list_a @ X @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Ys ) ) )
         => ~ ( member_a @ A2 @ ( set_a2 @ X ) ) )
     => ( ord_less_eq_set_a @ ( iKKBZ_8930711501659780123st_a_b @ Xs @ A2 ) @ ( iKKBZ_8930711501659780123st_a_b @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) @ A2 ) ) ) ).

% path_lverts_list_merge_supset_xs_notin
thf(fact_38_set__subset__Cons,axiom,
    ! [Xs: list_P2989903729216022225st_a_b,X2: produc5365630598061846603st_a_b] : ( ord_le8652052591400137291st_a_b @ ( set_Pr2913479409760856416st_a_b @ Xs ) @ ( set_Pr2913479409760856416st_a_b @ ( cons_P1986909248207666433st_a_b @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_39_set__subset__Cons,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,X2: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_40_set__subset__Cons,axiom,
    ! [Xs: list_b,X2: b] : ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_41_set__subset__Cons,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,X2: list_P2281458131032066856st_a_b] : ( ord_le2045451444129209640st_a_b @ ( set_li5246279850479030973st_a_b @ Xs ) @ ( set_li5246279850479030973st_a_b @ ( cons_l7524743740370447070st_a_b @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_42_set__subset__Cons,axiom,
    ! [Xs: list_P1351356133043840941st_a_b,X2: produc2313410840657238951st_a_b] : ( ord_le1374440848537019431st_a_b @ ( set_Pr5950792022122277180st_a_b @ Xs ) @ ( set_Pr5950792022122277180st_a_b @ ( cons_P5669161040283791197st_a_b @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_43_set__subset__Cons,axiom,
    ! [Xs: list_list_a,X2: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_44_set__subset__Cons,axiom,
    ! [Xs: list_set_list_a,X2: set_list_a] : ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ ( set_set_list_a2 @ ( cons_set_list_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_45_set__subset__Cons,axiom,
    ! [Xs: list_set_a,X2: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_46_set__subset__Cons,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] : ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_47_set__subset__Cons,axiom,
    ! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_48_image__eqI,axiom,
    ! [B2: a,F: a > a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_49_image__eqI,axiom,
    ! [B2: list_a,F: produc2579390649552321826st_a_b > list_a,X2: produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member8006451236149131979st_a_b @ X2 @ A )
       => ( member_list_a @ B2 @ ( image_4065140400681997415list_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_50_image__eqI,axiom,
    ! [B2: list_a,F: a > list_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member_list_a @ B2 @ ( image_a_list_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_51_image__eqI,axiom,
    ! [B2: set_a,F: a > set_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_52_image__eqI,axiom,
    ! [B2: a,F: list_a > a,X2: list_a,A: set_list_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_list_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_list_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_53_image__eqI,axiom,
    ! [B2: a,F: set_a > a,X2: set_a,A: set_set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_set_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_54_image__eqI,axiom,
    ! [B2: product_prod_a_a,F: a > product_prod_a_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member1426531477525435216od_a_a @ B2 @ ( image_7400625782589995694od_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_55_image__eqI,axiom,
    ! [B2: a,F: product_prod_a_a > a,X2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member1426531477525435216od_a_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_3437945252899457948_a_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_56_image__eqI,axiom,
    ! [B2: list_a,F: list_a > list_a,X2: list_a,A: set_list_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_list_a @ X2 @ A )
       => ( member_list_a @ B2 @ ( image_list_a_list_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_57_image__eqI,axiom,
    ! [B2: set_a,F: list_a > set_a,X2: list_a,A: set_list_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_list_a @ X2 @ A )
       => ( member_set_a @ B2 @ ( image_list_a_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_58_image__ident,axiom,
    ! [Y: set_a] :
      ( ( image_a_a
        @ ^ [X4: a] : X4
        @ Y )
      = Y ) ).

% image_ident
thf(fact_59_imageE,axiom,
    ! [B2: a,F: a > a,A: set_a] :
      ( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_60_imageE,axiom,
    ! [B2: list_a,F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( member_list_a @ B2 @ ( image_4065140400681997415list_a @ F @ A ) )
     => ~ ! [X: produc2579390649552321826st_a_b] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member8006451236149131979st_a_b @ X @ A ) ) ) ).

% imageE
thf(fact_61_imageE,axiom,
    ! [B2: a,F: list_a > a,A: set_list_a] :
      ( ( member_a @ B2 @ ( image_list_a_a @ F @ A ) )
     => ~ ! [X: list_a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_list_a @ X @ A ) ) ) ).

% imageE
thf(fact_62_imageE,axiom,
    ! [B2: a,F: set_a > a,A: set_set_a] :
      ( ( member_a @ B2 @ ( image_set_a_a @ F @ A ) )
     => ~ ! [X: set_a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_set_a @ X @ A ) ) ) ).

% imageE
thf(fact_63_imageE,axiom,
    ! [B2: list_a,F: a > list_a,A: set_a] :
      ( ( member_list_a @ B2 @ ( image_a_list_a @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_64_imageE,axiom,
    ! [B2: set_a,F: a > set_a,A: set_a] :
      ( ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_65_imageE,axiom,
    ! [B2: a,F: product_prod_a_a > a,A: set_Product_prod_a_a] :
      ( ( member_a @ B2 @ ( image_3437945252899457948_a_a_a @ F @ A ) )
     => ~ ! [X: product_prod_a_a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member1426531477525435216od_a_a @ X @ A ) ) ) ).

% imageE
thf(fact_66_imageE,axiom,
    ! [B2: product_prod_a_a,F: a > product_prod_a_a,A: set_a] :
      ( ( member1426531477525435216od_a_a @ B2 @ ( image_7400625782589995694od_a_a @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_67_imageE,axiom,
    ! [B2: list_a,F: list_a > list_a,A: set_list_a] :
      ( ( member_list_a @ B2 @ ( image_list_a_list_a @ F @ A ) )
     => ~ ! [X: list_a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_list_a @ X @ A ) ) ) ).

% imageE
thf(fact_68_imageE,axiom,
    ! [B2: list_a,F: set_a > list_a,A: set_set_a] :
      ( ( member_list_a @ B2 @ ( image_set_a_list_a @ F @ A ) )
     => ~ ! [X: set_a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_set_a @ X @ A ) ) ) ).

% imageE
thf(fact_69_imageI,axiom,
    ! [X2: a,A: set_a,F: a > a] :
      ( ( member_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_70_imageI,axiom,
    ! [X2: produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b,F: produc2579390649552321826st_a_b > list_a] :
      ( ( member8006451236149131979st_a_b @ X2 @ A )
     => ( member_list_a @ ( F @ X2 ) @ ( image_4065140400681997415list_a @ F @ A ) ) ) ).

% imageI
thf(fact_71_imageI,axiom,
    ! [X2: a,A: set_a,F: a > list_a] :
      ( ( member_a @ X2 @ A )
     => ( member_list_a @ ( F @ X2 ) @ ( image_a_list_a @ F @ A ) ) ) ).

% imageI
thf(fact_72_imageI,axiom,
    ! [X2: a,A: set_a,F: a > set_a] :
      ( ( member_a @ X2 @ A )
     => ( member_set_a @ ( F @ X2 ) @ ( image_a_set_a @ F @ A ) ) ) ).

% imageI
thf(fact_73_imageI,axiom,
    ! [X2: list_a,A: set_list_a,F: list_a > a] :
      ( ( member_list_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_list_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_74_imageI,axiom,
    ! [X2: set_a,A: set_set_a,F: set_a > a] :
      ( ( member_set_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_set_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_75_imageI,axiom,
    ! [X2: a,A: set_a,F: a > product_prod_a_a] :
      ( ( member_a @ X2 @ A )
     => ( member1426531477525435216od_a_a @ ( F @ X2 ) @ ( image_7400625782589995694od_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_76_imageI,axiom,
    ! [X2: product_prod_a_a,A: set_Product_prod_a_a,F: product_prod_a_a > a] :
      ( ( member1426531477525435216od_a_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_3437945252899457948_a_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_77_imageI,axiom,
    ! [X2: list_a,A: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ X2 @ A )
     => ( member_list_a @ ( F @ X2 ) @ ( image_list_a_list_a @ F @ A ) ) ) ).

% imageI
thf(fact_78_imageI,axiom,
    ! [X2: list_a,A: set_list_a,F: list_a > set_a] :
      ( ( member_list_a @ X2 @ A )
     => ( member_set_a @ ( F @ X2 ) @ ( image_list_a_set_a @ F @ A ) ) ) ).

% imageI
thf(fact_79_image__iff,axiom,
    ! [Z: produc2579390649552321826st_a_b,F: a > produc2579390649552321826st_a_b,A: set_a] :
      ( ( member8006451236149131979st_a_b @ Z @ ( image_1109531753604471593st_a_b @ F @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_80_image__iff,axiom,
    ! [Z: a,F: a > a,A: set_a] :
      ( ( member_a @ Z @ ( image_a_a @ F @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_81_image__iff,axiom,
    ! [Z: set_Pr9033091269622839554st_a_b,F: a > set_Pr9033091269622839554st_a_b,A: set_a] :
      ( ( member358942219636986539st_a_b @ Z @ ( image_8858725927978154761st_a_b @ F @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_82_image__iff,axiom,
    ! [Z: set_a,F: ( a > $o ) > set_a,A: set_a_o] :
      ( ( member_set_a @ Z @ ( image_a_o_set_a @ F @ A ) )
      = ( ? [X4: a > $o] :
            ( ( member_a_o @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_83_image__iff,axiom,
    ! [Z: set_a,F: a > set_a,A: set_a] :
      ( ( member_set_a @ Z @ ( image_a_set_a @ F @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_84_image__iff,axiom,
    ! [Z: list_a,F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( member_list_a @ Z @ ( image_4065140400681997415list_a @ F @ A ) )
      = ( ? [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_85_bex__imageD,axiom,
    ! [F: ( a > $o ) > set_a,A: set_a_o,P: set_a > $o] :
      ( ? [X3: set_a] :
          ( ( member_set_a @ X3 @ ( image_a_o_set_a @ F @ A ) )
          & ( P @ X3 ) )
     => ? [X: a > $o] :
          ( ( member_a_o @ X @ A )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_86_bex__imageD,axiom,
    ! [F: a > produc2579390649552321826st_a_b,A: set_a,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( image_1109531753604471593st_a_b @ F @ A ) )
          & ( P @ X3 ) )
     => ? [X: a] :
          ( ( member_a @ X @ A )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_87_bex__imageD,axiom,
    ! [F: a > set_Pr9033091269622839554st_a_b,A: set_a,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ? [X3: set_Pr9033091269622839554st_a_b] :
          ( ( member358942219636986539st_a_b @ X3 @ ( image_8858725927978154761st_a_b @ F @ A ) )
          & ( P @ X3 ) )
     => ? [X: a] :
          ( ( member_a @ X @ A )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_88_bex__imageD,axiom,
    ! [F: a > set_a,A: set_a,P: set_a > $o] :
      ( ? [X3: set_a] :
          ( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A ) )
          & ( P @ X3 ) )
     => ? [X: a] :
          ( ( member_a @ X @ A )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_89_bex__imageD,axiom,
    ! [F: a > a,A: set_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
          & ( P @ X3 ) )
     => ? [X: a] :
          ( ( member_a @ X @ A )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_90_bex__imageD,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,P: list_a > $o] :
      ( ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( image_4065140400681997415list_a @ F @ A ) )
          & ( P @ X3 ) )
     => ? [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ A )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_91_image__cong,axiom,
    ! [M: set_a_o,N: set_a_o,F: ( a > $o ) > set_a,G: ( a > $o ) > set_a] :
      ( ( M = N )
     => ( ! [X: a > $o] :
            ( ( member_a_o @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_a_o_set_a @ F @ M )
          = ( image_a_o_set_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_92_image__cong,axiom,
    ! [M: set_a,N: set_a,F: a > produc2579390649552321826st_a_b,G: a > produc2579390649552321826st_a_b] :
      ( ( M = N )
     => ( ! [X: a] :
            ( ( member_a @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_1109531753604471593st_a_b @ F @ M )
          = ( image_1109531753604471593st_a_b @ G @ N ) ) ) ) ).

% image_cong
thf(fact_93_image__cong,axiom,
    ! [M: set_a,N: set_a,F: a > set_Pr9033091269622839554st_a_b,G: a > set_Pr9033091269622839554st_a_b] :
      ( ( M = N )
     => ( ! [X: a] :
            ( ( member_a @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_8858725927978154761st_a_b @ F @ M )
          = ( image_8858725927978154761st_a_b @ G @ N ) ) ) ) ).

% image_cong
thf(fact_94_image__cong,axiom,
    ! [M: set_a,N: set_a,F: a > set_a,G: a > set_a] :
      ( ( M = N )
     => ( ! [X: a] :
            ( ( member_a @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_a_set_a @ F @ M )
          = ( image_a_set_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_95_image__cong,axiom,
    ! [M: set_a,N: set_a,F: a > a,G: a > a] :
      ( ( M = N )
     => ( ! [X: a] :
            ( ( member_a @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_a_a @ F @ M )
          = ( image_a_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_96_image__cong,axiom,
    ! [M: set_Pr9033091269622839554st_a_b,N: set_Pr9033091269622839554st_a_b,F: produc2579390649552321826st_a_b > list_a,G: produc2579390649552321826st_a_b > list_a] :
      ( ( M = N )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_4065140400681997415list_a @ F @ M )
          = ( image_4065140400681997415list_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_97_ball__imageD,axiom,
    ! [F: ( a > $o ) > set_a,A: set_a_o,P: set_a > $o] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ ( image_a_o_set_a @ F @ A ) )
         => ( P @ X ) )
     => ! [X3: a > $o] :
          ( ( member_a_o @ X3 @ A )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_98_ball__imageD,axiom,
    ! [F: a > produc2579390649552321826st_a_b,A: set_a,P: produc2579390649552321826st_a_b > $o] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ ( image_1109531753604471593st_a_b @ F @ A ) )
         => ( P @ X ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_99_ball__imageD,axiom,
    ! [F: a > set_Pr9033091269622839554st_a_b,A: set_a,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ! [X: set_Pr9033091269622839554st_a_b] :
          ( ( member358942219636986539st_a_b @ X @ ( image_8858725927978154761st_a_b @ F @ A ) )
         => ( P @ X ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_100_ball__imageD,axiom,
    ! [F: a > set_a,A: set_a,P: set_a > $o] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ ( image_a_set_a @ F @ A ) )
         => ( P @ X ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_101_ball__imageD,axiom,
    ! [F: a > a,A: set_a,P: a > $o] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( image_a_a @ F @ A ) )
         => ( P @ X ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_102_ball__imageD,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,P: list_a > $o] :
      ( ! [X: list_a] :
          ( ( member_list_a @ X @ ( image_4065140400681997415list_a @ F @ A ) )
         => ( P @ X ) )
     => ! [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ A )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_103_image__image,axiom,
    ! [F: list_a > list_a,G: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( image_list_a_list_a @ F @ ( image_4065140400681997415list_a @ G @ A ) )
      = ( image_4065140400681997415list_a
        @ ^ [X4: produc2579390649552321826st_a_b] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_104_image__image,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,G: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( image_4065140400681997415list_a @ F @ ( image_7587371051487426309st_a_b @ G @ A ) )
      = ( image_4065140400681997415list_a
        @ ^ [X4: produc2579390649552321826st_a_b] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_105_image__image,axiom,
    ! [F: a > a,G: a > a,A: set_a] :
      ( ( image_a_a @ F @ ( image_a_a @ G @ A ) )
      = ( image_a_a
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_106_image__image,axiom,
    ! [F: set_a > a,G: a > set_a,A: set_a] :
      ( ( image_set_a_a @ F @ ( image_a_set_a @ G @ A ) )
      = ( image_a_a
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_107_image__image,axiom,
    ! [F: a > set_a,G: a > a,A: set_a] :
      ( ( image_a_set_a @ F @ ( image_a_a @ G @ A ) )
      = ( image_a_set_a
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_108_image__image,axiom,
    ! [F: set_a > set_a,G: a > set_a,A: set_a] :
      ( ( image_set_a_set_a @ F @ ( image_a_set_a @ G @ A ) )
      = ( image_a_set_a
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_109_image__image,axiom,
    ! [F: produc2579390649552321826st_a_b > a,G: a > produc2579390649552321826st_a_b,A: set_a] :
      ( ( image_4255047928038901089_a_b_a @ F @ ( image_1109531753604471593st_a_b @ G @ A ) )
      = ( image_a_a
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_110_image__image,axiom,
    ! [F: ( a > $o ) > set_a,G: a > a > $o,A: set_a] :
      ( ( image_a_o_set_a @ F @ ( image_a_a_o @ G @ A ) )
      = ( image_a_set_a
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_111_image__image,axiom,
    ! [F: a > produc2579390649552321826st_a_b,G: a > a,A: set_a] :
      ( ( image_1109531753604471593st_a_b @ F @ ( image_a_a @ G @ A ) )
      = ( image_1109531753604471593st_a_b
        @ ^ [X4: a] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_112_image__image,axiom,
    ! [F: a > set_a,G: ( a > $o ) > a,A: set_a_o] :
      ( ( image_a_set_a @ F @ ( image_a_o_a @ G @ A ) )
      = ( image_a_o_set_a
        @ ^ [X4: a > $o] : ( F @ ( G @ X4 ) )
        @ A ) ) ).

% image_image
thf(fact_113_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: a,F: a > a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_114_rev__image__eqI,axiom,
    ! [X2: produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b,B2: list_a,F: produc2579390649552321826st_a_b > list_a] :
      ( ( member8006451236149131979st_a_b @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_list_a @ B2 @ ( image_4065140400681997415list_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_115_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: list_a,F: a > list_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_list_a @ B2 @ ( image_a_list_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_116_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: set_a,F: a > set_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_117_rev__image__eqI,axiom,
    ! [X2: list_a,A: set_list_a,B2: a,F: list_a > a] :
      ( ( member_list_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_list_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_118_rev__image__eqI,axiom,
    ! [X2: set_a,A: set_set_a,B2: a,F: set_a > a] :
      ( ( member_set_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_119_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: product_prod_a_a,F: a > product_prod_a_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member1426531477525435216od_a_a @ B2 @ ( image_7400625782589995694od_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_120_rev__image__eqI,axiom,
    ! [X2: product_prod_a_a,A: set_Product_prod_a_a,B2: a,F: product_prod_a_a > a] :
      ( ( member1426531477525435216od_a_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_3437945252899457948_a_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_121_rev__image__eqI,axiom,
    ! [X2: list_a,A: set_list_a,B2: list_a,F: list_a > list_a] :
      ( ( member_list_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_list_a @ B2 @ ( image_list_a_list_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_122_rev__image__eqI,axiom,
    ! [X2: list_a,A: set_list_a,B2: set_a,F: list_a > set_a] :
      ( ( member_list_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_set_a @ B2 @ ( image_list_a_set_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_123_Collect__subset,axiom,
    ! [A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
      ( ord_le746702958409616551od_a_a
      @ ( collec3336397797384452498od_a_a
        @ ^ [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_124_Collect__subset,axiom,
    ! [A: set_se1825328058903261410st_a_b,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ord_le4557451523740326018st_a_b
      @ ( collec4420230097056376813st_a_b
        @ ^ [X4: set_Pr9033091269622839554st_a_b] :
            ( ( member358942219636986539st_a_b @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_125_Collect__subset,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ord_le2879471331418251426st_a_b
      @ ( collec3957028476971440141st_a_b
        @ ^ [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_126_Collect__subset,axiom,
    ! [A: set_li7460651515946730888st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ord_le2045451444129209640st_a_b
      @ ( collec3283141540447581331st_a_b
        @ ^ [X4: list_P2281458131032066856st_a_b] :
            ( ( member1308482598881763409st_a_b @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_127_Collect__subset,axiom,
    ! [A: set_list_a,P: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_128_Collect__subset,axiom,
    ! [A: set_set_list_a,P: set_list_a > $o] :
      ( ord_le8877086941679407844list_a
      @ ( collect_set_list_a
        @ ^ [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_129_Collect__subset,axiom,
    ! [A: set_set_a,P: set_a > $o] :
      ( ord_le3724670747650509150_set_a
      @ ( collect_set_a
        @ ^ [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_130_Collect__subset,axiom,
    ! [A: set_a,P: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_131_Compr__image__eq,axiom,
    ! [F: a > a,A: set_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_132_Compr__image__eq,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,P: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ ( image_4065140400681997415list_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_4065140400681997415list_a @ F
        @ ( collec3957028476971440141st_a_b
          @ ^ [X4: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_133_Compr__image__eq,axiom,
    ! [F: a > set_a,A: set_a,P: set_a > $o] :
      ( ( collect_set_a
        @ ^ [X4: set_a] :
            ( ( member_set_a @ X4 @ ( image_a_set_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_a_set_a @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_134_Compr__image__eq,axiom,
    ! [F: set_a > a,A: set_set_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_set_a_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_set_a_a @ F
        @ ( collect_set_a
          @ ^ [X4: set_a] :
              ( ( member_set_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_135_Compr__image__eq,axiom,
    ! [F: list_a > a,A: set_list_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_list_a_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_list_a_a @ F
        @ ( collect_list_a
          @ ^ [X4: list_a] :
              ( ( member_list_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_136_Compr__image__eq,axiom,
    ! [F: a > list_a,A: set_a,P: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ ( image_a_list_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_a_list_a @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_137_Compr__image__eq,axiom,
    ! [F: set_a > set_a,A: set_set_a,P: set_a > $o] :
      ( ( collect_set_a
        @ ^ [X4: set_a] :
            ( ( member_set_a @ X4 @ ( image_set_a_set_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_set_a_set_a @ F
        @ ( collect_set_a
          @ ^ [X4: set_a] :
              ( ( member_set_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_138_Compr__image__eq,axiom,
    ! [F: a > product_prod_a_a,A: set_a,P: product_prod_a_a > $o] :
      ( ( collec3336397797384452498od_a_a
        @ ^ [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( image_7400625782589995694od_a_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_7400625782589995694od_a_a @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_139_Compr__image__eq,axiom,
    ! [F: list_a > set_a,A: set_list_a,P: set_a > $o] :
      ( ( collect_set_a
        @ ^ [X4: set_a] :
            ( ( member_set_a @ X4 @ ( image_list_a_set_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_list_a_set_a @ F
        @ ( collect_list_a
          @ ^ [X4: list_a] :
              ( ( member_list_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_140_Compr__image__eq,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_3437945252899457948_a_a_a @ F @ A ) )
            & ( P @ X4 ) ) )
      = ( image_3437945252899457948_a_a_a @ F
        @ ( collec3336397797384452498od_a_a
          @ ^ [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ A )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_141_less__eq__set__def,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
          ( ord_le1591150415168442102_a_a_o
          @ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A3 )
          @ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_142_less__eq__set__def,axiom,
    ( ord_le2045451444129209640st_a_b
    = ( ^ [A3: set_li7460651515946730888st_a_b,B3: set_li7460651515946730888st_a_b] :
          ( ord_le8242720074258346933_a_b_o
          @ ^ [X4: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ X4 @ A3 )
          @ ^ [X4: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_143_less__eq__set__def,axiom,
    ( ord_le4557451523740326018st_a_b
    = ( ^ [A3: set_se1825328058903261410st_a_b,B3: set_se1825328058903261410st_a_b] :
          ( ord_le1523511841465843419_a_b_o
          @ ^ [X4: set_Pr9033091269622839554st_a_b] : ( member358942219636986539st_a_b @ X4 @ A3 )
          @ ^ [X4: set_Pr9033091269622839554st_a_b] : ( member358942219636986539st_a_b @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_144_less__eq__set__def,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B3: set_list_a] :
          ( ord_less_eq_list_a_o
          @ ^ [X4: list_a] : ( member_list_a @ X4 @ A3 )
          @ ^ [X4: list_a] : ( member_list_a @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_145_less__eq__set__def,axiom,
    ( ord_le8877086941679407844list_a
    = ( ^ [A3: set_set_list_a,B3: set_set_list_a] :
          ( ord_le897266612844759801st_a_o
          @ ^ [X4: set_list_a] : ( member_set_list_a @ X4 @ A3 )
          @ ^ [X4: set_list_a] : ( member_set_list_a @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_146_less__eq__set__def,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A3: set_set_a,B3: set_set_a] :
          ( ord_less_eq_set_a_o
          @ ^ [X4: set_a] : ( member_set_a @ X4 @ A3 )
          @ ^ [X4: set_a] : ( member_set_a @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_147_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X4: a] : ( member_a @ X4 @ A3 )
          @ ^ [X4: a] : ( member_a @ X4 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_148_subset__image__iff,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_149_subset__image__iff,axiom,
    ! [B: set_list_a,F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_4065140400681997415list_a @ F @ A ) )
      = ( ? [AA: set_Pr9033091269622839554st_a_b] :
            ( ( ord_le2879471331418251426st_a_b @ AA @ A )
            & ( B
              = ( image_4065140400681997415list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_150_subset__image__iff,axiom,
    ! [B: set_a,F: list_a > a,A: set_list_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_list_a_a @ F @ A ) )
      = ( ? [AA: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ AA @ A )
            & ( B
              = ( image_list_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_151_subset__image__iff,axiom,
    ! [B: set_a,F: set_a > a,A: set_set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
      = ( ? [AA: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ AA @ A )
            & ( B
              = ( image_set_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_152_subset__image__iff,axiom,
    ! [B: set_list_a,F: a > list_a,A: set_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_a_list_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_153_subset__image__iff,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_154_subset__image__iff,axiom,
    ! [B: set_a,F: set_list_a > a,A: set_set_list_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_list_a_a @ F @ A ) )
      = ( ? [AA: set_set_list_a] :
            ( ( ord_le8877086941679407844list_a @ AA @ A )
            & ( B
              = ( image_set_list_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_155_subset__image__iff,axiom,
    ! [B: set_list_a,F: list_a > list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_list_a_list_a @ F @ A ) )
      = ( ? [AA: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ AA @ A )
            & ( B
              = ( image_list_a_list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_156_subset__image__iff,axiom,
    ! [B: set_list_a,F: set_a > list_a,A: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_set_a_list_a @ F @ A ) )
      = ( ? [AA: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ AA @ A )
            & ( B
              = ( image_set_a_list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_157_subset__image__iff,axiom,
    ! [B: set_set_list_a,F: a > set_list_a,A: set_a] :
      ( ( ord_le8877086941679407844list_a @ B @ ( image_a_set_list_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_set_list_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_158_image__subset__iff,axiom,
    ! [F: a > produc2579390649552321826st_a_b,A: set_a,B: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le2879471331418251426st_a_b @ ( image_1109531753604471593st_a_b @ F @ A ) @ B )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( member8006451236149131979st_a_b @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_159_image__subset__iff,axiom,
    ! [F: a > set_Pr9033091269622839554st_a_b,A: set_a,B: set_se1825328058903261410st_a_b] :
      ( ( ord_le4557451523740326018st_a_b @ ( image_8858725927978154761st_a_b @ F @ A ) @ B )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( member358942219636986539st_a_b @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_160_image__subset__iff,axiom,
    ! [F: a > a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( member_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_161_image__subset__iff,axiom,
    ! [F: ( a > $o ) > set_a,A: set_a_o,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( image_a_o_set_a @ F @ A ) @ B )
      = ( ! [X4: a > $o] :
            ( ( member_a_o @ X4 @ A )
           => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_162_image__subset__iff,axiom,
    ! [F: a > set_a,A: set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( member_set_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_163_image__subset__iff,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( image_4065140400681997415list_a @ F @ A ) @ B )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ A )
           => ( member_list_a @ ( F @ X4 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_164_subset__imageE,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_165_subset__imageE,axiom,
    ! [B: set_list_a,F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_4065140400681997415list_a @ F @ A ) )
     => ~ ! [C: set_Pr9033091269622839554st_a_b] :
            ( ( ord_le2879471331418251426st_a_b @ C @ A )
           => ( B
             != ( image_4065140400681997415list_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_166_subset__imageE,axiom,
    ! [B: set_a,F: list_a > a,A: set_list_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_list_a_a @ F @ A ) )
     => ~ ! [C: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C @ A )
           => ( B
             != ( image_list_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_167_subset__imageE,axiom,
    ! [B: set_a,F: set_a > a,A: set_set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
     => ~ ! [C: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ C @ A )
           => ( B
             != ( image_set_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_168_subset__imageE,axiom,
    ! [B: set_list_a,F: a > list_a,A: set_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_a_list_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_list_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_169_subset__imageE,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_set_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_170_subset__imageE,axiom,
    ! [B: set_a,F: set_list_a > a,A: set_set_list_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_set_list_a_a @ F @ A ) )
     => ~ ! [C: set_set_list_a] :
            ( ( ord_le8877086941679407844list_a @ C @ A )
           => ( B
             != ( image_set_list_a_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_171_subset__imageE,axiom,
    ! [B: set_list_a,F: list_a > list_a,A: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_list_a_list_a @ F @ A ) )
     => ~ ! [C: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C @ A )
           => ( B
             != ( image_list_a_list_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_172_subset__imageE,axiom,
    ! [B: set_list_a,F: set_a > list_a,A: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ B @ ( image_set_a_list_a @ F @ A ) )
     => ~ ! [C: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ C @ A )
           => ( B
             != ( image_set_a_list_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_173_subset__imageE,axiom,
    ! [B: set_set_list_a,F: a > set_list_a,A: set_a] :
      ( ( ord_le8877086941679407844list_a @ B @ ( image_a_set_list_a @ F @ A ) )
     => ~ ! [C: set_a] :
            ( ( ord_less_eq_set_a @ C @ A )
           => ( B
             != ( image_a_set_list_a @ F @ C ) ) ) ) ).

% subset_imageE
thf(fact_174_image__subsetI,axiom,
    ! [A: set_a,F: a > a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_175_image__subsetI,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,F: produc2579390649552321826st_a_b > list_a,B: set_list_a] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ A )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_4065140400681997415list_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_176_image__subsetI,axiom,
    ! [A: set_list_a,F: list_a > a,B: set_a] :
      ( ! [X: list_a] :
          ( ( member_list_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_177_image__subsetI,axiom,
    ! [A: set_set_a,F: set_a > a,B: set_a] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_178_image__subsetI,axiom,
    ! [A: set_a,F: a > list_a,B: set_list_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_179_image__subsetI,axiom,
    ! [A: set_a,F: a > set_a,B: set_set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_set_a @ ( F @ X ) @ B ) )
     => ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_180_image__subsetI,axiom,
    ! [A: set_a,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member1426531477525435216od_a_a @ ( F @ X ) @ B ) )
     => ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_181_image__subsetI,axiom,
    ! [A: set_Product_prod_a_a,F: product_prod_a_a > a,B: set_a] :
      ( ! [X: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_3437945252899457948_a_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_182_image__subsetI,axiom,
    ! [A: set_list_a,F: list_a > list_a,B: set_list_a] :
      ( ! [X: list_a] :
          ( ( member_list_a @ X @ A )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_183_image__subsetI,axiom,
    ! [A: set_set_a,F: set_a > list_a,B: set_list_a] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ A )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_set_a_list_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_184_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_185_image__mono,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,B: set_Pr9033091269622839554st_a_b,F: produc2579390649552321826st_a_b > list_a] :
      ( ( ord_le2879471331418251426st_a_b @ A @ B )
     => ( ord_le8861187494160871172list_a @ ( image_4065140400681997415list_a @ F @ A ) @ ( image_4065140400681997415list_a @ F @ B ) ) ) ).

% image_mono
thf(fact_186_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > list_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A ) @ ( image_a_list_a @ F @ B ) ) ) ).

% image_mono
thf(fact_187_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_188_image__mono,axiom,
    ! [A: set_list_a,B: set_list_a,F: list_a > a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A ) @ ( image_list_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_189_image__mono,axiom,
    ! [A: set_set_a,B: set_set_a,F: set_a > a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_190_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_list_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le8877086941679407844list_a @ ( image_a_set_list_a @ F @ A ) @ ( image_a_set_list_a @ F @ B ) ) ) ).

% image_mono
thf(fact_191_image__mono,axiom,
    ! [A: set_list_a,B: set_list_a,F: list_a > list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A ) @ ( image_list_a_list_a @ F @ B ) ) ) ).

% image_mono
thf(fact_192_image__mono,axiom,
    ! [A: set_list_a,B: set_list_a,F: list_a > set_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ A ) @ ( image_list_a_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_193_image__mono,axiom,
    ! [A: set_set_list_a,B: set_set_list_a,F: set_list_a > a] :
      ( ( ord_le8877086941679407844list_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_set_list_a_a @ F @ A ) @ ( image_set_list_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_194_set__ConsD,axiom,
    ! [Y2: set_Pr9033091269622839554st_a_b,X2: set_Pr9033091269622839554st_a_b,Xs: list_s2207380573456591624st_a_b] :
      ( ( member358942219636986539st_a_b @ Y2 @ ( set_se3802960479674350359st_a_b @ ( cons_s9161922236425393080st_a_b @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member358942219636986539st_a_b @ Y2 @ ( set_se3802960479674350359st_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_195_set__ConsD,axiom,
    ! [Y2: set_a,X2: set_a,Xs: list_set_a] :
      ( ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_196_set__ConsD,axiom,
    ! [Y2: produc5365630598061846603st_a_b,X2: produc5365630598061846603st_a_b,Xs: list_P2989903729216022225st_a_b] :
      ( ( member2248238449149680884st_a_b @ Y2 @ ( set_Pr2913479409760856416st_a_b @ ( cons_P1986909248207666433st_a_b @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member2248238449149680884st_a_b @ Y2 @ ( set_Pr2913479409760856416st_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_197_set__ConsD,axiom,
    ! [Y2: product_prod_a_a,X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_198_set__ConsD,axiom,
    ! [Y2: b,X2: b,Xs: list_b] :
      ( ( member_b @ Y2 @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member_b @ Y2 @ ( set_b2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_199_set__ConsD,axiom,
    ! [Y2: list_P2281458131032066856st_a_b,X2: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b] :
      ( ( member1308482598881763409st_a_b @ Y2 @ ( set_li5246279850479030973st_a_b @ ( cons_l7524743740370447070st_a_b @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member1308482598881763409st_a_b @ Y2 @ ( set_li5246279850479030973st_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_200_set__ConsD,axiom,
    ! [Y2: produc2313410840657238951st_a_b,X2: produc2313410840657238951st_a_b,Xs: list_P1351356133043840941st_a_b] :
      ( ( member7191766420993298128st_a_b @ Y2 @ ( set_Pr5950792022122277180st_a_b @ ( cons_P5669161040283791197st_a_b @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member7191766420993298128st_a_b @ Y2 @ ( set_Pr5950792022122277180st_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_201_set__ConsD,axiom,
    ! [Y2: list_a,X2: list_a,Xs: list_list_a] :
      ( ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_202_set__ConsD,axiom,
    ! [Y2: a,X2: a,Xs: list_a] :
      ( ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member_a @ Y2 @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_203_set__ConsD,axiom,
    ! [Y2: produc2579390649552321826st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) )
     => ( ( Y2 = X2 )
        | ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_204_list_Oset__cases,axiom,
    ! [E: set_Pr9033091269622839554st_a_b,A2: list_s2207380573456591624st_a_b] :
      ( ( member358942219636986539st_a_b @ E @ ( set_se3802960479674350359st_a_b @ A2 ) )
     => ( ! [Z2: list_s2207380573456591624st_a_b] :
            ( A2
           != ( cons_s9161922236425393080st_a_b @ E @ Z2 ) )
       => ~ ! [Z1: set_Pr9033091269622839554st_a_b,Z2: list_s2207380573456591624st_a_b] :
              ( ( A2
                = ( cons_s9161922236425393080st_a_b @ Z1 @ Z2 ) )
             => ~ ( member358942219636986539st_a_b @ E @ ( set_se3802960479674350359st_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_205_list_Oset__cases,axiom,
    ! [E: set_a,A2: list_set_a] :
      ( ( member_set_a @ E @ ( set_set_a2 @ A2 ) )
     => ( ! [Z2: list_set_a] :
            ( A2
           != ( cons_set_a @ E @ Z2 ) )
       => ~ ! [Z1: set_a,Z2: list_set_a] :
              ( ( A2
                = ( cons_set_a @ Z1 @ Z2 ) )
             => ~ ( member_set_a @ E @ ( set_set_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_206_list_Oset__cases,axiom,
    ! [E: produc5365630598061846603st_a_b,A2: list_P2989903729216022225st_a_b] :
      ( ( member2248238449149680884st_a_b @ E @ ( set_Pr2913479409760856416st_a_b @ A2 ) )
     => ( ! [Z2: list_P2989903729216022225st_a_b] :
            ( A2
           != ( cons_P1986909248207666433st_a_b @ E @ Z2 ) )
       => ~ ! [Z1: produc5365630598061846603st_a_b,Z2: list_P2989903729216022225st_a_b] :
              ( ( A2
                = ( cons_P1986909248207666433st_a_b @ Z1 @ Z2 ) )
             => ~ ( member2248238449149680884st_a_b @ E @ ( set_Pr2913479409760856416st_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_207_list_Oset__cases,axiom,
    ! [E: product_prod_a_a,A2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A2 ) )
     => ( ! [Z2: list_P1396940483166286381od_a_a] :
            ( A2
           != ( cons_P7316939126706565853od_a_a @ E @ Z2 ) )
       => ~ ! [Z1: product_prod_a_a,Z2: list_P1396940483166286381od_a_a] :
              ( ( A2
                = ( cons_P7316939126706565853od_a_a @ Z1 @ Z2 ) )
             => ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_208_list_Oset__cases,axiom,
    ! [E: b,A2: list_b] :
      ( ( member_b @ E @ ( set_b2 @ A2 ) )
     => ( ! [Z2: list_b] :
            ( A2
           != ( cons_b @ E @ Z2 ) )
       => ~ ! [Z1: b,Z2: list_b] :
              ( ( A2
                = ( cons_b @ Z1 @ Z2 ) )
             => ~ ( member_b @ E @ ( set_b2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_209_list_Oset__cases,axiom,
    ! [E: list_P2281458131032066856st_a_b,A2: list_l3733004638376709294st_a_b] :
      ( ( member1308482598881763409st_a_b @ E @ ( set_li5246279850479030973st_a_b @ A2 ) )
     => ( ! [Z2: list_l3733004638376709294st_a_b] :
            ( A2
           != ( cons_l7524743740370447070st_a_b @ E @ Z2 ) )
       => ~ ! [Z1: list_P2281458131032066856st_a_b,Z2: list_l3733004638376709294st_a_b] :
              ( ( A2
                = ( cons_l7524743740370447070st_a_b @ Z1 @ Z2 ) )
             => ~ ( member1308482598881763409st_a_b @ E @ ( set_li5246279850479030973st_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_210_list_Oset__cases,axiom,
    ! [E: produc2313410840657238951st_a_b,A2: list_P1351356133043840941st_a_b] :
      ( ( member7191766420993298128st_a_b @ E @ ( set_Pr5950792022122277180st_a_b @ A2 ) )
     => ( ! [Z2: list_P1351356133043840941st_a_b] :
            ( A2
           != ( cons_P5669161040283791197st_a_b @ E @ Z2 ) )
       => ~ ! [Z1: produc2313410840657238951st_a_b,Z2: list_P1351356133043840941st_a_b] :
              ( ( A2
                = ( cons_P5669161040283791197st_a_b @ Z1 @ Z2 ) )
             => ~ ( member7191766420993298128st_a_b @ E @ ( set_Pr5950792022122277180st_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_211_list_Oset__cases,axiom,
    ! [E: list_a,A2: list_list_a] :
      ( ( member_list_a @ E @ ( set_list_a2 @ A2 ) )
     => ( ! [Z2: list_list_a] :
            ( A2
           != ( cons_list_a @ E @ Z2 ) )
       => ~ ! [Z1: list_a,Z2: list_list_a] :
              ( ( A2
                = ( cons_list_a @ Z1 @ Z2 ) )
             => ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_212_list_Oset__cases,axiom,
    ! [E: a,A2: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A2 ) )
     => ( ! [Z2: list_a] :
            ( A2
           != ( cons_a @ E @ Z2 ) )
       => ~ ! [Z1: a,Z2: list_a] :
              ( ( A2
                = ( cons_a @ Z1 @ Z2 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_213_list_Oset__cases,axiom,
    ! [E: produc2579390649552321826st_a_b,A2: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ E @ ( set_Pr6156699709417984823st_a_b @ A2 ) )
     => ( ! [Z2: list_P2281458131032066856st_a_b] :
            ( A2
           != ( cons_P2018802354021969880st_a_b @ E @ Z2 ) )
       => ~ ! [Z1: produc2579390649552321826st_a_b,Z2: list_P2281458131032066856st_a_b] :
              ( ( A2
                = ( cons_P2018802354021969880st_a_b @ Z1 @ Z2 ) )
             => ~ ( member8006451236149131979st_a_b @ E @ ( set_Pr6156699709417984823st_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_214_list_Oset__intros_I1_J,axiom,
    ! [X21: set_Pr9033091269622839554st_a_b,X22: list_s2207380573456591624st_a_b] : ( member358942219636986539st_a_b @ X21 @ ( set_se3802960479674350359st_a_b @ ( cons_s9161922236425393080st_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_215_list_Oset__intros_I1_J,axiom,
    ! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_216_list_Oset__intros_I1_J,axiom,
    ! [X21: produc5365630598061846603st_a_b,X22: list_P2989903729216022225st_a_b] : ( member2248238449149680884st_a_b @ X21 @ ( set_Pr2913479409760856416st_a_b @ ( cons_P1986909248207666433st_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_217_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_218_list_Oset__intros_I1_J,axiom,
    ! [X21: b,X22: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_219_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P2281458131032066856st_a_b,X22: list_l3733004638376709294st_a_b] : ( member1308482598881763409st_a_b @ X21 @ ( set_li5246279850479030973st_a_b @ ( cons_l7524743740370447070st_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_220_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2313410840657238951st_a_b,X22: list_P1351356133043840941st_a_b] : ( member7191766420993298128st_a_b @ X21 @ ( set_Pr5950792022122277180st_a_b @ ( cons_P5669161040283791197st_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_221_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_222_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_223_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] : ( member8006451236149131979st_a_b @ X21 @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_224_list_Oset__intros_I2_J,axiom,
    ! [Y2: set_Pr9033091269622839554st_a_b,X22: list_s2207380573456591624st_a_b,X21: set_Pr9033091269622839554st_a_b] :
      ( ( member358942219636986539st_a_b @ Y2 @ ( set_se3802960479674350359st_a_b @ X22 ) )
     => ( member358942219636986539st_a_b @ Y2 @ ( set_se3802960479674350359st_a_b @ ( cons_s9161922236425393080st_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_225_list_Oset__intros_I2_J,axiom,
    ! [Y2: set_a,X22: list_set_a,X21: set_a] :
      ( ( member_set_a @ Y2 @ ( set_set_a2 @ X22 ) )
     => ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_226_list_Oset__intros_I2_J,axiom,
    ! [Y2: produc5365630598061846603st_a_b,X22: list_P2989903729216022225st_a_b,X21: produc5365630598061846603st_a_b] :
      ( ( member2248238449149680884st_a_b @ Y2 @ ( set_Pr2913479409760856416st_a_b @ X22 ) )
     => ( member2248238449149680884st_a_b @ Y2 @ ( set_Pr2913479409760856416st_a_b @ ( cons_P1986909248207666433st_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_227_list_Oset__intros_I2_J,axiom,
    ! [Y2: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ X22 ) )
     => ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_228_list_Oset__intros_I2_J,axiom,
    ! [Y2: b,X22: list_b,X21: b] :
      ( ( member_b @ Y2 @ ( set_b2 @ X22 ) )
     => ( member_b @ Y2 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_229_list_Oset__intros_I2_J,axiom,
    ! [Y2: list_P2281458131032066856st_a_b,X22: list_l3733004638376709294st_a_b,X21: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ Y2 @ ( set_li5246279850479030973st_a_b @ X22 ) )
     => ( member1308482598881763409st_a_b @ Y2 @ ( set_li5246279850479030973st_a_b @ ( cons_l7524743740370447070st_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_230_list_Oset__intros_I2_J,axiom,
    ! [Y2: produc2313410840657238951st_a_b,X22: list_P1351356133043840941st_a_b,X21: produc2313410840657238951st_a_b] :
      ( ( member7191766420993298128st_a_b @ Y2 @ ( set_Pr5950792022122277180st_a_b @ X22 ) )
     => ( member7191766420993298128st_a_b @ Y2 @ ( set_Pr5950792022122277180st_a_b @ ( cons_P5669161040283791197st_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_231_list_Oset__intros_I2_J,axiom,
    ! [Y2: list_a,X22: list_list_a,X21: list_a] :
      ( ( member_list_a @ Y2 @ ( set_list_a2 @ X22 ) )
     => ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_232_list_Oset__intros_I2_J,axiom,
    ! [Y2: a,X22: list_a,X21: a] :
      ( ( member_a @ Y2 @ ( set_a2 @ X22 ) )
     => ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_233_list_Oset__intros_I2_J,axiom,
    ! [Y2: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b,X21: produc2579390649552321826st_a_b] :
      ( ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ X22 ) )
     => ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_234_subset__code_I1_J,axiom,
    ! [Xs: list_s2207380573456591624st_a_b,B: set_se1825328058903261410st_a_b] :
      ( ( ord_le4557451523740326018st_a_b @ ( set_se3802960479674350359st_a_b @ Xs ) @ B )
      = ( ! [X4: set_Pr9033091269622839554st_a_b] :
            ( ( member358942219636986539st_a_b @ X4 @ ( set_se3802960479674350359st_a_b @ Xs ) )
           => ( member358942219636986539st_a_b @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_235_subset__code_I1_J,axiom,
    ! [Xs: list_P2989903729216022225st_a_b,B: set_Pr4431676530321648299st_a_b] :
      ( ( ord_le8652052591400137291st_a_b @ ( set_Pr2913479409760856416st_a_b @ Xs ) @ B )
      = ( ! [X4: produc5365630598061846603st_a_b] :
            ( ( member2248238449149680884st_a_b @ X4 @ ( set_Pr2913479409760856416st_a_b @ Xs ) )
           => ( member2248238449149680884st_a_b @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_236_subset__code_I1_J,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B )
      = ( ! [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
           => ( member1426531477525435216od_a_a @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_237_subset__code_I1_J,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,B: set_li7460651515946730888st_a_b] :
      ( ( ord_le2045451444129209640st_a_b @ ( set_li5246279850479030973st_a_b @ Xs ) @ B )
      = ( ! [X4: list_P2281458131032066856st_a_b] :
            ( ( member1308482598881763409st_a_b @ X4 @ ( set_li5246279850479030973st_a_b @ Xs ) )
           => ( member1308482598881763409st_a_b @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_238_subset__code_I1_J,axiom,
    ! [Xs: list_b,B: set_b] :
      ( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B )
      = ( ! [X4: b] :
            ( ( member_b @ X4 @ ( set_b2 @ Xs ) )
           => ( member_b @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_239_subset__code_I1_J,axiom,
    ! [Xs: list_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
      = ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
           => ( member_list_a @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_240_subset__code_I1_J,axiom,
    ! [Xs: list_set_list_a,B: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ B )
      = ( ! [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ ( set_set_list_a2 @ Xs ) )
           => ( member_set_list_a @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_241_subset__code_I1_J,axiom,
    ! [Xs: list_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B )
      = ( ! [X4: set_a] :
            ( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
           => ( member_set_a @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_242_subset__code_I1_J,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,B: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ B )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ( member8006451236149131979st_a_b @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_243_subset__code_I1_J,axiom,
    ! [Xs: list_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( member_a @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_244_order__antisym__conv,axiom,
    ! [Y2: set_list_a,X2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ Y2 @ X2 )
     => ( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_245_order__antisym__conv,axiom,
    ! [Y2: set_set_list_a,X2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ Y2 @ X2 )
     => ( ( ord_le8877086941679407844list_a @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_246_order__antisym__conv,axiom,
    ! [Y2: set_set_a,X2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ Y2 @ X2 )
     => ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_247_order__antisym__conv,axiom,
    ! [Y2: list_a > b > $o,X2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ Y2 @ X2 )
     => ( ( ord_le8305133717419367763_a_b_o @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_248_order__antisym__conv,axiom,
    ! [Y2: a > $o,X2: a > $o] :
      ( ( ord_less_eq_a_o @ Y2 @ X2 )
     => ( ( ord_less_eq_a_o @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_249_order__antisym__conv,axiom,
    ! [Y2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y2 @ X2 )
     => ( ( ord_less_eq_set_a @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_250_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_list_a,C2: set_list_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_252_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_set_a,C2: set_set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_253_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > a > $o,C2: a > $o] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_a_o @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_a_o @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_254_ord__le__eq__subst,axiom,
    ! [A2: set_list_a,B2: set_list_a,F: set_list_a > set_a,C2: set_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_255_ord__le__eq__subst,axiom,
    ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_set_a,Y3: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_256_ord__le__eq__subst,axiom,
    ! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C2: set_a] :
      ( ( ord_less_eq_a_o @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: a > $o,Y3: a > $o] :
              ( ( ord_less_eq_a_o @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_257_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_set_list_a,C2: set_set_list_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8877086941679407844list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8877086941679407844list_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_258_ord__le__eq__subst,axiom,
    ! [A2: set_list_a,B2: set_list_a,F: set_list_a > set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_259_ord__le__eq__subst,axiom,
    ! [A2: set_list_a,B2: set_list_a,F: set_list_a > set_set_a,C2: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_260_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_261_ord__eq__le__subst,axiom,
    ! [A2: set_list_a,F: set_a > set_list_a,B2: set_a,C2: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_262_ord__eq__le__subst,axiom,
    ! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C2: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_263_ord__eq__le__subst,axiom,
    ! [A2: a > $o,F: set_a > a > $o,B2: set_a,C2: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_a_o @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_a_o @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_264_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_list_a > set_a,B2: set_list_a,C2: set_list_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_265_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C2: set_set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
       => ( ! [X: set_set_a,Y3: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_266_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C2: a > $o] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_a_o @ B2 @ C2 )
       => ( ! [X: a > $o,Y3: a > $o] :
              ( ( ord_less_eq_a_o @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_267_ord__eq__le__subst,axiom,
    ! [A2: set_set_list_a,F: set_a > set_set_list_a,B2: set_a,C2: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8877086941679407844list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8877086941679407844list_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_268_ord__eq__le__subst,axiom,
    ! [A2: set_list_a,F: set_list_a > set_list_a,B2: set_list_a,C2: set_list_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_269_ord__eq__le__subst,axiom,
    ! [A2: set_set_a,F: set_list_a > set_set_a,B2: set_list_a,C2: set_list_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_270_order__eq__refl,axiom,
    ! [X2: set_list_a,Y2: set_list_a] :
      ( ( X2 = Y2 )
     => ( ord_le8861187494160871172list_a @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_271_order__eq__refl,axiom,
    ! [X2: set_set_list_a,Y2: set_set_list_a] :
      ( ( X2 = Y2 )
     => ( ord_le8877086941679407844list_a @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_272_order__eq__refl,axiom,
    ! [X2: set_set_a,Y2: set_set_a] :
      ( ( X2 = Y2 )
     => ( ord_le3724670747650509150_set_a @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_273_order__eq__refl,axiom,
    ! [X2: list_a > b > $o,Y2: list_a > b > $o] :
      ( ( X2 = Y2 )
     => ( ord_le8305133717419367763_a_b_o @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_274_order__eq__refl,axiom,
    ! [X2: a > $o,Y2: a > $o] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_a_o @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_275_order__eq__refl,axiom,
    ! [X2: set_a,Y2: set_a] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_276_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_277_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_list_a,C2: set_list_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_le8861187494160871172list_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_278_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_set_a,C2: set_set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_279_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > a > $o,C2: a > $o] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_a_o @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_a_o @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_a_o @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_280_order__subst2,axiom,
    ! [A2: set_list_a,B2: set_list_a,F: set_list_a > set_a,C2: set_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_281_order__subst2,axiom,
    ! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_set_a,Y3: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_282_order__subst2,axiom,
    ! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C2: set_a] :
      ( ( ord_less_eq_a_o @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: a > $o,Y3: a > $o] :
              ( ( ord_less_eq_a_o @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_283_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_set_list_a,C2: set_set_list_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_le8877086941679407844list_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8877086941679407844list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8877086941679407844list_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_284_order__subst2,axiom,
    ! [A2: set_list_a,B2: set_list_a,F: set_list_a > set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ord_le8861187494160871172list_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_285_order__subst2,axiom,
    ! [A2: set_list_a,B2: set_list_a,F: set_list_a > set_set_a,C2: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_286_mem__Collect__eq,axiom,
    ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_287_mem__Collect__eq,axiom,
    ! [A2: set_Pr9033091269622839554st_a_b,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ( member358942219636986539st_a_b @ A2 @ ( collec4420230097056376813st_a_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_288_mem__Collect__eq,axiom,
    ! [A2: set_a,P: set_a > $o] :
      ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_289_mem__Collect__eq,axiom,
    ! [A2: list_a,P: list_a > $o] :
      ( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_290_mem__Collect__eq,axiom,
    ! [A2: produc2579390649552321826st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ( member8006451236149131979st_a_b @ A2 @ ( collec3957028476971440141st_a_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_291_mem__Collect__eq,axiom,
    ! [A2: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ( member1308482598881763409st_a_b @ A2 @ ( collec3283141540447581331st_a_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_292_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_293_Collect__mem__eq,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( collec3336397797384452498od_a_a
        @ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_294_Collect__mem__eq,axiom,
    ! [A: set_se1825328058903261410st_a_b] :
      ( ( collec4420230097056376813st_a_b
        @ ^ [X4: set_Pr9033091269622839554st_a_b] : ( member358942219636986539st_a_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_295_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_296_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_297_Collect__mem__eq,axiom,
    ! [A: set_Pr9033091269622839554st_a_b] :
      ( ( collec3957028476971440141st_a_b
        @ ^ [X4: produc2579390649552321826st_a_b] : ( member8006451236149131979st_a_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_298_Collect__mem__eq,axiom,
    ! [A: set_li7460651515946730888st_a_b] :
      ( ( collec3283141540447581331st_a_b
        @ ^ [X4: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_299_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_300_Collect__cong,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X: a] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_a @ P )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_301_Collect__cong,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X: list_a] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_list_a @ P )
        = ( collect_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_302_Collect__cong,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Q: produc2579390649552321826st_a_b > $o] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collec3957028476971440141st_a_b @ P )
        = ( collec3957028476971440141st_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_303_Collect__cong,axiom,
    ! [P: list_P2281458131032066856st_a_b > $o,Q: list_P2281458131032066856st_a_b > $o] :
      ( ! [X: list_P2281458131032066856st_a_b] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collec3283141540447581331st_a_b @ P )
        = ( collec3283141540447581331st_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_304_order__subst1,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_305_order__subst1,axiom,
    ! [A2: set_a,F: set_list_a > set_a,B2: set_list_a,C2: set_list_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_306_order__subst1,axiom,
    ! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C2: set_set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
       => ( ! [X: set_set_a,Y3: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_307_order__subst1,axiom,
    ! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C2: a > $o] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_a_o @ B2 @ C2 )
       => ( ! [X: a > $o,Y3: a > $o] :
              ( ( ord_less_eq_a_o @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_308_order__subst1,axiom,
    ! [A2: set_list_a,F: set_a > set_list_a,B2: set_a,C2: set_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_309_order__subst1,axiom,
    ! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_310_order__subst1,axiom,
    ! [A2: a > $o,F: set_a > a > $o,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_a_o @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_a_o @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_a_o @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_311_order__subst1,axiom,
    ! [A2: set_a,F: set_set_list_a > set_a,B2: set_set_list_a,C2: set_set_list_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le8877086941679407844list_a @ B2 @ C2 )
       => ( ! [X: set_set_list_a,Y3: set_set_list_a] :
              ( ( ord_le8877086941679407844list_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_312_order__subst1,axiom,
    ! [A2: set_list_a,F: set_list_a > set_list_a,B2: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ! [X: set_list_a,Y3: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_313_order__subst1,axiom,
    ! [A2: set_list_a,F: set_set_a > set_list_a,B2: set_set_a,C2: set_set_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
       => ( ! [X: set_set_a,Y3: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
             => ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le8861187494160871172list_a @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_314_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_list_a,Z3: set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
          & ( ord_le8861187494160871172list_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_315_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_set_list_a,Z3: set_set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_set_list_a,B4: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ A4 @ B4 )
          & ( ord_le8877086941679407844list_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_316_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_set_a,B4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
          & ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_317_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: list_a > b > $o,Z3: list_a > b > $o] : ( Y4 = Z3 ) )
    = ( ^ [A4: list_a > b > $o,B4: list_a > b > $o] :
          ( ( ord_le8305133717419367763_a_b_o @ A4 @ B4 )
          & ( ord_le8305133717419367763_a_b_o @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_318_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: a > $o,Z3: a > $o] : ( Y4 = Z3 ) )
    = ( ^ [A4: a > $o,B4: a > $o] :
          ( ( ord_less_eq_a_o @ A4 @ B4 )
          & ( ord_less_eq_a_o @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_319_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_320_le__fun__def,axiom,
    ( ord_le8305133717419367763_a_b_o
    = ( ^ [F2: list_a > b > $o,G2: list_a > b > $o] :
        ! [X4: list_a] : ( ord_less_eq_b_o @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).

% le_fun_def
thf(fact_321_le__fun__def,axiom,
    ( ord_less_eq_a_o
    = ( ^ [F2: a > $o,G2: a > $o] :
        ! [X4: a] : ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).

% le_fun_def
thf(fact_322_le__funI,axiom,
    ! [F: list_a > b > $o,G: list_a > b > $o] :
      ( ! [X: list_a] : ( ord_less_eq_b_o @ ( F @ X ) @ ( G @ X ) )
     => ( ord_le8305133717419367763_a_b_o @ F @ G ) ) ).

% le_funI
thf(fact_323_le__funI,axiom,
    ! [F: a > $o,G: a > $o] :
      ( ! [X: a] : ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) )
     => ( ord_less_eq_a_o @ F @ G ) ) ).

% le_funI
thf(fact_324_le__funE,axiom,
    ! [F: list_a > b > $o,G: list_a > b > $o,X2: list_a] :
      ( ( ord_le8305133717419367763_a_b_o @ F @ G )
     => ( ord_less_eq_b_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).

% le_funE
thf(fact_325_le__funE,axiom,
    ! [F: a > $o,G: a > $o,X2: a] :
      ( ( ord_less_eq_a_o @ F @ G )
     => ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).

% le_funE
thf(fact_326_le__funD,axiom,
    ! [F: list_a > b > $o,G: list_a > b > $o,X2: list_a] :
      ( ( ord_le8305133717419367763_a_b_o @ F @ G )
     => ( ord_less_eq_b_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).

% le_funD
thf(fact_327_le__funD,axiom,
    ! [F: a > $o,G: a > $o,X2: a] :
      ( ( ord_less_eq_a_o @ F @ G )
     => ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).

% le_funD
thf(fact_328_antisym,axiom,
    ! [A2: set_list_a,B2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_329_antisym,axiom,
    ! [A2: set_set_list_a,B2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B2 )
     => ( ( ord_le8877086941679407844list_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_330_antisym,axiom,
    ! [A2: set_set_a,B2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_331_antisym,axiom,
    ! [A2: list_a > b > $o,B2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ A2 @ B2 )
     => ( ( ord_le8305133717419367763_a_b_o @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_332_antisym,axiom,
    ! [A2: a > $o,B2: a > $o] :
      ( ( ord_less_eq_a_o @ A2 @ B2 )
     => ( ( ord_less_eq_a_o @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_333_antisym,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_334_dual__order_Otrans,axiom,
    ! [B2: set_list_a,A2: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
     => ( ( ord_le8861187494160871172list_a @ C2 @ B2 )
       => ( ord_le8861187494160871172list_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_335_dual__order_Otrans,axiom,
    ! [B2: set_set_list_a,A2: set_set_list_a,C2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ B2 @ A2 )
     => ( ( ord_le8877086941679407844list_a @ C2 @ B2 )
       => ( ord_le8877086941679407844list_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_336_dual__order_Otrans,axiom,
    ! [B2: set_set_a,A2: set_set_a,C2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ C2 @ B2 )
       => ( ord_le3724670747650509150_set_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_337_dual__order_Otrans,axiom,
    ! [B2: list_a > b > $o,A2: list_a > b > $o,C2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ B2 @ A2 )
     => ( ( ord_le8305133717419367763_a_b_o @ C2 @ B2 )
       => ( ord_le8305133717419367763_a_b_o @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_338_dual__order_Otrans,axiom,
    ! [B2: a > $o,A2: a > $o,C2: a > $o] :
      ( ( ord_less_eq_a_o @ B2 @ A2 )
     => ( ( ord_less_eq_a_o @ C2 @ B2 )
       => ( ord_less_eq_a_o @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_339_dual__order_Otrans,axiom,
    ! [B2: set_a,A2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C2 @ B2 )
       => ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_340_dual__order_Oantisym,axiom,
    ! [B2: set_list_a,A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_341_dual__order_Oantisym,axiom,
    ! [B2: set_set_list_a,A2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ B2 @ A2 )
     => ( ( ord_le8877086941679407844list_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_342_dual__order_Oantisym,axiom,
    ! [B2: set_set_a,A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_343_dual__order_Oantisym,axiom,
    ! [B2: list_a > b > $o,A2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ B2 @ A2 )
     => ( ( ord_le8305133717419367763_a_b_o @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_344_dual__order_Oantisym,axiom,
    ! [B2: a > $o,A2: a > $o] :
      ( ( ord_less_eq_a_o @ B2 @ A2 )
     => ( ( ord_less_eq_a_o @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_345_dual__order_Oantisym,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_346_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_list_a,Z3: set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_list_a,B4: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ B4 @ A4 )
          & ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_347_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_set_list_a,Z3: set_set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_set_list_a,B4: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ B4 @ A4 )
          & ( ord_le8877086941679407844list_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_348_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_set_a,B4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
          & ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_349_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: list_a > b > $o,Z3: list_a > b > $o] : ( Y4 = Z3 ) )
    = ( ^ [A4: list_a > b > $o,B4: list_a > b > $o] :
          ( ( ord_le8305133717419367763_a_b_o @ B4 @ A4 )
          & ( ord_le8305133717419367763_a_b_o @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_350_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: a > $o,Z3: a > $o] : ( Y4 = Z3 ) )
    = ( ^ [A4: a > $o,B4: a > $o] :
          ( ( ord_less_eq_a_o @ B4 @ A4 )
          & ( ord_less_eq_a_o @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_351_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_352_order__trans,axiom,
    ! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
     => ( ( ord_le8861187494160871172list_a @ Y2 @ Z )
       => ( ord_le8861187494160871172list_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_353_order__trans,axiom,
    ! [X2: set_set_list_a,Y2: set_set_list_a,Z: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ X2 @ Y2 )
     => ( ( ord_le8877086941679407844list_a @ Y2 @ Z )
       => ( ord_le8877086941679407844list_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_354_order__trans,axiom,
    ! [X2: set_set_a,Y2: set_set_a,Z: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
     => ( ( ord_le3724670747650509150_set_a @ Y2 @ Z )
       => ( ord_le3724670747650509150_set_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_355_order__trans,axiom,
    ! [X2: list_a > b > $o,Y2: list_a > b > $o,Z: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ X2 @ Y2 )
     => ( ( ord_le8305133717419367763_a_b_o @ Y2 @ Z )
       => ( ord_le8305133717419367763_a_b_o @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_356_order__trans,axiom,
    ! [X2: a > $o,Y2: a > $o,Z: a > $o] :
      ( ( ord_less_eq_a_o @ X2 @ Y2 )
     => ( ( ord_less_eq_a_o @ Y2 @ Z )
       => ( ord_less_eq_a_o @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_357_order__trans,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( ord_less_eq_set_a @ Y2 @ Z )
       => ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_358_order_Otrans,axiom,
    ! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_359_order_Otrans,axiom,
    ! [A2: set_set_list_a,B2: set_set_list_a,C2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B2 )
     => ( ( ord_le8877086941679407844list_a @ B2 @ C2 )
       => ( ord_le8877086941679407844list_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_360_order_Otrans,axiom,
    ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
       => ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_361_order_Otrans,axiom,
    ! [A2: list_a > b > $o,B2: list_a > b > $o,C2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ A2 @ B2 )
     => ( ( ord_le8305133717419367763_a_b_o @ B2 @ C2 )
       => ( ord_le8305133717419367763_a_b_o @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_362_order_Otrans,axiom,
    ! [A2: a > $o,B2: a > $o,C2: a > $o] :
      ( ( ord_less_eq_a_o @ A2 @ B2 )
     => ( ( ord_less_eq_a_o @ B2 @ C2 )
       => ( ord_less_eq_a_o @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_363_order_Otrans,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_364_order__antisym,axiom,
    ! [X2: set_list_a,Y2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
     => ( ( ord_le8861187494160871172list_a @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_365_order__antisym,axiom,
    ! [X2: set_set_list_a,Y2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ X2 @ Y2 )
     => ( ( ord_le8877086941679407844list_a @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_366_order__antisym,axiom,
    ! [X2: set_set_a,Y2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
     => ( ( ord_le3724670747650509150_set_a @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_367_order__antisym,axiom,
    ! [X2: list_a > b > $o,Y2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ X2 @ Y2 )
     => ( ( ord_le8305133717419367763_a_b_o @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_368_order__antisym,axiom,
    ! [X2: a > $o,Y2: a > $o] :
      ( ( ord_less_eq_a_o @ X2 @ Y2 )
     => ( ( ord_less_eq_a_o @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_369_order__antisym,axiom,
    ! [X2: set_a,Y2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( ord_less_eq_set_a @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_370_ord__le__eq__trans,axiom,
    ! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_371_ord__le__eq__trans,axiom,
    ! [A2: set_set_list_a,B2: set_set_list_a,C2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le8877086941679407844list_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_372_ord__le__eq__trans,axiom,
    ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_373_ord__le__eq__trans,axiom,
    ! [A2: list_a > b > $o,B2: list_a > b > $o,C2: list_a > b > $o] :
      ( ( ord_le8305133717419367763_a_b_o @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le8305133717419367763_a_b_o @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_374_ord__le__eq__trans,axiom,
    ! [A2: a > $o,B2: a > $o,C2: a > $o] :
      ( ( ord_less_eq_a_o @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_a_o @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_375_ord__le__eq__trans,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_376_ord__eq__le__trans,axiom,
    ! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
      ( ( A2 = B2 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
       => ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_377_ord__eq__le__trans,axiom,
    ! [A2: set_set_list_a,B2: set_set_list_a,C2: set_set_list_a] :
      ( ( A2 = B2 )
     => ( ( ord_le8877086941679407844list_a @ B2 @ C2 )
       => ( ord_le8877086941679407844list_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_378_ord__eq__le__trans,axiom,
    ! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
      ( ( A2 = B2 )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
       => ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_379_ord__eq__le__trans,axiom,
    ! [A2: list_a > b > $o,B2: list_a > b > $o,C2: list_a > b > $o] :
      ( ( A2 = B2 )
     => ( ( ord_le8305133717419367763_a_b_o @ B2 @ C2 )
       => ( ord_le8305133717419367763_a_b_o @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_380_ord__eq__le__trans,axiom,
    ! [A2: a > $o,B2: a > $o,C2: a > $o] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_a_o @ B2 @ C2 )
       => ( ord_less_eq_a_o @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_381_ord__eq__le__trans,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C2 )
       => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_382_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_list_a,Z3: set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_list_a,Y5: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ X4 @ Y5 )
          & ( ord_le8861187494160871172list_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_383_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_set_list_a,Z3: set_set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_set_list_a,Y5: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ X4 @ Y5 )
          & ( ord_le8877086941679407844list_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_384_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_set_a,Y5: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
          & ( ord_le3724670747650509150_set_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_385_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: list_a > b > $o,Z3: list_a > b > $o] : ( Y4 = Z3 ) )
    = ( ^ [X4: list_a > b > $o,Y5: list_a > b > $o] :
          ( ( ord_le8305133717419367763_a_b_o @ X4 @ Y5 )
          & ( ord_le8305133717419367763_a_b_o @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_386_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: a > $o,Z3: a > $o] : ( Y4 = Z3 ) )
    = ( ^ [X4: a > $o,Y5: a > $o] :
          ( ( ord_less_eq_a_o @ X4 @ Y5 )
          & ( ord_less_eq_a_o @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_387_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y5 )
          & ( ord_less_eq_set_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_388_not__Cons__self2,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( cons_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_389_not__Cons__self2,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b] :
      ( ( cons_l7524743740370447070st_a_b @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_390_not__Cons__self2,axiom,
    ! [X2: produc2313410840657238951st_a_b,Xs: list_P1351356133043840941st_a_b] :
      ( ( cons_P5669161040283791197st_a_b @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_391_not__Cons__self2,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( cons_list_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_392_not__Cons__self2,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_393_Collect__mono__iff,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Q: produc2579390649552321826st_a_b > $o] :
      ( ( ord_le2879471331418251426st_a_b @ ( collec3957028476971440141st_a_b @ P ) @ ( collec3957028476971440141st_a_b @ Q ) )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_394_Collect__mono__iff,axiom,
    ! [P: list_P2281458131032066856st_a_b > $o,Q: list_P2281458131032066856st_a_b > $o] :
      ( ( ord_le2045451444129209640st_a_b @ ( collec3283141540447581331st_a_b @ P ) @ ( collec3283141540447581331st_a_b @ Q ) )
      = ( ! [X4: list_P2281458131032066856st_a_b] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_395_Collect__mono__iff,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
      = ( ! [X4: list_a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_396_Collect__mono__iff,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) )
      = ( ! [X4: set_list_a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_397_Collect__mono__iff,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
      = ( ! [X4: set_a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_398_Collect__mono__iff,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
      = ( ! [X4: a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_399_set__eq__subset,axiom,
    ( ( ^ [Y4: set_list_a,Z3: set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_list_a,B3: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
          & ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_400_set__eq__subset,axiom,
    ( ( ^ [Y4: set_set_list_a,Z3: set_set_list_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_set_list_a,B3: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ A3 @ B3 )
          & ( ord_le8877086941679407844list_a @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_401_set__eq__subset,axiom,
    ( ( ^ [Y4: set_set_a,Z3: set_set_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_set_a,B3: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
          & ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_402_set__eq__subset,axiom,
    ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B3 )
          & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_403_subset__trans,axiom,
    ! [A: set_list_a,B: set_list_a,C3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ( ord_le8861187494160871172list_a @ B @ C3 )
       => ( ord_le8861187494160871172list_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_404_subset__trans,axiom,
    ! [A: set_set_list_a,B: set_set_list_a,C3: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B )
     => ( ( ord_le8877086941679407844list_a @ B @ C3 )
       => ( ord_le8877086941679407844list_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_405_subset__trans,axiom,
    ! [A: set_set_a,B: set_set_a,C3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ C3 )
       => ( ord_le3724670747650509150_set_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_406_subset__trans,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_407_Collect__mono,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Q: produc2579390649552321826st_a_b > $o] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le2879471331418251426st_a_b @ ( collec3957028476971440141st_a_b @ P ) @ ( collec3957028476971440141st_a_b @ Q ) ) ) ).

% Collect_mono
thf(fact_408_Collect__mono,axiom,
    ! [P: list_P2281458131032066856st_a_b > $o,Q: list_P2281458131032066856st_a_b > $o] :
      ( ! [X: list_P2281458131032066856st_a_b] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le2045451444129209640st_a_b @ ( collec3283141540447581331st_a_b @ P ) @ ( collec3283141540447581331st_a_b @ Q ) ) ) ).

% Collect_mono
thf(fact_409_Collect__mono,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X: list_a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).

% Collect_mono
thf(fact_410_Collect__mono,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ! [X: set_list_a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) ) ) ).

% Collect_mono
thf(fact_411_Collect__mono,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ! [X: set_a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).

% Collect_mono
thf(fact_412_Collect__mono,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_413_subset__refl,axiom,
    ! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).

% subset_refl
thf(fact_414_subset__refl,axiom,
    ! [A: set_set_list_a] : ( ord_le8877086941679407844list_a @ A @ A ) ).

% subset_refl
thf(fact_415_subset__refl,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).

% subset_refl
thf(fact_416_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_417_subset__iff,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
        ! [T: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ T @ A3 )
         => ( member1426531477525435216od_a_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_418_subset__iff,axiom,
    ( ord_le2045451444129209640st_a_b
    = ( ^ [A3: set_li7460651515946730888st_a_b,B3: set_li7460651515946730888st_a_b] :
        ! [T: list_P2281458131032066856st_a_b] :
          ( ( member1308482598881763409st_a_b @ T @ A3 )
         => ( member1308482598881763409st_a_b @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_419_subset__iff,axiom,
    ( ord_le4557451523740326018st_a_b
    = ( ^ [A3: set_se1825328058903261410st_a_b,B3: set_se1825328058903261410st_a_b] :
        ! [T: set_Pr9033091269622839554st_a_b] :
          ( ( member358942219636986539st_a_b @ T @ A3 )
         => ( member358942219636986539st_a_b @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_420_subset__iff,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B3: set_list_a] :
        ! [T: list_a] :
          ( ( member_list_a @ T @ A3 )
         => ( member_list_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_421_subset__iff,axiom,
    ( ord_le8877086941679407844list_a
    = ( ^ [A3: set_set_list_a,B3: set_set_list_a] :
        ! [T: set_list_a] :
          ( ( member_set_list_a @ T @ A3 )
         => ( member_set_list_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_422_subset__iff,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A3: set_set_a,B3: set_set_a] :
        ! [T: set_a] :
          ( ( member_set_a @ T @ A3 )
         => ( member_set_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_423_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A3 )
         => ( member_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_424_equalityD2,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( A = B )
     => ( ord_le8861187494160871172list_a @ B @ A ) ) ).

% equalityD2
thf(fact_425_equalityD2,axiom,
    ! [A: set_set_list_a,B: set_set_list_a] :
      ( ( A = B )
     => ( ord_le8877086941679407844list_a @ B @ A ) ) ).

% equalityD2
thf(fact_426_equalityD2,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A = B )
     => ( ord_le3724670747650509150_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_427_equalityD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_428_equalityD1,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( A = B )
     => ( ord_le8861187494160871172list_a @ A @ B ) ) ).

% equalityD1
thf(fact_429_equalityD1,axiom,
    ! [A: set_set_list_a,B: set_set_list_a] :
      ( ( A = B )
     => ( ord_le8877086941679407844list_a @ A @ B ) ) ).

% equalityD1
thf(fact_430_equalityD1,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A = B )
     => ( ord_le3724670747650509150_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_431_equalityD1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_432_subset__eq,axiom,
    ( ord_le746702958409616551od_a_a
    = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
        ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A3 )
         => ( member1426531477525435216od_a_a @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_433_subset__eq,axiom,
    ( ord_le2045451444129209640st_a_b
    = ( ^ [A3: set_li7460651515946730888st_a_b,B3: set_li7460651515946730888st_a_b] :
        ! [X4: list_P2281458131032066856st_a_b] :
          ( ( member1308482598881763409st_a_b @ X4 @ A3 )
         => ( member1308482598881763409st_a_b @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_434_subset__eq,axiom,
    ( ord_le4557451523740326018st_a_b
    = ( ^ [A3: set_se1825328058903261410st_a_b,B3: set_se1825328058903261410st_a_b] :
        ! [X4: set_Pr9033091269622839554st_a_b] :
          ( ( member358942219636986539st_a_b @ X4 @ A3 )
         => ( member358942219636986539st_a_b @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_435_subset__eq,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A3: set_list_a,B3: set_list_a] :
        ! [X4: list_a] :
          ( ( member_list_a @ X4 @ A3 )
         => ( member_list_a @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_436_subset__eq,axiom,
    ( ord_le8877086941679407844list_a
    = ( ^ [A3: set_set_list_a,B3: set_set_list_a] :
        ! [X4: set_list_a] :
          ( ( member_set_list_a @ X4 @ A3 )
         => ( member_set_list_a @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_437_subset__eq,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A3: set_set_a,B3: set_set_a] :
        ! [X4: set_a] :
          ( ( member_set_a @ X4 @ A3 )
         => ( member_set_a @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_438_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A3 )
         => ( member_a @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_439_equalityE,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( A = B )
     => ~ ( ( ord_le8861187494160871172list_a @ A @ B )
         => ~ ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).

% equalityE
thf(fact_440_equalityE,axiom,
    ! [A: set_set_list_a,B: set_set_list_a] :
      ( ( A = B )
     => ~ ( ( ord_le8877086941679407844list_a @ A @ B )
         => ~ ( ord_le8877086941679407844list_a @ B @ A ) ) ) ).

% equalityE
thf(fact_441_equalityE,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A = B )
     => ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
         => ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_442_equalityE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_443_subsetD,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B )
     => ( ( member1426531477525435216od_a_a @ C2 @ A )
       => ( member1426531477525435216od_a_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_444_subsetD,axiom,
    ! [A: set_li7460651515946730888st_a_b,B: set_li7460651515946730888st_a_b,C2: list_P2281458131032066856st_a_b] :
      ( ( ord_le2045451444129209640st_a_b @ A @ B )
     => ( ( member1308482598881763409st_a_b @ C2 @ A )
       => ( member1308482598881763409st_a_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_445_subsetD,axiom,
    ! [A: set_se1825328058903261410st_a_b,B: set_se1825328058903261410st_a_b,C2: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le4557451523740326018st_a_b @ A @ B )
     => ( ( member358942219636986539st_a_b @ C2 @ A )
       => ( member358942219636986539st_a_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_446_subsetD,axiom,
    ! [A: set_list_a,B: set_list_a,C2: list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ( member_list_a @ C2 @ A )
       => ( member_list_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_447_subsetD,axiom,
    ! [A: set_set_list_a,B: set_set_list_a,C2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B )
     => ( ( member_set_list_a @ C2 @ A )
       => ( member_set_list_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_448_subsetD,axiom,
    ! [A: set_set_a,B: set_set_a,C2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( member_set_a @ C2 @ A )
       => ( member_set_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_449_subsetD,axiom,
    ! [A: set_a,B: set_a,C2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_450_in__mono,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X2: product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B )
     => ( ( member1426531477525435216od_a_a @ X2 @ A )
       => ( member1426531477525435216od_a_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_451_in__mono,axiom,
    ! [A: set_li7460651515946730888st_a_b,B: set_li7460651515946730888st_a_b,X2: list_P2281458131032066856st_a_b] :
      ( ( ord_le2045451444129209640st_a_b @ A @ B )
     => ( ( member1308482598881763409st_a_b @ X2 @ A )
       => ( member1308482598881763409st_a_b @ X2 @ B ) ) ) ).

% in_mono
thf(fact_452_in__mono,axiom,
    ! [A: set_se1825328058903261410st_a_b,B: set_se1825328058903261410st_a_b,X2: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le4557451523740326018st_a_b @ A @ B )
     => ( ( member358942219636986539st_a_b @ X2 @ A )
       => ( member358942219636986539st_a_b @ X2 @ B ) ) ) ).

% in_mono
thf(fact_453_in__mono,axiom,
    ! [A: set_list_a,B: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ A @ B )
     => ( ( member_list_a @ X2 @ A )
       => ( member_list_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_454_in__mono,axiom,
    ! [A: set_set_list_a,B: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A @ B )
     => ( ( member_set_list_a @ X2 @ A )
       => ( member_set_list_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_455_in__mono,axiom,
    ! [A: set_set_a,B: set_set_a,X2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( member_set_a @ X2 @ A )
       => ( member_set_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_456_in__mono,axiom,
    ! [A: set_a,B: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X2 @ A )
       => ( member_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_457_merge_Osimps_I3_J,axiom,
    ! [Cmp: compar6837619475480278830st_a_b,X2: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b] :
      ( ( ( ( compar5222392469468919210st_a_b @ Cmp @ X2 @ Y2 )
          = greater )
       => ( ( sortin6459637776524370667st_a_b @ Cmp @ ( cons_l7524743740370447070st_a_b @ X2 @ Xs ) @ ( cons_l7524743740370447070st_a_b @ Y2 @ Ys ) )
          = ( cons_l7524743740370447070st_a_b @ Y2 @ ( sortin6459637776524370667st_a_b @ Cmp @ ( cons_l7524743740370447070st_a_b @ X2 @ Xs ) @ Ys ) ) ) )
      & ( ( ( compar5222392469468919210st_a_b @ Cmp @ X2 @ Y2 )
         != greater )
       => ( ( sortin6459637776524370667st_a_b @ Cmp @ ( cons_l7524743740370447070st_a_b @ X2 @ Xs ) @ ( cons_l7524743740370447070st_a_b @ Y2 @ Ys ) )
          = ( cons_l7524743740370447070st_a_b @ X2 @ ( sortin6459637776524370667st_a_b @ Cmp @ Xs @ ( cons_l7524743740370447070st_a_b @ Y2 @ Ys ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_458_merge_Osimps_I3_J,axiom,
    ! [Cmp: compar70190622580527661st_a_b,X2: produc2313410840657238951st_a_b,Y2: produc2313410840657238951st_a_b,Xs: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( ( ( compar2130703882349896233st_a_b @ Cmp @ X2 @ Y2 )
          = greater )
       => ( ( sortin5193368813060177258st_a_b @ Cmp @ ( cons_P5669161040283791197st_a_b @ X2 @ Xs ) @ ( cons_P5669161040283791197st_a_b @ Y2 @ Ys ) )
          = ( cons_P5669161040283791197st_a_b @ Y2 @ ( sortin5193368813060177258st_a_b @ Cmp @ ( cons_P5669161040283791197st_a_b @ X2 @ Xs ) @ Ys ) ) ) )
      & ( ( ( compar2130703882349896233st_a_b @ Cmp @ X2 @ Y2 )
         != greater )
       => ( ( sortin5193368813060177258st_a_b @ Cmp @ ( cons_P5669161040283791197st_a_b @ X2 @ Xs ) @ ( cons_P5669161040283791197st_a_b @ Y2 @ Ys ) )
          = ( cons_P5669161040283791197st_a_b @ X2 @ ( sortin5193368813060177258st_a_b @ Cmp @ Xs @ ( cons_P5669161040283791197st_a_b @ Y2 @ Ys ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_459_merge_Osimps_I3_J,axiom,
    ! [Cmp: comparator_list_a,X2: list_a,Y2: list_a,Xs: list_list_a,Ys: list_list_a] :
      ( ( ( ( compare_list_a @ Cmp @ X2 @ Y2 )
          = greater )
       => ( ( sorting_merge_list_a @ Cmp @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ Y2 @ Ys ) )
          = ( cons_list_a @ Y2 @ ( sorting_merge_list_a @ Cmp @ ( cons_list_a @ X2 @ Xs ) @ Ys ) ) ) )
      & ( ( ( compare_list_a @ Cmp @ X2 @ Y2 )
         != greater )
       => ( ( sorting_merge_list_a @ Cmp @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ Y2 @ Ys ) )
          = ( cons_list_a @ X2 @ ( sorting_merge_list_a @ Cmp @ Xs @ ( cons_list_a @ Y2 @ Ys ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_460_merge_Osimps_I3_J,axiom,
    ! [Cmp: comparator_a,X2: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( ( ( compare_a @ Cmp @ X2 @ Y2 )
          = greater )
       => ( ( sorting_merge_a @ Cmp @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys ) )
          = ( cons_a @ Y2 @ ( sorting_merge_a @ Cmp @ ( cons_a @ X2 @ Xs ) @ Ys ) ) ) )
      & ( ( ( compare_a @ Cmp @ X2 @ Y2 )
         != greater )
       => ( ( sorting_merge_a @ Cmp @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys ) )
          = ( cons_a @ X2 @ ( sorting_merge_a @ Cmp @ Xs @ ( cons_a @ Y2 @ Ys ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_461_merge_Osimps_I3_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: produc2579390649552321826st_a_b,Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ X2 @ Y2 )
          = greater )
       => ( ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
          = ( cons_P2018802354021969880st_a_b @ Y2 @ ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) ) ) )
      & ( ( ( compar8165832444956594340st_a_b @ Cmp @ X2 @ Y2 )
         != greater )
       => ( ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
          = ( cons_P2018802354021969880st_a_b @ X2 @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_462_prod_Oinject,axiom,
    ! [X1: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,X23: list_P2281458131032066856st_a_b,Y1: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,Y23: list_P2281458131032066856st_a_b] :
      ( ( ( produc3504008754983458001st_a_b @ X1 @ X23 )
        = ( produc3504008754983458001st_a_b @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_463_prod_Oinject,axiom,
    ! [X1: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,X23: list_P2281458131032066856st_a_b,Y1: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,Y23: list_P2281458131032066856st_a_b] :
      ( ( ( produc6916920643058086483st_a_b @ X1 @ X23 )
        = ( produc6916920643058086483st_a_b @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_464_prod_Oinject,axiom,
    ! [X1: b,X23: produc2579390649552321826st_a_b,Y1: b,Y23: produc2579390649552321826st_a_b] :
      ( ( ( produc2237782585831962490st_a_b @ X1 @ X23 )
        = ( produc2237782585831962490st_a_b @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_465_prod_Oinject,axiom,
    ! [X1: a,X23: produc2579390649552321826st_a_b,Y1: a,Y23: produc2579390649552321826st_a_b] :
      ( ( ( produc7754680479886915515st_a_b @ X1 @ X23 )
        = ( produc7754680479886915515st_a_b @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_466_prod_Oinject,axiom,
    ! [X1: a,X23: a,Y1: a,Y23: a] :
      ( ( ( product_Pair_a_a @ X1 @ X23 )
        = ( product_Pair_a_a @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_467_prod_Oinject,axiom,
    ! [X1: list_a,X23: b,Y1: list_a,Y23: b] :
      ( ( ( produc4781227316648555538st_a_b @ X1 @ X23 )
        = ( produc4781227316648555538st_a_b @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_468_old_Oprod_Oinject,axiom,
    ! [A2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,B2: list_P2281458131032066856st_a_b,A5: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,B5: list_P2281458131032066856st_a_b] :
      ( ( ( produc3504008754983458001st_a_b @ A2 @ B2 )
        = ( produc3504008754983458001st_a_b @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_469_old_Oprod_Oinject,axiom,
    ! [A2: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,B2: list_P2281458131032066856st_a_b,A5: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,B5: list_P2281458131032066856st_a_b] :
      ( ( ( produc6916920643058086483st_a_b @ A2 @ B2 )
        = ( produc6916920643058086483st_a_b @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_470_old_Oprod_Oinject,axiom,
    ! [A2: b,B2: produc2579390649552321826st_a_b,A5: b,B5: produc2579390649552321826st_a_b] :
      ( ( ( produc2237782585831962490st_a_b @ A2 @ B2 )
        = ( produc2237782585831962490st_a_b @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_471_old_Oprod_Oinject,axiom,
    ! [A2: a,B2: produc2579390649552321826st_a_b,A5: a,B5: produc2579390649552321826st_a_b] :
      ( ( ( produc7754680479886915515st_a_b @ A2 @ B2 )
        = ( produc7754680479886915515st_a_b @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_472_old_Oprod_Oinject,axiom,
    ! [A2: a,B2: a,A5: a,B5: a] :
      ( ( ( product_Pair_a_a @ A2 @ B2 )
        = ( product_Pair_a_a @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_473_old_Oprod_Oinject,axiom,
    ! [A2: list_a,B2: b,A5: list_a,B5: b] :
      ( ( ( produc4781227316648555538st_a_b @ A2 @ B2 )
        = ( produc4781227316648555538st_a_b @ A5 @ B5 ) )
      = ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_474_image__Collect__subsetI,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,F: produc2579390649552321826st_a_b > list_a,B: set_list_a] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( P @ X )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_4065140400681997415list_a @ F @ ( collec3957028476971440141st_a_b @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_475_image__Collect__subsetI,axiom,
    ! [P: a > $o,F: a > a,B: set_a] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_476_image__Collect__subsetI,axiom,
    ! [P: list_a > $o,F: list_a > a,B: set_a] :
      ( ! [X: list_a] :
          ( ( P @ X )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ ( collect_list_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_477_image__Collect__subsetI,axiom,
    ! [P: a > $o,F: a > list_a,B: set_list_a] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ ( collect_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_478_image__Collect__subsetI,axiom,
    ! [P: a > $o,F: a > set_a,B: set_set_a] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( member_set_a @ ( F @ X ) @ B ) )
     => ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( collect_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_479_image__Collect__subsetI,axiom,
    ! [P: a > $o,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( member1426531477525435216od_a_a @ ( F @ X ) @ B ) )
     => ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F @ ( collect_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_480_image__Collect__subsetI,axiom,
    ! [P: list_a > $o,F: list_a > list_a,B: set_list_a] :
      ( ! [X: list_a] :
          ( ( P @ X )
         => ( member_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ ( collect_list_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_481_image__Collect__subsetI,axiom,
    ! [P: a > $o,F: a > set_list_a,B: set_set_list_a] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( member_set_list_a @ ( F @ X ) @ B ) )
     => ( ord_le8877086941679407844list_a @ ( image_a_set_list_a @ F @ ( collect_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_482_image__Collect__subsetI,axiom,
    ! [P: list_a > $o,F: list_a > set_a,B: set_set_a] :
      ( ! [X: list_a] :
          ( ( P @ X )
         => ( member_set_a @ ( F @ X ) @ B ) )
     => ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ ( collect_list_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_483_image__Collect__subsetI,axiom,
    ! [P: a > $o,F: a > produc2579390649552321826st_a_b,B: set_Pr9033091269622839554st_a_b] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( member8006451236149131979st_a_b @ ( F @ X ) @ B ) )
     => ( ord_le2879471331418251426st_a_b @ ( image_1109531753604471593st_a_b @ F @ ( collect_a @ P ) ) @ B ) ) ).

% image_Collect_subsetI
thf(fact_484_compare_Oasym__greater,axiom,
    ! [Cmp: comparator_a,A2: a,B2: a] :
      ( ( ( compare_a @ Cmp @ A2 @ B2 )
        = greater )
     => ( ( compare_a @ Cmp @ B2 @ A2 )
       != greater ) ) ).

% compare.asym_greater
thf(fact_485_compare_Oasym__greater,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,A2: produc2579390649552321826st_a_b,B2: produc2579390649552321826st_a_b] :
      ( ( ( compar8165832444956594340st_a_b @ Cmp @ A2 @ B2 )
        = greater )
     => ( ( compar8165832444956594340st_a_b @ Cmp @ B2 @ A2 )
       != greater ) ) ).

% compare.asym_greater
thf(fact_486_compare_Otrans__greater,axiom,
    ! [Cmp: comparator_a,A2: a,B2: a,C2: a] :
      ( ( ( compare_a @ Cmp @ A2 @ B2 )
        = greater )
     => ( ( ( compare_a @ Cmp @ B2 @ C2 )
          = greater )
       => ( ( compare_a @ Cmp @ A2 @ C2 )
          = greater ) ) ) ).

% compare.trans_greater
thf(fact_487_compare_Otrans__greater,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,A2: produc2579390649552321826st_a_b,B2: produc2579390649552321826st_a_b,C2: produc2579390649552321826st_a_b] :
      ( ( ( compar8165832444956594340st_a_b @ Cmp @ A2 @ B2 )
        = greater )
     => ( ( ( compar8165832444956594340st_a_b @ Cmp @ B2 @ C2 )
          = greater )
       => ( ( compar8165832444956594340st_a_b @ Cmp @ A2 @ C2 )
          = greater ) ) ) ).

% compare.trans_greater
thf(fact_488_compare_Otrans__not__greater,axiom,
    ! [Cmp: comparator_a,A2: a,B2: a,C2: a] :
      ( ( ( compare_a @ Cmp @ A2 @ B2 )
       != greater )
     => ( ( ( compare_a @ Cmp @ B2 @ C2 )
         != greater )
       => ( ( compare_a @ Cmp @ A2 @ C2 )
         != greater ) ) ) ).

% compare.trans_not_greater
thf(fact_489_compare_Otrans__not__greater,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,A2: produc2579390649552321826st_a_b,B2: produc2579390649552321826st_a_b,C2: produc2579390649552321826st_a_b] :
      ( ( ( compar8165832444956594340st_a_b @ Cmp @ A2 @ B2 )
       != greater )
     => ( ( ( compar8165832444956594340st_a_b @ Cmp @ B2 @ C2 )
         != greater )
       => ( ( compar8165832444956594340st_a_b @ Cmp @ A2 @ C2 )
         != greater ) ) ) ).

% compare.trans_not_greater
thf(fact_490_fstI,axiom,
    ! [X2: produc8664631368769065441st_a_b,Y2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,Z: list_P2281458131032066856st_a_b] :
      ( ( X2
        = ( produc3504008754983458001st_a_b @ Y2 @ Z ) )
     => ( ( produc1290499411302820541st_a_b @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_491_fstI,axiom,
    ! [X2: produc6292069877431205281st_a_b,Y2: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,Z: list_P2281458131032066856st_a_b] :
      ( ( X2
        = ( produc6916920643058086483st_a_b @ Y2 @ Z ) )
     => ( ( produc6225685312319343463st_a_b @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_492_fstI,axiom,
    ! [X2: produc9072104740861669386st_a_b,Y2: b,Z: produc2579390649552321826st_a_b] :
      ( ( X2
        = ( produc2237782585831962490st_a_b @ Y2 @ Z ) )
     => ( ( produc3358030408229276518st_a_b @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_493_fstI,axiom,
    ! [X2: produc5365630598061846603st_a_b,Y2: a,Z: produc2579390649552321826st_a_b] :
      ( ( X2
        = ( produc7754680479886915515st_a_b @ Y2 @ Z ) )
     => ( ( produc8874928302284229543st_a_b @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_494_fstI,axiom,
    ! [X2: product_prod_a_a,Y2: a,Z: a] :
      ( ( X2
        = ( product_Pair_a_a @ Y2 @ Z ) )
     => ( ( product_fst_a_a @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_495_fstI,axiom,
    ! [X2: produc2313410840657238951st_a_b,Y2: produc2579390649552321826st_a_b,Z: produc2579390649552321826st_a_b] :
      ( ( X2
        = ( produc7813282991251521687st_a_b @ Y2 @ Z ) )
     => ( ( produc6544331879426417283st_a_b @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_496_fstI,axiom,
    ! [X2: produc2579390649552321826st_a_b,Y2: list_a,Z: b] :
      ( ( X2
        = ( produc4781227316648555538st_a_b @ Y2 @ Z ) )
     => ( ( product_fst_list_a_b @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_497_fst__eqD,axiom,
    ! [X2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,Y2: list_P2281458131032066856st_a_b,A2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o] :
      ( ( ( produc1290499411302820541st_a_b @ ( produc3504008754983458001st_a_b @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_498_fst__eqD,axiom,
    ! [X2: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,Y2: list_P2281458131032066856st_a_b,A2: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real] :
      ( ( ( produc6225685312319343463st_a_b @ ( produc6916920643058086483st_a_b @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_499_fst__eqD,axiom,
    ! [X2: b,Y2: produc2579390649552321826st_a_b,A2: b] :
      ( ( ( produc3358030408229276518st_a_b @ ( produc2237782585831962490st_a_b @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_500_fst__eqD,axiom,
    ! [X2: a,Y2: produc2579390649552321826st_a_b,A2: a] :
      ( ( ( produc8874928302284229543st_a_b @ ( produc7754680479886915515st_a_b @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_501_fst__eqD,axiom,
    ! [X2: a,Y2: a,A2: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_502_fst__eqD,axiom,
    ! [X2: produc2579390649552321826st_a_b,Y2: produc2579390649552321826st_a_b,A2: produc2579390649552321826st_a_b] :
      ( ( ( produc6544331879426417283st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_503_fst__eqD,axiom,
    ! [X2: list_a,Y2: b,A2: list_a] :
      ( ( ( product_fst_list_a_b @ ( produc4781227316648555538st_a_b @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_504_fst__conv,axiom,
    ! [X1: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,X23: list_P2281458131032066856st_a_b] :
      ( ( produc1290499411302820541st_a_b @ ( produc3504008754983458001st_a_b @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_505_fst__conv,axiom,
    ! [X1: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,X23: list_P2281458131032066856st_a_b] :
      ( ( produc6225685312319343463st_a_b @ ( produc6916920643058086483st_a_b @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_506_fst__conv,axiom,
    ! [X1: b,X23: produc2579390649552321826st_a_b] :
      ( ( produc3358030408229276518st_a_b @ ( produc2237782585831962490st_a_b @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_507_fst__conv,axiom,
    ! [X1: a,X23: produc2579390649552321826st_a_b] :
      ( ( produc8874928302284229543st_a_b @ ( produc7754680479886915515st_a_b @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_508_fst__conv,axiom,
    ! [X1: a,X23: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_509_fst__conv,axiom,
    ! [X1: produc2579390649552321826st_a_b,X23: produc2579390649552321826st_a_b] :
      ( ( produc6544331879426417283st_a_b @ ( produc7813282991251521687st_a_b @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_510_fst__conv,axiom,
    ! [X1: list_a,X23: b] :
      ( ( product_fst_list_a_b @ ( produc4781227316648555538st_a_b @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_511_eq__fst__iff,axiom,
    ! [A2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,P2: produc8664631368769065441st_a_b] :
      ( ( A2
        = ( produc1290499411302820541st_a_b @ P2 ) )
      = ( ? [B4: list_P2281458131032066856st_a_b] :
            ( P2
            = ( produc3504008754983458001st_a_b @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_512_eq__fst__iff,axiom,
    ! [A2: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,P2: produc6292069877431205281st_a_b] :
      ( ( A2
        = ( produc6225685312319343463st_a_b @ P2 ) )
      = ( ? [B4: list_P2281458131032066856st_a_b] :
            ( P2
            = ( produc6916920643058086483st_a_b @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_513_eq__fst__iff,axiom,
    ! [A2: b,P2: produc9072104740861669386st_a_b] :
      ( ( A2
        = ( produc3358030408229276518st_a_b @ P2 ) )
      = ( ? [B4: produc2579390649552321826st_a_b] :
            ( P2
            = ( produc2237782585831962490st_a_b @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_514_eq__fst__iff,axiom,
    ! [A2: a,P2: produc5365630598061846603st_a_b] :
      ( ( A2
        = ( produc8874928302284229543st_a_b @ P2 ) )
      = ( ? [B4: produc2579390649552321826st_a_b] :
            ( P2
            = ( produc7754680479886915515st_a_b @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_515_eq__fst__iff,axiom,
    ! [A2: a,P2: product_prod_a_a] :
      ( ( A2
        = ( product_fst_a_a @ P2 ) )
      = ( ? [B4: a] :
            ( P2
            = ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_516_eq__fst__iff,axiom,
    ! [A2: produc2579390649552321826st_a_b,P2: produc2313410840657238951st_a_b] :
      ( ( A2
        = ( produc6544331879426417283st_a_b @ P2 ) )
      = ( ? [B4: produc2579390649552321826st_a_b] :
            ( P2
            = ( produc7813282991251521687st_a_b @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_517_eq__fst__iff,axiom,
    ! [A2: list_a,P2: produc2579390649552321826st_a_b] :
      ( ( A2
        = ( product_fst_list_a_b @ P2 ) )
      = ( ? [B4: b] :
            ( P2
            = ( produc4781227316648555538st_a_b @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_518_all__subset__image,axiom,
    ! [F: a > a,A: set_a,P: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A )
           => ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_519_all__subset__image,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,P: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_4065140400681997415list_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_Pr9033091269622839554st_a_b] :
            ( ( ord_le2879471331418251426st_a_b @ B3 @ A )
           => ( P @ ( image_4065140400681997415list_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_520_all__subset__image,axiom,
    ! [F: list_a > a,A: set_list_a,P: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ A )
           => ( P @ ( image_list_a_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_521_all__subset__image,axiom,
    ! [F: set_a > a,A: set_set_a,P: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ A )
           => ( P @ ( image_set_a_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_522_all__subset__image,axiom,
    ! [F: a > list_a,A: set_a,P: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A )
           => ( P @ ( image_a_list_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_523_all__subset__image,axiom,
    ! [F: a > set_a,A: set_a,P: set_set_a > $o] :
      ( ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A )
           => ( P @ ( image_a_set_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_524_all__subset__image,axiom,
    ! [F: set_list_a > a,A: set_set_list_a,P: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_set_list_a_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_set_list_a] :
            ( ( ord_le8877086941679407844list_a @ B3 @ A )
           => ( P @ ( image_set_list_a_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_525_all__subset__image,axiom,
    ! [F: list_a > list_a,A: set_list_a,P: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ A )
           => ( P @ ( image_list_a_list_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_526_all__subset__image,axiom,
    ! [F: set_a > list_a,A: set_set_a,P: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_set_a_list_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ A )
           => ( P @ ( image_set_a_list_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_527_all__subset__image,axiom,
    ! [F: a > set_list_a,A: set_a,P: set_set_list_a > $o] :
      ( ( ! [B3: set_set_list_a] :
            ( ( ord_le8877086941679407844list_a @ B3 @ ( image_a_set_list_a @ F @ A ) )
           => ( P @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A )
           => ( P @ ( image_a_set_list_a @ F @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_528_predicate1I,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_a_o @ P @ Q ) ) ).

% predicate1I
thf(fact_529_predicate1D,axiom,
    ! [P: a > $o,Q: a > $o,X2: a] :
      ( ( ord_less_eq_a_o @ P @ Q )
     => ( ( P @ X2 )
       => ( Q @ X2 ) ) ) ).

% predicate1D
thf(fact_530_rev__predicate1D,axiom,
    ! [P: a > $o,X2: a,Q: a > $o] :
      ( ( P @ X2 )
     => ( ( ord_less_eq_a_o @ P @ Q )
       => ( Q @ X2 ) ) ) ).

% rev_predicate1D
thf(fact_531_ssubst__Pair__rhs,axiom,
    ! [R: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,S: list_P2281458131032066856st_a_b,R2: set_Pr7060751113198729409st_a_b,S2: list_P2281458131032066856st_a_b] :
      ( ( member3277860331849966602st_a_b @ ( produc3504008754983458001st_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3277860331849966602st_a_b @ ( produc3504008754983458001st_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_532_ssubst__Pair__rhs,axiom,
    ! [R: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,S: list_P2281458131032066856st_a_b,R2: set_Pr3822622692308018519st_a_b,S2: list_P2281458131032066856st_a_b] :
      ( ( member2006299769585415352st_a_b @ ( produc6916920643058086483st_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2006299769585415352st_a_b @ ( produc6916920643058086483st_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_533_ssubst__Pair__rhs,axiom,
    ! [R: b,S: produc2579390649552321826st_a_b,R2: set_Pr5427516116882750186st_a_b,S2: produc2579390649552321826st_a_b] :
      ( ( member5954712591949503667st_a_b @ ( produc2237782585831962490st_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5954712591949503667st_a_b @ ( produc2237782585831962490st_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_534_ssubst__Pair__rhs,axiom,
    ! [R: a,S: produc2579390649552321826st_a_b,R2: set_Pr4431676530321648299st_a_b,S2: produc2579390649552321826st_a_b] :
      ( ( member2248238449149680884st_a_b @ ( produc7754680479886915515st_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2248238449149680884st_a_b @ ( produc7754680479886915515st_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_535_ssubst__Pair__rhs,axiom,
    ! [R: a,S: a,R2: set_Product_prod_a_a,S2: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_536_ssubst__Pair__rhs,axiom,
    ! [R: list_a,S: b,R2: set_Pr9033091269622839554st_a_b,S2: b] :
      ( ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_537_prod__induct3,axiom,
    ! [P: produc9072104740861669386st_a_b > $o,X2: produc9072104740861669386st_a_b] :
      ( ! [A6: b,B6: list_a,C4: b] : ( P @ ( produc2237782585831962490st_a_b @ A6 @ ( produc4781227316648555538st_a_b @ B6 @ C4 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_538_prod__induct3,axiom,
    ! [P: produc5365630598061846603st_a_b > $o,X2: produc5365630598061846603st_a_b] :
      ( ! [A6: a,B6: list_a,C4: b] : ( P @ ( produc7754680479886915515st_a_b @ A6 @ ( produc4781227316648555538st_a_b @ B6 @ C4 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_539_prod__cases3,axiom,
    ! [Y2: produc9072104740861669386st_a_b] :
      ~ ! [A6: b,B6: list_a,C4: b] :
          ( Y2
         != ( produc2237782585831962490st_a_b @ A6 @ ( produc4781227316648555538st_a_b @ B6 @ C4 ) ) ) ).

% prod_cases3
thf(fact_540_prod__cases3,axiom,
    ! [Y2: produc5365630598061846603st_a_b] :
      ~ ! [A6: a,B6: list_a,C4: b] :
          ( Y2
         != ( produc7754680479886915515st_a_b @ A6 @ ( produc4781227316648555538st_a_b @ B6 @ C4 ) ) ) ).

% prod_cases3
thf(fact_541_Pair__inject,axiom,
    ! [A2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,B2: list_P2281458131032066856st_a_b,A5: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,B5: list_P2281458131032066856st_a_b] :
      ( ( ( produc3504008754983458001st_a_b @ A2 @ B2 )
        = ( produc3504008754983458001st_a_b @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_542_Pair__inject,axiom,
    ! [A2: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,B2: list_P2281458131032066856st_a_b,A5: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,B5: list_P2281458131032066856st_a_b] :
      ( ( ( produc6916920643058086483st_a_b @ A2 @ B2 )
        = ( produc6916920643058086483st_a_b @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_543_Pair__inject,axiom,
    ! [A2: b,B2: produc2579390649552321826st_a_b,A5: b,B5: produc2579390649552321826st_a_b] :
      ( ( ( produc2237782585831962490st_a_b @ A2 @ B2 )
        = ( produc2237782585831962490st_a_b @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_544_Pair__inject,axiom,
    ! [A2: a,B2: produc2579390649552321826st_a_b,A5: a,B5: produc2579390649552321826st_a_b] :
      ( ( ( produc7754680479886915515st_a_b @ A2 @ B2 )
        = ( produc7754680479886915515st_a_b @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_545_Pair__inject,axiom,
    ! [A2: a,B2: a,A5: a,B5: a] :
      ( ( ( product_Pair_a_a @ A2 @ B2 )
        = ( product_Pair_a_a @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_546_Pair__inject,axiom,
    ! [A2: list_a,B2: b,A5: list_a,B5: b] :
      ( ( ( produc4781227316648555538st_a_b @ A2 @ B2 )
        = ( produc4781227316648555538st_a_b @ A5 @ B5 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_547_prod__cases,axiom,
    ! [P: produc8664631368769065441st_a_b > $o,P2: produc8664631368769065441st_a_b] :
      ( ! [A6: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,B6: list_P2281458131032066856st_a_b] : ( P @ ( produc3504008754983458001st_a_b @ A6 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_548_prod__cases,axiom,
    ! [P: produc6292069877431205281st_a_b > $o,P2: produc6292069877431205281st_a_b] :
      ( ! [A6: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,B6: list_P2281458131032066856st_a_b] : ( P @ ( produc6916920643058086483st_a_b @ A6 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_549_prod__cases,axiom,
    ! [P: produc9072104740861669386st_a_b > $o,P2: produc9072104740861669386st_a_b] :
      ( ! [A6: b,B6: produc2579390649552321826st_a_b] : ( P @ ( produc2237782585831962490st_a_b @ A6 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_550_prod__cases,axiom,
    ! [P: produc5365630598061846603st_a_b > $o,P2: produc5365630598061846603st_a_b] :
      ( ! [A6: a,B6: produc2579390649552321826st_a_b] : ( P @ ( produc7754680479886915515st_a_b @ A6 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_551_prod__cases,axiom,
    ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
      ( ! [A6: a,B6: a] : ( P @ ( product_Pair_a_a @ A6 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_552_prod__cases,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,P2: produc2579390649552321826st_a_b] :
      ( ! [A6: list_a,B6: b] : ( P @ ( produc4781227316648555538st_a_b @ A6 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_553_surj__pair,axiom,
    ! [P2: produc8664631368769065441st_a_b] :
    ? [X: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,Y3: list_P2281458131032066856st_a_b] :
      ( P2
      = ( produc3504008754983458001st_a_b @ X @ Y3 ) ) ).

% surj_pair
thf(fact_554_surj__pair,axiom,
    ! [P2: produc6292069877431205281st_a_b] :
    ? [X: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,Y3: list_P2281458131032066856st_a_b] :
      ( P2
      = ( produc6916920643058086483st_a_b @ X @ Y3 ) ) ).

% surj_pair
thf(fact_555_surj__pair,axiom,
    ! [P2: produc9072104740861669386st_a_b] :
    ? [X: b,Y3: produc2579390649552321826st_a_b] :
      ( P2
      = ( produc2237782585831962490st_a_b @ X @ Y3 ) ) ).

% surj_pair
thf(fact_556_surj__pair,axiom,
    ! [P2: produc5365630598061846603st_a_b] :
    ? [X: a,Y3: produc2579390649552321826st_a_b] :
      ( P2
      = ( produc7754680479886915515st_a_b @ X @ Y3 ) ) ).

% surj_pair
thf(fact_557_surj__pair,axiom,
    ! [P2: product_prod_a_a] :
    ? [X: a,Y3: a] :
      ( P2
      = ( product_Pair_a_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_558_surj__pair,axiom,
    ! [P2: produc2579390649552321826st_a_b] :
    ? [X: list_a,Y3: b] :
      ( P2
      = ( produc4781227316648555538st_a_b @ X @ Y3 ) ) ).

% surj_pair
thf(fact_559_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc8664631368769065441st_a_b] :
      ~ ! [A6: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,B6: list_P2281458131032066856st_a_b] :
          ( Y2
         != ( produc3504008754983458001st_a_b @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_560_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc6292069877431205281st_a_b] :
      ~ ! [A6: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,B6: list_P2281458131032066856st_a_b] :
          ( Y2
         != ( produc6916920643058086483st_a_b @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_561_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc9072104740861669386st_a_b] :
      ~ ! [A6: b,B6: produc2579390649552321826st_a_b] :
          ( Y2
         != ( produc2237782585831962490st_a_b @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_562_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc5365630598061846603st_a_b] :
      ~ ! [A6: a,B6: produc2579390649552321826st_a_b] :
          ( Y2
         != ( produc7754680479886915515st_a_b @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_563_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_a_a] :
      ~ ! [A6: a,B6: a] :
          ( Y2
         != ( product_Pair_a_a @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_564_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc2579390649552321826st_a_b] :
      ~ ! [A6: list_a,B6: b] :
          ( Y2
         != ( produc4781227316648555538st_a_b @ A6 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_565_compare__inject,axiom,
    ! [X2: comparator_a,Y2: comparator_a] :
      ( ( ( compare_a @ X2 )
        = ( compare_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% compare_inject
thf(fact_566_compare__inject,axiom,
    ! [X2: compar7542523840845723048st_a_b,Y2: compar7542523840845723048st_a_b] :
      ( ( ( compar8165832444956594340st_a_b @ X2 )
        = ( compar8165832444956594340st_a_b @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% compare_inject
thf(fact_567_Collect__restrict,axiom,
    ! [X5: set_Product_prod_a_a,P: product_prod_a_a > $o] :
      ( ord_le746702958409616551od_a_a
      @ ( collec3336397797384452498od_a_a
        @ ^ [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_568_Collect__restrict,axiom,
    ! [X5: set_se1825328058903261410st_a_b,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ord_le4557451523740326018st_a_b
      @ ( collec4420230097056376813st_a_b
        @ ^ [X4: set_Pr9033091269622839554st_a_b] :
            ( ( member358942219636986539st_a_b @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_569_Collect__restrict,axiom,
    ! [X5: set_Pr9033091269622839554st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ord_le2879471331418251426st_a_b
      @ ( collec3957028476971440141st_a_b
        @ ^ [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_570_Collect__restrict,axiom,
    ! [X5: set_li7460651515946730888st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ord_le2045451444129209640st_a_b
      @ ( collec3283141540447581331st_a_b
        @ ^ [X4: list_P2281458131032066856st_a_b] :
            ( ( member1308482598881763409st_a_b @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_571_Collect__restrict,axiom,
    ! [X5: set_list_a,P: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X4: list_a] :
            ( ( member_list_a @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_572_Collect__restrict,axiom,
    ! [X5: set_set_list_a,P: set_list_a > $o] :
      ( ord_le8877086941679407844list_a
      @ ( collect_set_list_a
        @ ^ [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_573_Collect__restrict,axiom,
    ! [X5: set_set_a,P: set_a > $o] :
      ( ord_le3724670747650509150_set_a
      @ ( collect_set_a
        @ ^ [X4: set_a] :
            ( ( member_set_a @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_574_Collect__restrict,axiom,
    ! [X5: set_a,P: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_575_prop__restrict,axiom,
    ! [X2: product_prod_a_a,Z4: set_Product_prod_a_a,X5: set_Product_prod_a_a,P: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ X2 @ Z4 )
     => ( ( ord_le746702958409616551od_a_a @ Z4
          @ ( collec3336397797384452498od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_576_prop__restrict,axiom,
    ! [X2: set_Pr9033091269622839554st_a_b,Z4: set_se1825328058903261410st_a_b,X5: set_se1825328058903261410st_a_b,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ( member358942219636986539st_a_b @ X2 @ Z4 )
     => ( ( ord_le4557451523740326018st_a_b @ Z4
          @ ( collec4420230097056376813st_a_b
            @ ^ [X4: set_Pr9033091269622839554st_a_b] :
                ( ( member358942219636986539st_a_b @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_577_prop__restrict,axiom,
    ! [X2: produc2579390649552321826st_a_b,Z4: set_Pr9033091269622839554st_a_b,X5: set_Pr9033091269622839554st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ( member8006451236149131979st_a_b @ X2 @ Z4 )
     => ( ( ord_le2879471331418251426st_a_b @ Z4
          @ ( collec3957028476971440141st_a_b
            @ ^ [X4: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_578_prop__restrict,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Z4: set_li7460651515946730888st_a_b,X5: set_li7460651515946730888st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ( member1308482598881763409st_a_b @ X2 @ Z4 )
     => ( ( ord_le2045451444129209640st_a_b @ Z4
          @ ( collec3283141540447581331st_a_b
            @ ^ [X4: list_P2281458131032066856st_a_b] :
                ( ( member1308482598881763409st_a_b @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_579_prop__restrict,axiom,
    ! [X2: list_a,Z4: set_list_a,X5: set_list_a,P: list_a > $o] :
      ( ( member_list_a @ X2 @ Z4 )
     => ( ( ord_le8861187494160871172list_a @ Z4
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_580_prop__restrict,axiom,
    ! [X2: set_list_a,Z4: set_set_list_a,X5: set_set_list_a,P: set_list_a > $o] :
      ( ( member_set_list_a @ X2 @ Z4 )
     => ( ( ord_le8877086941679407844list_a @ Z4
          @ ( collect_set_list_a
            @ ^ [X4: set_list_a] :
                ( ( member_set_list_a @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_581_prop__restrict,axiom,
    ! [X2: set_a,Z4: set_set_a,X5: set_set_a,P: set_a > $o] :
      ( ( member_set_a @ X2 @ Z4 )
     => ( ( ord_le3724670747650509150_set_a @ Z4
          @ ( collect_set_a
            @ ^ [X4: set_a] :
                ( ( member_set_a @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_582_prop__restrict,axiom,
    ! [X2: a,Z4: set_a,X5: set_a,P: a > $o] :
      ( ( member_a @ X2 @ Z4 )
     => ( ( ord_less_eq_set_a @ Z4
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_583_pred__subset__eq,axiom,
    ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
      ( ( ord_le1591150415168442102_a_a_o
        @ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ R2 )
        @ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ S3 ) )
      = ( ord_le746702958409616551od_a_a @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_584_pred__subset__eq,axiom,
    ! [R2: set_li7460651515946730888st_a_b,S3: set_li7460651515946730888st_a_b] :
      ( ( ord_le8242720074258346933_a_b_o
        @ ^ [X4: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ X4 @ R2 )
        @ ^ [X4: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ X4 @ S3 ) )
      = ( ord_le2045451444129209640st_a_b @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_585_pred__subset__eq,axiom,
    ! [R2: set_se1825328058903261410st_a_b,S3: set_se1825328058903261410st_a_b] :
      ( ( ord_le1523511841465843419_a_b_o
        @ ^ [X4: set_Pr9033091269622839554st_a_b] : ( member358942219636986539st_a_b @ X4 @ R2 )
        @ ^ [X4: set_Pr9033091269622839554st_a_b] : ( member358942219636986539st_a_b @ X4 @ S3 ) )
      = ( ord_le4557451523740326018st_a_b @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_586_pred__subset__eq,axiom,
    ! [R2: set_list_a,S3: set_list_a] :
      ( ( ord_less_eq_list_a_o
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ R2 )
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ S3 ) )
      = ( ord_le8861187494160871172list_a @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_587_pred__subset__eq,axiom,
    ! [R2: set_set_list_a,S3: set_set_list_a] :
      ( ( ord_le897266612844759801st_a_o
        @ ^ [X4: set_list_a] : ( member_set_list_a @ X4 @ R2 )
        @ ^ [X4: set_list_a] : ( member_set_list_a @ X4 @ S3 ) )
      = ( ord_le8877086941679407844list_a @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_588_pred__subset__eq,axiom,
    ! [R2: set_set_a,S3: set_set_a] :
      ( ( ord_less_eq_set_a_o
        @ ^ [X4: set_a] : ( member_set_a @ X4 @ R2 )
        @ ^ [X4: set_a] : ( member_set_a @ X4 @ S3 ) )
      = ( ord_le3724670747650509150_set_a @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_589_pred__subset__eq,axiom,
    ! [R2: set_a,S3: set_a] :
      ( ( ord_less_eq_a_o
        @ ^ [X4: a] : ( member_a @ X4 @ R2 )
        @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
      = ( ord_less_eq_set_a @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_590_cons2__if__2elems,axiom,
    ! [X2: set_Pr9033091269622839554st_a_b,Xs: list_s2207380573456591624st_a_b,Y2: set_Pr9033091269622839554st_a_b] :
      ( ( member358942219636986539st_a_b @ X2 @ ( set_se3802960479674350359st_a_b @ Xs ) )
     => ( ( member358942219636986539st_a_b @ Y2 @ ( set_se3802960479674350359st_a_b @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: set_Pr9033091269622839554st_a_b,Z5: set_Pr9033091269622839554st_a_b,Zs2: list_s2207380573456591624st_a_b] :
              ( Xs
              = ( cons_s9161922236425393080st_a_b @ Y3 @ ( cons_s9161922236425393080st_a_b @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_591_cons2__if__2elems,axiom,
    ! [X2: set_a,Xs: list_set_a,Y2: set_a] :
      ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
     => ( ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: set_a,Z5: set_a,Zs2: list_set_a] :
              ( Xs
              = ( cons_set_a @ Y3 @ ( cons_set_a @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_592_cons2__if__2elems,axiom,
    ! [X2: produc5365630598061846603st_a_b,Xs: list_P2989903729216022225st_a_b,Y2: produc5365630598061846603st_a_b] :
      ( ( member2248238449149680884st_a_b @ X2 @ ( set_Pr2913479409760856416st_a_b @ Xs ) )
     => ( ( member2248238449149680884st_a_b @ Y2 @ ( set_Pr2913479409760856416st_a_b @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: produc5365630598061846603st_a_b,Z5: produc5365630598061846603st_a_b,Zs2: list_P2989903729216022225st_a_b] :
              ( Xs
              = ( cons_P1986909248207666433st_a_b @ Y3 @ ( cons_P1986909248207666433st_a_b @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_593_cons2__if__2elems,axiom,
    ! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Y2: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
     => ( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: product_prod_a_a,Z5: product_prod_a_a,Zs2: list_P1396940483166286381od_a_a] :
              ( Xs
              = ( cons_P7316939126706565853od_a_a @ Y3 @ ( cons_P7316939126706565853od_a_a @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_594_cons2__if__2elems,axiom,
    ! [X2: b,Xs: list_b,Y2: b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ( member_b @ Y2 @ ( set_b2 @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: b,Z5: b,Zs2: list_b] :
              ( Xs
              = ( cons_b @ Y3 @ ( cons_b @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_595_cons2__if__2elems,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ X2 @ ( set_li5246279850479030973st_a_b @ Xs ) )
     => ( ( member1308482598881763409st_a_b @ Y2 @ ( set_li5246279850479030973st_a_b @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: list_P2281458131032066856st_a_b,Z5: list_P2281458131032066856st_a_b,Zs2: list_l3733004638376709294st_a_b] :
              ( Xs
              = ( cons_l7524743740370447070st_a_b @ Y3 @ ( cons_l7524743740370447070st_a_b @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_596_cons2__if__2elems,axiom,
    ! [X2: produc2313410840657238951st_a_b,Xs: list_P1351356133043840941st_a_b,Y2: produc2313410840657238951st_a_b] :
      ( ( member7191766420993298128st_a_b @ X2 @ ( set_Pr5950792022122277180st_a_b @ Xs ) )
     => ( ( member7191766420993298128st_a_b @ Y2 @ ( set_Pr5950792022122277180st_a_b @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: produc2313410840657238951st_a_b,Z5: produc2313410840657238951st_a_b,Zs2: list_P1351356133043840941st_a_b] :
              ( Xs
              = ( cons_P5669161040283791197st_a_b @ Y3 @ ( cons_P5669161040283791197st_a_b @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_597_cons2__if__2elems,axiom,
    ! [X2: list_a,Xs: list_list_a,Y2: list_a] :
      ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: list_a,Z5: list_a,Zs2: list_list_a] :
              ( Xs
              = ( cons_list_a @ Y3 @ ( cons_list_a @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_598_cons2__if__2elems,axiom,
    ! [X2: a,Xs: list_a,Y2: a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: a,Z5: a,Zs2: list_a] :
              ( Xs
              = ( cons_a @ Y3 @ ( cons_a @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_599_cons2__if__2elems,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
       => ( ( X2 != Y2 )
         => ? [Y3: produc2579390649552321826st_a_b,Z5: produc2579390649552321826st_a_b,Zs2: list_P2281458131032066856st_a_b] :
              ( Xs
              = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ Z5 @ Zs2 ) ) ) ) ) ) ).

% cons2_if_2elems
thf(fact_600_merge__split__supset__fst_H,axiom,
    ! [R: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,Cmp: compar845056488427145319st_a_b,Xs: list_P7619746292468751847st_a_b,Ys: list_P7619746292468751847st_a_b] :
      ( ( member3358527249890354327_a_b_o @ R @ ( image_6364829808187590098_a_b_o @ produc1290499411302820541st_a_b @ ( set_Pr6459581840787452534st_a_b @ ( sortin1486583550604052644st_a_b @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P7619746292468751847st_a_b,E3: list_P2281458131032066856st_a_b,Bs: list_P7619746292468751847st_a_b,As2: list_P7619746292468751847st_a_b,Bs2: list_P7619746292468751847st_a_b] :
          ( ( ( append8824326365066233212st_a_b @ As @ ( cons_P548349758394933143st_a_b @ ( produc3504008754983458001st_a_b @ R @ E3 ) @ Bs ) )
            = ( sortin1486583550604052644st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le3067793322193530977st_a_b @ ( set_Pr6459581840787452534st_a_b @ Bs2 ) @ ( set_Pr6459581840787452534st_a_b @ Bs ) )
          & ( ( ( append8824326365066233212st_a_b @ As2 @ ( cons_P548349758394933143st_a_b @ ( produc3504008754983458001st_a_b @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append8824326365066233212st_a_b @ As2 @ ( cons_P548349758394933143st_a_b @ ( produc3504008754983458001st_a_b @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_601_merge__split__supset__fst_H,axiom,
    ! [R: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,Cmp: compar2911641280750143793st_a_b,Xs: list_P2123825271394620337st_a_b,Ys: list_P2123825271394620337st_a_b] :
      ( ( member2196815172658074515b_real @ R @ ( image_5843572149806333988b_real @ produc6225685312319343463st_a_b @ ( set_Pr5623640687540303180st_a_b @ ( sortin8478709104958712990st_a_b @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P2123825271394620337st_a_b,E3: list_P2281458131032066856st_a_b,Bs: list_P2123825271394620337st_a_b,As2: list_P2123825271394620337st_a_b,Bs2: list_P2123825271394620337st_a_b] :
          ( ( ( append7445173226576399814st_a_b @ As @ ( cons_P5662456763204687723st_a_b @ ( produc6916920643058086483st_a_b @ R @ E3 ) @ Bs ) )
            = ( sortin8478709104958712990st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le6429597907857667511st_a_b @ ( set_Pr5623640687540303180st_a_b @ Bs2 ) @ ( set_Pr5623640687540303180st_a_b @ Bs ) )
          & ( ( ( append7445173226576399814st_a_b @ As2 @ ( cons_P5662456763204687723st_a_b @ ( produc6916920643058086483st_a_b @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append7445173226576399814st_a_b @ As2 @ ( cons_P5662456763204687723st_a_b @ ( produc6916920643058086483st_a_b @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_602_merge__split__supset__fst_H,axiom,
    ! [R: b,Cmp: compar2383660659220439440st_a_b,Xs: list_P3985743315777124112st_a_b,Ys: list_P3985743315777124112st_a_b] :
      ( ( member_b @ R @ ( image_3754579970968438394_a_b_b @ produc3358030408229276518st_a_b @ ( set_Pr6619953552560679199st_a_b @ ( sortin7420787910063966797st_a_b @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P3985743315777124112st_a_b,E3: produc2579390649552321826st_a_b,Bs: list_P3985743315777124112st_a_b,As2: list_P3985743315777124112st_a_b,Bs2: list_P3985743315777124112st_a_b] :
          ( ( ( append78464452443947813st_a_b @ As @ ( cons_P5693383391007489216st_a_b @ ( produc2237782585831962490st_a_b @ R @ E3 ) @ Bs ) )
            = ( sortin7420787910063966797st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le424520141106463370st_a_b @ ( set_Pr6619953552560679199st_a_b @ Bs2 ) @ ( set_Pr6619953552560679199st_a_b @ Bs ) )
          & ( ( ( append78464452443947813st_a_b @ As2 @ ( cons_P5693383391007489216st_a_b @ ( produc2237782585831962490st_a_b @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append78464452443947813st_a_b @ As2 @ ( cons_P5693383391007489216st_a_b @ ( produc2237782585831962490st_a_b @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_603_merge__split__supset__fst_H,axiom,
    ! [R: a,Cmp: compar1387821072659337553st_a_b,Xs: list_P2989903729216022225st_a_b,Ys: list_P2989903729216022225st_a_b] :
      ( ( member_a @ R @ ( image_4591392417036834936_a_b_a @ produc8874928302284229543st_a_b @ ( set_Pr2913479409760856416st_a_b @ ( sortin3714313767264144014st_a_b @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P2989903729216022225st_a_b,E3: produc2579390649552321826st_a_b,Bs: list_P2989903729216022225st_a_b,As2: list_P2989903729216022225st_a_b,Bs2: list_P2989903729216022225st_a_b] :
          ( ( ( append5595362346498900838st_a_b @ As @ ( cons_P1986909248207666433st_a_b @ ( produc7754680479886915515st_a_b @ R @ E3 ) @ Bs ) )
            = ( sortin3714313767264144014st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le8652052591400137291st_a_b @ ( set_Pr2913479409760856416st_a_b @ Bs2 ) @ ( set_Pr2913479409760856416st_a_b @ Bs ) )
          & ( ( ( append5595362346498900838st_a_b @ As2 @ ( cons_P1986909248207666433st_a_b @ ( produc7754680479886915515st_a_b @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append5595362346498900838st_a_b @ As2 @ ( cons_P1986909248207666433st_a_b @ ( produc7754680479886915515st_a_b @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_604_merge__split__supset__fst_H,axiom,
    ! [R: a,Cmp: compar5643800712925338797od_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
      ( ( member_a @ R @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ ( sortin327075415733043178od_a_a @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P1396940483166286381od_a_a,E3: a,Bs: list_P1396940483166286381od_a_a,As2: list_P1396940483166286381od_a_a,Bs2: list_P1396940483166286381od_a_a] :
          ( ( ( append5335208819046833346od_a_a @ As @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ R @ E3 ) @ Bs ) )
            = ( sortin327075415733043178od_a_a @ Cmp @ Xs @ Ys ) )
          & ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Bs2 ) @ ( set_Product_prod_a_a2 @ Bs ) )
          & ( ( ( append5335208819046833346od_a_a @ As2 @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append5335208819046833346od_a_a @ As2 @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_605_merge__split__supset__fst_H,axiom,
    ! [R: produc2579390649552321826st_a_b,Cmp: compar70190622580527661st_a_b,Xs: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( member8006451236149131979st_a_b @ R @ ( image_2887695028685365312st_a_b @ produc6544331879426417283st_a_b @ ( set_Pr5950792022122277180st_a_b @ ( sortin5193368813060177258st_a_b @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P1351356133043840941st_a_b,E3: produc2579390649552321826st_a_b,Bs: list_P1351356133043840941st_a_b,As2: list_P1351356133043840941st_a_b,Bs2: list_P1351356133043840941st_a_b] :
          ( ( ( append301769967221396034st_a_b @ As @ ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ R @ E3 ) @ Bs ) )
            = ( sortin5193368813060177258st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le1374440848537019431st_a_b @ ( set_Pr5950792022122277180st_a_b @ Bs2 ) @ ( set_Pr5950792022122277180st_a_b @ Bs ) )
          & ( ( ( append301769967221396034st_a_b @ As2 @ ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append301769967221396034st_a_b @ As2 @ ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_606_merge__split__supset__fst_H,axiom,
    ! [R: list_a,Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( member_list_a @ R @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) ) ) )
     => ? [As: list_P2281458131032066856st_a_b,E3: b,Bs: list_P2281458131032066856st_a_b,As2: list_P2281458131032066856st_a_b,Bs2: list_P2281458131032066856st_a_b] :
          ( ( ( append8668862642614166333st_a_b @ As @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R @ E3 ) @ Bs ) )
            = ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Bs2 ) @ ( set_Pr6156699709417984823st_a_b @ Bs ) )
          & ( ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R @ E3 ) @ Bs2 ) )
              = Xs )
            | ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R @ E3 ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst'
thf(fact_607_not__in__set__insert,axiom,
    ! [X2: set_Pr9033091269622839554st_a_b,Xs: list_s2207380573456591624st_a_b] :
      ( ~ ( member358942219636986539st_a_b @ X2 @ ( set_se3802960479674350359st_a_b @ Xs ) )
     => ( ( insert4611608076054913054st_a_b @ X2 @ Xs )
        = ( cons_s9161922236425393080st_a_b @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_608_not__in__set__insert,axiom,
    ! [X2: set_a,Xs: list_set_a] :
      ( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
     => ( ( insert_set_a @ X2 @ Xs )
        = ( cons_set_a @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_609_not__in__set__insert,axiom,
    ! [X2: produc5365630598061846603st_a_b,Xs: list_P2989903729216022225st_a_b] :
      ( ~ ( member2248238449149680884st_a_b @ X2 @ ( set_Pr2913479409760856416st_a_b @ Xs ) )
     => ( ( insert3206060309516159847st_a_b @ X2 @ Xs )
        = ( cons_P1986909248207666433st_a_b @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_610_not__in__set__insert,axiom,
    ! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
     => ( ( insert7736115120964043331od_a_a @ X2 @ Xs )
        = ( cons_P7316939126706565853od_a_a @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_611_not__in__set__insert,axiom,
    ! [X2: b,Xs: list_b] :
      ( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ( insert_b @ X2 @ Xs )
        = ( cons_b @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_612_not__in__set__insert,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b] :
      ( ~ ( member1308482598881763409st_a_b @ X2 @ ( set_li5246279850479030973st_a_b @ Xs ) )
     => ( ( insert3558307482101359684st_a_b @ X2 @ Xs )
        = ( cons_l7524743740370447070st_a_b @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_613_not__in__set__insert,axiom,
    ! [X2: produc2313410840657238951st_a_b,Xs: list_P1351356133043840941st_a_b] :
      ( ~ ( member7191766420993298128st_a_b @ X2 @ ( set_Pr5950792022122277180st_a_b @ Xs ) )
     => ( ( insert6398464614709525699st_a_b @ X2 @ Xs )
        = ( cons_P5669161040283791197st_a_b @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_614_not__in__set__insert,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( insert_list_a @ X2 @ Xs )
        = ( cons_list_a @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_615_not__in__set__insert,axiom,
    ! [X2: a,Xs: list_a] :
      ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( insert_a @ X2 @ Xs )
        = ( cons_a @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_616_not__in__set__insert,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( ( insert6036452702741353534st_a_b @ X2 @ Xs )
        = ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_617_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,E: list_P2281458131032066856st_a_b,Xs: list_P7619746292468751847st_a_b] :
      ( ( wf_lis74079735226913945st_a_b @ ( cons_P548349758394933143st_a_b @ ( produc3504008754983458001st_a_b @ V @ E ) @ Xs ) )
      = ( ~ ( member3358527249890354327_a_b_o @ V @ ( image_6364829808187590098_a_b_o @ produc1290499411302820541st_a_b @ ( set_Pr6459581840787452534st_a_b @ Xs ) ) )
        & ( wf_lis74079735226913945st_a_b @ Xs ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_618_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,E: list_P2281458131032066856st_a_b,Xs: list_P2123825271394620337st_a_b] :
      ( ( wf_lis6206885346534431883st_a_b @ ( cons_P5662456763204687723st_a_b @ ( produc6916920643058086483st_a_b @ V @ E ) @ Xs ) )
      = ( ~ ( member2196815172658074515b_real @ V @ ( image_5843572149806333988b_real @ produc6225685312319343463st_a_b @ ( set_Pr5623640687540303180st_a_b @ Xs ) ) )
        & ( wf_lis6206885346534431883st_a_b @ Xs ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_619_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: b,E: produc2579390649552321826st_a_b,Xs: list_P3985743315777124112st_a_b] :
      ( ( wf_lis3940338229446200130st_a_b @ ( cons_P5693383391007489216st_a_b @ ( produc2237782585831962490st_a_b @ V @ E ) @ Xs ) )
      = ( ~ ( member_b @ V @ ( image_3754579970968438394_a_b_b @ produc3358030408229276518st_a_b @ ( set_Pr6619953552560679199st_a_b @ Xs ) ) )
        & ( wf_lis3940338229446200130st_a_b @ Xs ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_620_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: a,E: produc2579390649552321826st_a_b,Xs: list_P2989903729216022225st_a_b] :
      ( ( wf_lis233864086646377347st_a_b @ ( cons_P1986909248207666433st_a_b @ ( produc7754680479886915515st_a_b @ V @ E ) @ Xs ) )
      = ( ~ ( member_a @ V @ ( image_4591392417036834936_a_b_a @ produc8874928302284229543st_a_b @ ( set_Pr2913479409760856416st_a_b @ Xs ) ) )
        & ( wf_lis233864086646377347st_a_b @ Xs ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_621_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: a,E: a,Xs: list_P1396940483166286381od_a_a] :
      ( ( wf_list_verts_a_a @ ( 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.simps(2)
thf(fact_622_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: produc2579390649552321826st_a_b,E: produc2579390649552321826st_a_b,Xs: list_P1351356133043840941st_a_b] :
      ( ( wf_lis1516364317032777823st_a_b @ ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ V @ E ) @ Xs ) )
      = ( ~ ( member8006451236149131979st_a_b @ V @ ( image_2887695028685365312st_a_b @ produc6544331879426417283st_a_b @ ( set_Pr5950792022122277180st_a_b @ Xs ) ) )
        & ( wf_lis1516364317032777823st_a_b @ Xs ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_623_wf__list__verts_Osimps_I2_J,axiom,
    ! [V: list_a,E: b,Xs: list_P2281458131032066856st_a_b] :
      ( ( wf_lis5250910623959142362st_a_b @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V @ E ) @ Xs ) )
      = ( ~ ( member_list_a @ V @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) )
        & ( wf_lis5250910623959142362st_a_b @ Xs ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_624_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P7619746292468751847st_a_b] :
      ( ~ ( wf_lis74079735226913945st_a_b @ X2 )
     => ~ ! [V3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,E3: list_P2281458131032066856st_a_b,Xs2: list_P7619746292468751847st_a_b] :
            ( ( X2
              = ( cons_P548349758394933143st_a_b @ ( produc3504008754983458001st_a_b @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member3358527249890354327_a_b_o @ V3 @ ( image_6364829808187590098_a_b_o @ produc1290499411302820541st_a_b @ ( set_Pr6459581840787452534st_a_b @ Xs2 ) ) )
              & ( wf_lis74079735226913945st_a_b @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_625_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P2123825271394620337st_a_b] :
      ( ~ ( wf_lis6206885346534431883st_a_b @ X2 )
     => ~ ! [V3: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,E3: list_P2281458131032066856st_a_b,Xs2: list_P2123825271394620337st_a_b] :
            ( ( X2
              = ( cons_P5662456763204687723st_a_b @ ( produc6916920643058086483st_a_b @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member2196815172658074515b_real @ V3 @ ( image_5843572149806333988b_real @ produc6225685312319343463st_a_b @ ( set_Pr5623640687540303180st_a_b @ Xs2 ) ) )
              & ( wf_lis6206885346534431883st_a_b @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_626_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P3985743315777124112st_a_b] :
      ( ~ ( wf_lis3940338229446200130st_a_b @ X2 )
     => ~ ! [V3: b,E3: produc2579390649552321826st_a_b,Xs2: list_P3985743315777124112st_a_b] :
            ( ( X2
              = ( cons_P5693383391007489216st_a_b @ ( produc2237782585831962490st_a_b @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member_b @ V3 @ ( image_3754579970968438394_a_b_b @ produc3358030408229276518st_a_b @ ( set_Pr6619953552560679199st_a_b @ Xs2 ) ) )
              & ( wf_lis3940338229446200130st_a_b @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_627_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P2989903729216022225st_a_b] :
      ( ~ ( wf_lis233864086646377347st_a_b @ X2 )
     => ~ ! [V3: a,E3: produc2579390649552321826st_a_b,Xs2: list_P2989903729216022225st_a_b] :
            ( ( X2
              = ( cons_P1986909248207666433st_a_b @ ( produc7754680479886915515st_a_b @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member_a @ V3 @ ( image_4591392417036834936_a_b_a @ produc8874928302284229543st_a_b @ ( set_Pr2913479409760856416st_a_b @ Xs2 ) ) )
              & ( wf_lis233864086646377347st_a_b @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_628_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P1396940483166286381od_a_a] :
      ( ~ ( wf_list_verts_a_a @ X2 )
     => ~ ! [V3: a,E3: a,Xs2: list_P1396940483166286381od_a_a] :
            ( ( X2
              = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member_a @ V3 @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xs2 ) ) )
              & ( wf_list_verts_a_a @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_629_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P1351356133043840941st_a_b] :
      ( ~ ( wf_lis1516364317032777823st_a_b @ X2 )
     => ~ ! [V3: produc2579390649552321826st_a_b,E3: produc2579390649552321826st_a_b,Xs2: list_P1351356133043840941st_a_b] :
            ( ( X2
              = ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member8006451236149131979st_a_b @ V3 @ ( image_2887695028685365312st_a_b @ produc6544331879426417283st_a_b @ ( set_Pr5950792022122277180st_a_b @ Xs2 ) ) )
              & ( wf_lis1516364317032777823st_a_b @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_630_wf__list__verts_Oelims_I3_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ~ ( wf_lis5250910623959142362st_a_b @ X2 )
     => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
            ( ( X2
              = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
           => ( ~ ( member_list_a @ V3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
              & ( wf_lis5250910623959142362st_a_b @ Xs2 ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_631_merge_Oelims,axiom,
    ! [Cmp: compar6837619475480278830st_a_b,X2: list_l3733004638376709294st_a_b,Xa: list_l3733004638376709294st_a_b,Y2: list_l3733004638376709294st_a_b] :
      ( ( ( sortin6459637776524370667st_a_b @ Cmp @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_li5528507983314571054st_a_b )
         => ( Y2 != Xa ) )
       => ( ( ( Xa = nil_li5528507983314571054st_a_b )
           => ( Y2 != X2 ) )
         => ~ ! [X: list_P2281458131032066856st_a_b,Xs2: list_l3733004638376709294st_a_b] :
                ( ( X2
                  = ( cons_l7524743740370447070st_a_b @ X @ Xs2 ) )
               => ! [Y3: list_P2281458131032066856st_a_b,Ys2: list_l3733004638376709294st_a_b] :
                    ( ( Xa
                      = ( cons_l7524743740370447070st_a_b @ Y3 @ Ys2 ) )
                   => ~ ( ( ( ( compar5222392469468919210st_a_b @ Cmp @ X @ Y3 )
                            = greater )
                         => ( Y2
                            = ( cons_l7524743740370447070st_a_b @ Y3 @ ( sortin6459637776524370667st_a_b @ Cmp @ ( cons_l7524743740370447070st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) )
                        & ( ( ( compar5222392469468919210st_a_b @ Cmp @ X @ Y3 )
                           != greater )
                         => ( Y2
                            = ( cons_l7524743740370447070st_a_b @ X @ ( sortin6459637776524370667st_a_b @ Cmp @ Xs2 @ ( cons_l7524743740370447070st_a_b @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_632_merge_Oelims,axiom,
    ! [Cmp: compar70190622580527661st_a_b,X2: list_P1351356133043840941st_a_b,Xa: list_P1351356133043840941st_a_b,Y2: list_P1351356133043840941st_a_b] :
      ( ( ( sortin5193368813060177258st_a_b @ Cmp @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_Pr6552475826380092333st_a_b )
         => ( Y2 != Xa ) )
       => ( ( ( Xa = nil_Pr6552475826380092333st_a_b )
           => ( Y2 != X2 ) )
         => ~ ! [X: produc2313410840657238951st_a_b,Xs2: list_P1351356133043840941st_a_b] :
                ( ( X2
                  = ( cons_P5669161040283791197st_a_b @ X @ Xs2 ) )
               => ! [Y3: produc2313410840657238951st_a_b,Ys2: list_P1351356133043840941st_a_b] :
                    ( ( Xa
                      = ( cons_P5669161040283791197st_a_b @ Y3 @ Ys2 ) )
                   => ~ ( ( ( ( compar2130703882349896233st_a_b @ Cmp @ X @ Y3 )
                            = greater )
                         => ( Y2
                            = ( cons_P5669161040283791197st_a_b @ Y3 @ ( sortin5193368813060177258st_a_b @ Cmp @ ( cons_P5669161040283791197st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) )
                        & ( ( ( compar2130703882349896233st_a_b @ Cmp @ X @ Y3 )
                           != greater )
                         => ( Y2
                            = ( cons_P5669161040283791197st_a_b @ X @ ( sortin5193368813060177258st_a_b @ Cmp @ Xs2 @ ( cons_P5669161040283791197st_a_b @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_633_merge_Oelims,axiom,
    ! [Cmp: comparator_list_a,X2: list_list_a,Xa: list_list_a,Y2: list_list_a] :
      ( ( ( sorting_merge_list_a @ Cmp @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_list_a )
         => ( Y2 != Xa ) )
       => ( ( ( Xa = nil_list_a )
           => ( Y2 != X2 ) )
         => ~ ! [X: list_a,Xs2: list_list_a] :
                ( ( X2
                  = ( cons_list_a @ X @ Xs2 ) )
               => ! [Y3: list_a,Ys2: list_list_a] :
                    ( ( Xa
                      = ( cons_list_a @ Y3 @ Ys2 ) )
                   => ~ ( ( ( ( compare_list_a @ Cmp @ X @ Y3 )
                            = greater )
                         => ( Y2
                            = ( cons_list_a @ Y3 @ ( sorting_merge_list_a @ Cmp @ ( cons_list_a @ X @ Xs2 ) @ Ys2 ) ) ) )
                        & ( ( ( compare_list_a @ Cmp @ X @ Y3 )
                           != greater )
                         => ( Y2
                            = ( cons_list_a @ X @ ( sorting_merge_list_a @ Cmp @ Xs2 @ ( cons_list_a @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_634_merge_Oelims,axiom,
    ! [Cmp: comparator_a,X2: list_a,Xa: list_a,Y2: list_a] :
      ( ( ( sorting_merge_a @ Cmp @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_a )
         => ( Y2 != Xa ) )
       => ( ( ( Xa = nil_a )
           => ( Y2 != X2 ) )
         => ~ ! [X: a,Xs2: list_a] :
                ( ( X2
                  = ( cons_a @ X @ Xs2 ) )
               => ! [Y3: a,Ys2: list_a] :
                    ( ( Xa
                      = ( cons_a @ Y3 @ Ys2 ) )
                   => ~ ( ( ( ( compare_a @ Cmp @ X @ Y3 )
                            = greater )
                         => ( Y2
                            = ( cons_a @ Y3 @ ( sorting_merge_a @ Cmp @ ( cons_a @ X @ Xs2 ) @ Ys2 ) ) ) )
                        & ( ( ( compare_a @ Cmp @ X @ Y3 )
                           != greater )
                         => ( Y2
                            = ( cons_a @ X @ ( sorting_merge_a @ Cmp @ Xs2 @ ( cons_a @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_635_merge_Oelims,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: list_P2281458131032066856st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( ( sortin5994461831780029029st_a_b @ Cmp @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
         => ( Y2 != Xa ) )
       => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
           => ( Y2 != X2 ) )
         => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) )
               => ! [Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                    ( ( Xa
                      = ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
                   => ~ ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
                            = greater )
                         => ( Y2
                            = ( cons_P2018802354021969880st_a_b @ Y3 @ ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) )
                        & ( ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
                           != greater )
                         => ( Y2
                            = ( cons_P2018802354021969880st_a_b @ X @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_636_subset__Collect__iff,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,P: product_prod_a_a > $o] :
      ( ( ord_le746702958409616551od_a_a @ B @ A )
     => ( ( ord_le746702958409616551od_a_a @ B
          @ ( collec3336397797384452498od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_637_subset__Collect__iff,axiom,
    ! [B: set_se1825328058903261410st_a_b,A: set_se1825328058903261410st_a_b,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ( ord_le4557451523740326018st_a_b @ B @ A )
     => ( ( ord_le4557451523740326018st_a_b @ B
          @ ( collec4420230097056376813st_a_b
            @ ^ [X4: set_Pr9033091269622839554st_a_b] :
                ( ( member358942219636986539st_a_b @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: set_Pr9033091269622839554st_a_b] :
              ( ( member358942219636986539st_a_b @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_638_subset__Collect__iff,axiom,
    ! [B: set_Pr9033091269622839554st_a_b,A: set_Pr9033091269622839554st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ( ord_le2879471331418251426st_a_b @ B @ A )
     => ( ( ord_le2879471331418251426st_a_b @ B
          @ ( collec3957028476971440141st_a_b
            @ ^ [X4: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_639_subset__Collect__iff,axiom,
    ! [B: set_li7460651515946730888st_a_b,A: set_li7460651515946730888st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ( ord_le2045451444129209640st_a_b @ B @ A )
     => ( ( ord_le2045451444129209640st_a_b @ B
          @ ( collec3283141540447581331st_a_b
            @ ^ [X4: list_P2281458131032066856st_a_b] :
                ( ( member1308482598881763409st_a_b @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: list_P2281458131032066856st_a_b] :
              ( ( member1308482598881763409st_a_b @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_640_subset__Collect__iff,axiom,
    ! [B: set_list_a,A: set_list_a,P: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ B @ A )
     => ( ( ord_le8861187494160871172list_a @ B
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_641_subset__Collect__iff,axiom,
    ! [B: set_set_list_a,A: set_set_list_a,P: set_list_a > $o] :
      ( ( ord_le8877086941679407844list_a @ B @ A )
     => ( ( ord_le8877086941679407844list_a @ B
          @ ( collect_set_list_a
            @ ^ [X4: set_list_a] :
                ( ( member_set_list_a @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: set_list_a] :
              ( ( member_set_list_a @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_642_subset__Collect__iff,axiom,
    ! [B: set_set_a,A: set_set_a,P: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ B @ A )
     => ( ( ord_le3724670747650509150_set_a @ B
          @ ( collect_set_a
            @ ^ [X4: set_a] :
                ( ( member_set_a @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: set_a] :
              ( ( member_set_a @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_643_subset__Collect__iff,axiom,
    ! [B: set_a,A: set_a,P: a > $o] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( ord_less_eq_set_a @ B
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: a] :
              ( ( member_a @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_644_subset__CollectI,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,Q: product_prod_a_a > $o,P: product_prod_a_a > $o] :
      ( ( ord_le746702958409616551od_a_a @ B @ A )
     => ( ! [X: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le746702958409616551od_a_a
          @ ( collec3336397797384452498od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collec3336397797384452498od_a_a
            @ ^ [X4: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_645_subset__CollectI,axiom,
    ! [B: set_se1825328058903261410st_a_b,A: set_se1825328058903261410st_a_b,Q: set_Pr9033091269622839554st_a_b > $o,P: set_Pr9033091269622839554st_a_b > $o] :
      ( ( ord_le4557451523740326018st_a_b @ B @ A )
     => ( ! [X: set_Pr9033091269622839554st_a_b] :
            ( ( member358942219636986539st_a_b @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le4557451523740326018st_a_b
          @ ( collec4420230097056376813st_a_b
            @ ^ [X4: set_Pr9033091269622839554st_a_b] :
                ( ( member358942219636986539st_a_b @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collec4420230097056376813st_a_b
            @ ^ [X4: set_Pr9033091269622839554st_a_b] :
                ( ( member358942219636986539st_a_b @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_646_subset__CollectI,axiom,
    ! [B: set_Pr9033091269622839554st_a_b,A: set_Pr9033091269622839554st_a_b,Q: produc2579390649552321826st_a_b > $o,P: produc2579390649552321826st_a_b > $o] :
      ( ( ord_le2879471331418251426st_a_b @ B @ A )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le2879471331418251426st_a_b
          @ ( collec3957028476971440141st_a_b
            @ ^ [X4: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collec3957028476971440141st_a_b
            @ ^ [X4: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_647_subset__CollectI,axiom,
    ! [B: set_li7460651515946730888st_a_b,A: set_li7460651515946730888st_a_b,Q: list_P2281458131032066856st_a_b > $o,P: list_P2281458131032066856st_a_b > $o] :
      ( ( ord_le2045451444129209640st_a_b @ B @ A )
     => ( ! [X: list_P2281458131032066856st_a_b] :
            ( ( member1308482598881763409st_a_b @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le2045451444129209640st_a_b
          @ ( collec3283141540447581331st_a_b
            @ ^ [X4: list_P2281458131032066856st_a_b] :
                ( ( member1308482598881763409st_a_b @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collec3283141540447581331st_a_b
            @ ^ [X4: list_P2281458131032066856st_a_b] :
                ( ( member1308482598881763409st_a_b @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_648_subset__CollectI,axiom,
    ! [B: set_list_a,A: set_list_a,Q: list_a > $o,P: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ B @ A )
     => ( ! [X: list_a] :
            ( ( member_list_a @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le8861187494160871172list_a
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_list_a
            @ ^ [X4: list_a] :
                ( ( member_list_a @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_649_subset__CollectI,axiom,
    ! [B: set_set_list_a,A: set_set_list_a,Q: set_list_a > $o,P: set_list_a > $o] :
      ( ( ord_le8877086941679407844list_a @ B @ A )
     => ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le8877086941679407844list_a
          @ ( collect_set_list_a
            @ ^ [X4: set_list_a] :
                ( ( member_set_list_a @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_set_list_a
            @ ^ [X4: set_list_a] :
                ( ( member_set_list_a @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_650_subset__CollectI,axiom,
    ! [B: set_set_a,A: set_set_a,Q: set_a > $o,P: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ B @ A )
     => ( ! [X: set_a] :
            ( ( member_set_a @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le3724670747650509150_set_a
          @ ( collect_set_a
            @ ^ [X4: set_a] :
                ( ( member_set_a @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_set_a
            @ ^ [X4: set_a] :
                ( ( member_set_a @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_651_subset__CollectI,axiom,
    ! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ! [X: a] :
            ( ( member_a @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_less_eq_set_a
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_652_same__append__eq,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Xs @ Ys )
        = ( append8668862642614166333st_a_b @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_653_same__append__eq,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = ( append_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_654_same__append__eq,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b,Zs: list_l3733004638376709294st_a_b] :
      ( ( ( append2962543925774821315st_a_b @ Xs @ Ys )
        = ( append2962543925774821315st_a_b @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_655_append__same__eq,axiom,
    ! [Ys: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Ys @ Xs )
        = ( append8668862642614166333st_a_b @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_656_append__same__eq,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( append_a @ Ys @ Xs )
        = ( append_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_657_append__same__eq,axiom,
    ! [Ys: list_l3733004638376709294st_a_b,Xs: list_l3733004638376709294st_a_b,Zs: list_l3733004638376709294st_a_b] :
      ( ( ( append2962543925774821315st_a_b @ Ys @ Xs )
        = ( append2962543925774821315st_a_b @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_658_append__assoc,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( append8668862642614166333st_a_b @ ( append8668862642614166333st_a_b @ Xs @ Ys ) @ Zs )
      = ( append8668862642614166333st_a_b @ Xs @ ( append8668862642614166333st_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_659_append__assoc,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
      = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_660_append__assoc,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b,Zs: list_l3733004638376709294st_a_b] :
      ( ( append2962543925774821315st_a_b @ ( append2962543925774821315st_a_b @ Xs @ Ys ) @ Zs )
      = ( append2962543925774821315st_a_b @ Xs @ ( append2962543925774821315st_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_661_append_Oassoc,axiom,
    ! [A2: list_P2281458131032066856st_a_b,B2: list_P2281458131032066856st_a_b,C2: list_P2281458131032066856st_a_b] :
      ( ( append8668862642614166333st_a_b @ ( append8668862642614166333st_a_b @ A2 @ B2 ) @ C2 )
      = ( append8668862642614166333st_a_b @ A2 @ ( append8668862642614166333st_a_b @ B2 @ C2 ) ) ) ).

% append.assoc
thf(fact_662_append_Oassoc,axiom,
    ! [A2: list_a,B2: list_a,C2: list_a] :
      ( ( append_a @ ( append_a @ A2 @ B2 ) @ C2 )
      = ( append_a @ A2 @ ( append_a @ B2 @ C2 ) ) ) ).

% append.assoc
thf(fact_663_append_Oassoc,axiom,
    ! [A2: list_l3733004638376709294st_a_b,B2: list_l3733004638376709294st_a_b,C2: list_l3733004638376709294st_a_b] :
      ( ( append2962543925774821315st_a_b @ ( append2962543925774821315st_a_b @ A2 @ B2 ) @ C2 )
      = ( append2962543925774821315st_a_b @ A2 @ ( append2962543925774821315st_a_b @ B2 @ C2 ) ) ) ).

% append.assoc
thf(fact_664_append__is__Nil__conv,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Xs @ Ys )
        = nil_Pr523822035851181096st_a_b )
      = ( ( Xs = nil_Pr523822035851181096st_a_b )
        & ( Ys = nil_Pr523822035851181096st_a_b ) ) ) ).

% append_is_Nil_conv
thf(fact_665_append__is__Nil__conv,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b] :
      ( ( ( append2962543925774821315st_a_b @ Xs @ Ys )
        = nil_li5528507983314571054st_a_b )
      = ( ( Xs = nil_li5528507983314571054st_a_b )
        & ( Ys = nil_li5528507983314571054st_a_b ) ) ) ).

% append_is_Nil_conv
thf(fact_666_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_667_append__is__Nil__conv,axiom,
    ! [Xs: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( ( append301769967221396034st_a_b @ Xs @ Ys )
        = nil_Pr6552475826380092333st_a_b )
      = ( ( Xs = nil_Pr6552475826380092333st_a_b )
        & ( Ys = nil_Pr6552475826380092333st_a_b ) ) ) ).

% append_is_Nil_conv
thf(fact_668_Nil__is__append__conv,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( nil_Pr523822035851181096st_a_b
        = ( append8668862642614166333st_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr523822035851181096st_a_b )
        & ( Ys = nil_Pr523822035851181096st_a_b ) ) ) ).

% Nil_is_append_conv
thf(fact_669_Nil__is__append__conv,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b] :
      ( ( nil_li5528507983314571054st_a_b
        = ( append2962543925774821315st_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_li5528507983314571054st_a_b )
        & ( Ys = nil_li5528507983314571054st_a_b ) ) ) ).

% Nil_is_append_conv
thf(fact_670_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_671_Nil__is__append__conv,axiom,
    ! [Xs: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( nil_Pr6552475826380092333st_a_b
        = ( append301769967221396034st_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr6552475826380092333st_a_b )
        & ( Ys = nil_Pr6552475826380092333st_a_b ) ) ) ).

% Nil_is_append_conv
thf(fact_672_self__append__conv2,axiom,
    ! [Y2: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( Y2
        = ( append8668862642614166333st_a_b @ Xs @ Y2 ) )
      = ( Xs = nil_Pr523822035851181096st_a_b ) ) ).

% self_append_conv2
thf(fact_673_self__append__conv2,axiom,
    ! [Y2: list_l3733004638376709294st_a_b,Xs: list_l3733004638376709294st_a_b] :
      ( ( Y2
        = ( append2962543925774821315st_a_b @ Xs @ Y2 ) )
      = ( Xs = nil_li5528507983314571054st_a_b ) ) ).

% self_append_conv2
thf(fact_674_self__append__conv2,axiom,
    ! [Y2: list_a,Xs: list_a] :
      ( ( Y2
        = ( append_a @ Xs @ Y2 ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_675_self__append__conv2,axiom,
    ! [Y2: list_P1351356133043840941st_a_b,Xs: list_P1351356133043840941st_a_b] :
      ( ( Y2
        = ( append301769967221396034st_a_b @ Xs @ Y2 ) )
      = ( Xs = nil_Pr6552475826380092333st_a_b ) ) ).

% self_append_conv2
thf(fact_676_append__self__conv2,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr523822035851181096st_a_b ) ) ).

% append_self_conv2
thf(fact_677_append__self__conv2,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b] :
      ( ( ( append2962543925774821315st_a_b @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li5528507983314571054st_a_b ) ) ).

% append_self_conv2
thf(fact_678_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_679_append__self__conv2,axiom,
    ! [Xs: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( ( append301769967221396034st_a_b @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr6552475826380092333st_a_b ) ) ).

% append_self_conv2
thf(fact_680_self__append__conv,axiom,
    ! [Y2: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( Y2
        = ( append8668862642614166333st_a_b @ Y2 @ Ys ) )
      = ( Ys = nil_Pr523822035851181096st_a_b ) ) ).

% self_append_conv
thf(fact_681_self__append__conv,axiom,
    ! [Y2: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b] :
      ( ( Y2
        = ( append2962543925774821315st_a_b @ Y2 @ Ys ) )
      = ( Ys = nil_li5528507983314571054st_a_b ) ) ).

% self_append_conv
thf(fact_682_self__append__conv,axiom,
    ! [Y2: list_a,Ys: list_a] :
      ( ( Y2
        = ( append_a @ Y2 @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_683_self__append__conv,axiom,
    ! [Y2: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( Y2
        = ( append301769967221396034st_a_b @ Y2 @ Ys ) )
      = ( Ys = nil_Pr6552475826380092333st_a_b ) ) ).

% self_append_conv
thf(fact_684_append__self__conv,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr523822035851181096st_a_b ) ) ).

% append_self_conv
thf(fact_685_append__self__conv,axiom,
    ! [Xs: list_l3733004638376709294st_a_b,Ys: list_l3733004638376709294st_a_b] :
      ( ( ( append2962543925774821315st_a_b @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li5528507983314571054st_a_b ) ) ).

% append_self_conv
thf(fact_686_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_687_append__self__conv,axiom,
    ! [Xs: list_P1351356133043840941st_a_b,Ys: list_P1351356133043840941st_a_b] :
      ( ( ( append301769967221396034st_a_b @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr6552475826380092333st_a_b ) ) ).

% append_self_conv
thf(fact_688_append__Nil2,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( append8668862642614166333st_a_b @ Xs @ nil_Pr523822035851181096st_a_b )
      = Xs ) ).

% append_Nil2
thf(fact_689_append__Nil2,axiom,
    ! [Xs: list_l3733004638376709294st_a_b] :
      ( ( append2962543925774821315st_a_b @ Xs @ nil_li5528507983314571054st_a_b )
      = Xs ) ).

% append_Nil2
thf(fact_690_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_691_append__Nil2,axiom,
    ! [Xs: list_P1351356133043840941st_a_b] :
      ( ( append301769967221396034st_a_b @ Xs @ nil_Pr6552475826380092333st_a_b )
      = Xs ) ).

% append_Nil2
thf(fact_692_append_Oright__neutral,axiom,
    ! [A2: list_P2281458131032066856st_a_b] :
      ( ( append8668862642614166333st_a_b @ A2 @ nil_Pr523822035851181096st_a_b )
      = A2 ) ).

% append.right_neutral
thf(fact_693_append_Oright__neutral,axiom,
    ! [A2: list_l3733004638376709294st_a_b] :
      ( ( append2962543925774821315st_a_b @ A2 @ nil_li5528507983314571054st_a_b )
      = A2 ) ).

% append.right_neutral
thf(fact_694_append_Oright__neutral,axiom,
    ! [A2: list_a] :
      ( ( append_a @ A2 @ nil_a )
      = A2 ) ).

% append.right_neutral
thf(fact_695_append_Oright__neutral,axiom,
    ! [A2: list_P1351356133043840941st_a_b] :
      ( ( append301769967221396034st_a_b @ A2 @ nil_Pr6552475826380092333st_a_b )
      = A2 ) ).

% append.right_neutral
thf(fact_696_in__set__insert,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b] :
      ( ( member1308482598881763409st_a_b @ X2 @ ( set_li5246279850479030973st_a_b @ Xs ) )
     => ( ( insert3558307482101359684st_a_b @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_697_in__set__insert,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
     => ( ( insert_list_a @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_698_in__set__insert,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ( insert_b @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_699_in__set__insert,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( ( insert6036452702741353534st_a_b @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_700_in__set__insert,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( insert_a @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_701_append1__eq__conv,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) )
        = ( append8668862642614166333st_a_b @ Ys @ ( cons_P2018802354021969880st_a_b @ Y2 @ nil_Pr523822035851181096st_a_b ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_702_insert__Nil,axiom,
    ! [X2: produc2579390649552321826st_a_b] :
      ( ( insert6036452702741353534st_a_b @ X2 @ nil_Pr523822035851181096st_a_b )
      = ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ).

% insert_Nil
thf(fact_703_rev__nonempty__induct,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ! [X: produc2579390649552321826st_a_b] : ( P @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
       => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( Xs2 != nil_Pr523822035851181096st_a_b )
             => ( ( P @ Xs2 )
               => ( P @ ( append8668862642614166333st_a_b @ Xs2 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_704_append__eq__Cons__conv,axiom,
    ! [Ys: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Ys @ Zs )
        = ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( ( ( Ys = nil_Pr523822035851181096st_a_b )
          & ( Zs
            = ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) )
        | ? [Ys3: list_P2281458131032066856st_a_b] :
            ( ( Ys
              = ( cons_P2018802354021969880st_a_b @ X2 @ Ys3 ) )
            & ( ( append8668862642614166333st_a_b @ Ys3 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_705_Cons__eq__append__conv,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
        = ( append8668862642614166333st_a_b @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Pr523822035851181096st_a_b )
          & ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
            = Zs ) )
        | ? [Ys3: list_P2281458131032066856st_a_b] :
            ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Ys3 )
              = Ys )
            & ( Xs
              = ( append8668862642614166333st_a_b @ Ys3 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_706_rev__exhaust,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ~ ! [Ys2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b] :
            ( Xs
           != ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ nil_Pr523822035851181096st_a_b ) ) ) ) ).

% rev_exhaust
thf(fact_707_rev__induct,axiom,
    ! [P: list_P2281458131032066856st_a_b > $o,Xs: list_P2281458131032066856st_a_b] :
      ( ( P @ nil_Pr523822035851181096st_a_b )
     => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
            ( ( P @ Xs2 )
           => ( P @ ( append8668862642614166333st_a_b @ Xs2 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_708_transpose_Ocases,axiom,
    ! [X2: list_l3733004638376709294st_a_b] :
      ( ( X2 != nil_li5528507983314571054st_a_b )
     => ( ! [Xss: list_l3733004638376709294st_a_b] :
            ( X2
           != ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ Xss ) )
       => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Xss: list_l3733004638376709294st_a_b] :
              ( X2
             != ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_709_create__h__list_Ocases,axiom,
    ! [X2: produc6292069877431205281st_a_b] :
      ( ! [Uu: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real] :
          ( X2
         != ( produc6916920643058086483st_a_b @ Uu @ nil_Pr523822035851181096st_a_b ) )
     => ~ ! [H: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc6916920643058086483st_a_b @ H @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ).

% create_h_list.cases
thf(fact_710_c__list_H_Ocases,axiom,
    ! [X2: produc3161689529325124978st_a_b] :
      ( ! [Uu: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Uv: produc2579390649552321826st_a_b > real,Uw: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real] :
          ( X2
         != ( produc7946489679485025322st_a_b @ Uu @ ( produc633835039278819721st_a_b @ Uv @ ( produc6916920643058086483st_a_b @ Uw @ nil_Pr523822035851181096st_a_b ) ) ) )
     => ( ! [Ux: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Uy: produc2579390649552321826st_a_b > real,Uz: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b] :
            ( X2
           != ( produc7946489679485025322st_a_b @ Ux @ ( produc633835039278819721st_a_b @ Uy @ ( produc6916920643058086483st_a_b @ Uz @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) ) )
       => ~ ! [F3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Cf: produc2579390649552321826st_a_b > real,H: list_P2281458131032066856st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b,V3: produc2579390649552321826st_a_b,Va: list_P2281458131032066856st_a_b] :
              ( X2
             != ( produc7946489679485025322st_a_b @ F3 @ ( produc633835039278819721st_a_b @ Cf @ ( produc6916920643058086483st_a_b @ H @ ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ V3 @ Va ) ) ) ) ) ) ) ) ).

% c_list'.cases
thf(fact_711_c__list_Ocases,axiom,
    ! [X2: produc6211782426242234917st_a_b] :
      ( ! [Uu: produc2579390649552321826st_a_b > real,Uv: produc2579390649552321826st_a_b > real,Uw: produc2579390649552321826st_a_b > real,Ux: produc2579390649552321826st_a_b] :
          ( X2
         != ( produc3968337941225440917st_a_b @ Uu @ ( produc7968639511622629997st_a_b @ Uv @ ( produc2440457079528226757st_a_b @ Uw @ ( produc6648078606966411293st_a_b @ Ux @ nil_Pr523822035851181096st_a_b ) ) ) ) )
     => ( ! [Uy: produc2579390649552321826st_a_b > real,Uz: produc2579390649552321826st_a_b > real,H: produc2579390649552321826st_a_b > real,R3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b] :
            ( X2
           != ( produc3968337941225440917st_a_b @ Uy @ ( produc7968639511622629997st_a_b @ Uz @ ( produc2440457079528226757st_a_b @ H @ ( produc6648078606966411293st_a_b @ R3 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) ) ) )
       => ~ ! [Sf: produc2579390649552321826st_a_b > real,Cf: produc2579390649552321826st_a_b > real,H: produc2579390649552321826st_a_b > real,R3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,V3: produc2579390649552321826st_a_b,Va: list_P2281458131032066856st_a_b] :
              ( X2
             != ( produc3968337941225440917st_a_b @ Sf @ ( produc7968639511622629997st_a_b @ Cf @ ( produc2440457079528226757st_a_b @ H @ ( produc6648078606966411293st_a_b @ R3 @ ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ V3 @ Va ) ) ) ) ) ) ) ) ) ).

% c_list.cases
thf(fact_712_Cons__eq__appendI,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs1: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append8668862642614166333st_a_b @ Xs1 @ Zs ) )
       => ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
          = ( append8668862642614166333st_a_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_713_append__Cons,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( append8668862642614166333st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys )
      = ( cons_P2018802354021969880st_a_b @ X2 @ ( append8668862642614166333st_a_b @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_714_list__nonempty__induct,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ! [X: produc2579390649552321826st_a_b] : ( P @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
       => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( Xs2 != nil_Pr523822035851181096st_a_b )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_715_list__induct2_H,axiom,
    ! [P: list_P2281458131032066856st_a_b > list_P2281458131032066856st_a_b > $o,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( P @ nil_Pr523822035851181096st_a_b @ nil_Pr523822035851181096st_a_b )
     => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] : ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ nil_Pr523822035851181096st_a_b )
       => ( ! [Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] : ( P @ nil_Pr523822035851181096st_a_b @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
         => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_716_neq__Nil__conv,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
      = ( ? [Y5: produc2579390649552321826st_a_b,Ys4: list_P2281458131032066856st_a_b] :
            ( Xs
            = ( cons_P2018802354021969880st_a_b @ Y5 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_717_list_Oexhaust,axiom,
    ! [Y2: list_P2281458131032066856st_a_b] :
      ( ( Y2 != nil_Pr523822035851181096st_a_b )
     => ~ ! [X212: produc2579390649552321826st_a_b,X222: list_P2281458131032066856st_a_b] :
            ( Y2
           != ( cons_P2018802354021969880st_a_b @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_718_list_OdiscI,axiom,
    ! [List: list_P2281458131032066856st_a_b,X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] :
      ( ( List
        = ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) )
     => ( List != nil_Pr523822035851181096st_a_b ) ) ).

% list.discI
thf(fact_719_list_Odistinct_I1_J,axiom,
    ! [X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] :
      ( nil_Pr523822035851181096st_a_b
     != ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_720_sorted__wrt_Ocases,axiom,
    ! [X2: produc8664631368769065441st_a_b] :
      ( ! [P3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o] :
          ( X2
         != ( produc3504008754983458001st_a_b @ P3 @ nil_Pr523822035851181096st_a_b ) )
     => ~ ! [P3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,X: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc3504008754983458001st_a_b @ P3 @ ( cons_P2018802354021969880st_a_b @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_721_successively_Ocases,axiom,
    ! [X2: produc8664631368769065441st_a_b] :
      ( ! [P3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o] :
          ( X2
         != ( produc3504008754983458001st_a_b @ P3 @ nil_Pr523822035851181096st_a_b ) )
     => ( ! [P3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,X: produc2579390649552321826st_a_b] :
            ( X2
           != ( produc3504008754983458001st_a_b @ P3 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) )
       => ~ ! [P3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > $o,X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( X2
             != ( produc3504008754983458001st_a_b @ P3 @ ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_722_sorted_Ocases,axiom,
    ! [X2: produc5746198147368802087st_a_b] :
      ( ! [Cmp2: compar7542523840845723048st_a_b] :
          ( X2
         != ( produc4166231589323681047st_a_b @ Cmp2 @ nil_Pr523822035851181096st_a_b ) )
     => ( ! [Cmp2: compar7542523840845723048st_a_b,X: produc2579390649552321826st_a_b] :
            ( X2
           != ( produc4166231589323681047st_a_b @ Cmp2 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) )
       => ~ ! [Cmp2: compar7542523840845723048st_a_b,Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( X2
             != ( produc4166231589323681047st_a_b @ Cmp2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ) ).

% sorted.cases
thf(fact_723_merge_Ocases,axiom,
    ! [X2: produc2725289747333967015st_a_b] :
      ( ! [Ys2: list_P2281458131032066856st_a_b] :
          ( X2
         != ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
     => ( ! [Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc717113118580727703st_a_b @ Xs2 @ nil_Pr523822035851181096st_a_b ) )
       => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
              ( X2
             != ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ).

% merge.cases
thf(fact_724_input__empty__if__merge__empty,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys )
        = nil_Pr523822035851181096st_a_b )
     => ( ( Xs = nil_Pr523822035851181096st_a_b )
        & ( Ys = nil_Pr523822035851181096st_a_b ) ) ) ).

% input_empty_if_merge_empty
thf(fact_725_merge_Osimps_I1_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( sortin5994461831780029029st_a_b @ Cmp @ nil_Pr523822035851181096st_a_b @ Ys )
      = Ys ) ).

% merge.simps(1)
thf(fact_726_merge_Osimps_I2_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ nil_Pr523822035851181096st_a_b )
      = Xs ) ).

% merge.simps(2)
thf(fact_727_dtree__from__list_Ocases,axiom,
    ! [X2: produc4965795741157486795st_a_b] :
      ( ! [R3: list_a] :
          ( X2
         != ( produc4250525283383199291st_a_b @ R3 @ nil_Pr523822035851181096st_a_b ) )
     => ~ ! [R3: list_a,V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc4250525283383199291st_a_b @ R3 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) ) ) ) ).

% dtree_from_list.cases
thf(fact_728_wf__list__verts_Ocases,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( X2 != nil_Pr523822035851181096st_a_b )
     => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) ) ) ).

% wf_list_verts.cases
thf(fact_729_split__list__first__prop__iff,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys4: list_a,X4: a] :
            ( ? [Zs3: list_a] :
                ( Xs
                = ( append_a @ Ys4 @ ( cons_a @ X4 @ Zs3 ) ) )
            & ( P @ X4 )
            & ! [Y5: a] :
                ( ( member_a @ Y5 @ ( set_a2 @ Ys4 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_730_split__list__first__prop__iff,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ( ? [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys4: list_P2281458131032066856st_a_b,X4: produc2579390649552321826st_a_b] :
            ( ? [Zs3: list_P2281458131032066856st_a_b] :
                ( Xs
                = ( append8668862642614166333st_a_b @ Ys4 @ ( cons_P2018802354021969880st_a_b @ X4 @ Zs3 ) ) )
            & ( P @ X4 )
            & ! [Y5: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ Y5 @ ( set_Pr6156699709417984823st_a_b @ Ys4 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_731_split__list__last__prop__iff,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys4: list_a,X4: a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys4 @ ( cons_a @ X4 @ Zs3 ) ) )
            & ( P @ X4 )
            & ! [Y5: a] :
                ( ( member_a @ Y5 @ ( set_a2 @ Zs3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_732_split__list__last__prop__iff,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ( ? [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys4: list_P2281458131032066856st_a_b,X4: produc2579390649552321826st_a_b,Zs3: list_P2281458131032066856st_a_b] :
            ( ( Xs
              = ( append8668862642614166333st_a_b @ Ys4 @ ( cons_P2018802354021969880st_a_b @ X4 @ Zs3 ) ) )
            & ( P @ X4 )
            & ! [Y5: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ Y5 @ ( set_Pr6156699709417984823st_a_b @ Zs3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_733_in__set__conv__decomp__first,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
      = ( ? [Ys4: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
            & ~ ( member_a @ X2 @ ( set_a2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_734_in__set__conv__decomp__first,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( ? [Ys4: list_P2281458131032066856st_a_b,Zs3: list_P2281458131032066856st_a_b] :
            ( ( Xs
              = ( append8668862642614166333st_a_b @ Ys4 @ ( cons_P2018802354021969880st_a_b @ X2 @ Zs3 ) ) )
            & ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_735_in__set__conv__decomp__last,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
      = ( ? [Ys4: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
            & ~ ( member_a @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_736_in__set__conv__decomp__last,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( ? [Ys4: list_P2281458131032066856st_a_b,Zs3: list_P2281458131032066856st_a_b] :
            ( ( Xs
              = ( append8668862642614166333st_a_b @ Ys4 @ ( cons_P2018802354021969880st_a_b @ X2 @ Zs3 ) ) )
            & ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_737_split__list__first__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
          & ( P @ X3 ) )
     => ~ ! [Ys2: list_a,X: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_738_split__list__first__propE,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
          & ( P @ X3 ) )
     => ~ ! [Ys2: list_P2281458131032066856st_a_b,X: produc2579390649552321826st_a_b] :
            ( ? [Zs2: list_P2281458131032066856st_a_b] :
                ( Xs
                = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa2: produc2579390649552321826st_a_b] :
                    ( ( member8006451236149131979st_a_b @ Xa2 @ ( set_Pr6156699709417984823st_a_b @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_739_split__list__last__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
          & ( P @ X3 ) )
     => ~ ! [Ys2: list_a,X: a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_740_split__list__last__propE,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
          & ( P @ X3 ) )
     => ~ ! [Ys2: list_P2281458131032066856st_a_b,X: produc2579390649552321826st_a_b,Zs2: list_P2281458131032066856st_a_b] :
            ( ( Xs
              = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa2: produc2579390649552321826st_a_b] :
                    ( ( member8006451236149131979st_a_b @ Xa2 @ ( set_Pr6156699709417984823st_a_b @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_741_split__list__first__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
          & ( P @ X3 ) )
     => ? [Ys2: list_a,X: a] :
          ( ? [Zs2: list_a] :
              ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa2: a] :
              ( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_742_split__list__first__prop,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
          & ( P @ X3 ) )
     => ? [Ys2: list_P2281458131032066856st_a_b,X: produc2579390649552321826st_a_b] :
          ( ? [Zs2: list_P2281458131032066856st_a_b] :
              ( Xs
              = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa2: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ Xa2 @ ( set_Pr6156699709417984823st_a_b @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_743_split__list__last__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
          & ( P @ X3 ) )
     => ? [Ys2: list_a,X: a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa2: a] :
              ( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_744_split__list__last__prop,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
          & ( P @ X3 ) )
     => ? [Ys2: list_P2281458131032066856st_a_b,X: produc2579390649552321826st_a_b,Zs2: list_P2281458131032066856st_a_b] :
          ( ( Xs
            = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa2: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ Xa2 @ ( set_Pr6156699709417984823st_a_b @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_745_in__set__conv__decomp,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
      = ( ? [Ys4: list_a,Zs3: list_a] :
            ( Xs
            = ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_746_in__set__conv__decomp,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( ? [Ys4: list_P2281458131032066856st_a_b,Zs3: list_P2281458131032066856st_a_b] :
            ( Xs
            = ( append8668862642614166333st_a_b @ Ys4 @ ( cons_P2018802354021969880st_a_b @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_747_append__Cons__eq__iff,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys5: list_a] :
      ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
            = ( append_a @ Xs3 @ ( cons_a @ X2 @ Ys5 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_748_append__Cons__eq__iff,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Xs3: list_P2281458131032066856st_a_b,Ys5: list_P2281458131032066856st_a_b] :
      ( ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
       => ( ( ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ Ys ) )
            = ( append8668862642614166333st_a_b @ Xs3 @ ( cons_P2018802354021969880st_a_b @ X2 @ Ys5 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_749_split__list__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
          & ( P @ X3 ) )
     => ~ ! [Ys2: list_a,X: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_750_split__list__propE,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
          & ( P @ X3 ) )
     => ~ ! [Ys2: list_P2281458131032066856st_a_b,X: produc2579390649552321826st_a_b] :
            ( ? [Zs2: list_P2281458131032066856st_a_b] :
                ( Xs
                = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_751_split__list__first,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
          & ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_752_split__list__first,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ? [Ys2: list_P2281458131032066856st_a_b,Zs2: list_P2281458131032066856st_a_b] :
          ( ( Xs
            = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Zs2 ) ) )
          & ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_753_split__list__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
          & ( P @ X3 ) )
     => ? [Ys2: list_a,X: a] :
          ( ? [Zs2: list_a] :
              ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_754_split__list__prop,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ? [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
          & ( P @ X3 ) )
     => ? [Ys2: list_P2281458131032066856st_a_b,X: produc2579390649552321826st_a_b] :
          ( ? [Zs2: list_P2281458131032066856st_a_b] :
              ( Xs
              = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_755_split__list__last,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
          & ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_756_split__list__last,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ? [Ys2: list_P2281458131032066856st_a_b,Zs2: list_P2281458131032066856st_a_b] :
          ( ( Xs
            = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Zs2 ) ) )
          & ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_757_split__list,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( Xs
          = ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_758_split__list,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ? [Ys2: list_P2281458131032066856st_a_b,Zs2: list_P2281458131032066856st_a_b] :
          ( Xs
          = ( append8668862642614166333st_a_b @ Ys2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_759_wf__list__verts_Oelims_I2_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( wf_lis5250910623959142362st_a_b @ X2 )
     => ( ( X2 != nil_Pr523822035851181096st_a_b )
       => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( X2
                = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
             => ~ ( ~ ( member_list_a @ V3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
                  & ( wf_lis5250910623959142362st_a_b @ Xs2 ) ) ) ) ) ).

% wf_list_verts.elims(2)
thf(fact_760_wf__list__verts_Oelims_I1_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Y2: $o] :
      ( ( ( wf_lis5250910623959142362st_a_b @ X2 )
        = Y2 )
     => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
         => ~ Y2 )
       => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( X2
                = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
             => ( Y2
                = ( ~ ( ~ ( member_list_a @ V3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
                      & ( wf_lis5250910623959142362st_a_b @ Xs2 ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_761_subrelI,axiom,
    ! [R: set_Pr9033091269622839554st_a_b,S: set_Pr9033091269622839554st_a_b] :
      ( ! [X: list_a,Y3: b] :
          ( ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X @ Y3 ) @ R )
         => ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X @ Y3 ) @ S ) )
     => ( ord_le2879471331418251426st_a_b @ R @ S ) ) ).

% subrelI
thf(fact_762_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_763_List_Oinsert__def,axiom,
    ( insert6036452702741353534st_a_b
    = ( ^ [X4: produc2579390649552321826st_a_b,Xs4: list_P2281458131032066856st_a_b] : ( if_lis3463779252948870126st_a_b @ ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs4 ) ) @ Xs4 @ ( cons_P2018802354021969880st_a_b @ X4 @ Xs4 ) ) ) ) ).

% List.insert_def
thf(fact_764_merge__split__supset_H,axiom,
    ! [R: produc2579390649552321826st_a_b,Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ R @ ( set_Pr6156699709417984823st_a_b @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) ) )
     => ? [As: list_P2281458131032066856st_a_b,Bs: list_P2281458131032066856st_a_b,As2: list_P2281458131032066856st_a_b,Bs2: list_P2281458131032066856st_a_b] :
          ( ( ( append8668862642614166333st_a_b @ As @ ( cons_P2018802354021969880st_a_b @ R @ Bs ) )
            = ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) )
          & ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Bs2 ) @ ( set_Pr6156699709417984823st_a_b @ Bs ) )
          & ( ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ R @ Bs2 ) )
              = Xs )
            | ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ R @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset'
thf(fact_765_merge__split__supset_H,axiom,
    ! [R: a,Cmp: comparator_a,Xs: list_a,Ys: list_a] :
      ( ( member_a @ R @ ( set_a2 @ ( sorting_merge_a @ Cmp @ Xs @ Ys ) ) )
     => ? [As: list_a,Bs: list_a,As2: list_a,Bs2: list_a] :
          ( ( ( append_a @ As @ ( cons_a @ R @ Bs ) )
            = ( sorting_merge_a @ Cmp @ Xs @ Ys ) )
          & ( ord_less_eq_set_a @ ( set_a2 @ Bs2 ) @ ( set_a2 @ Bs ) )
          & ( ( ( append_a @ As2 @ ( cons_a @ R @ Bs2 ) )
              = Xs )
            | ( ( append_a @ As2 @ ( cons_a @ R @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset'
thf(fact_766_merge__split__supset,axiom,
    ! [As3: list_P2281458131032066856st_a_b,R: produc2579390649552321826st_a_b,Bs3: list_P2281458131032066856st_a_b,Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ As3 @ ( cons_P2018802354021969880st_a_b @ R @ Bs3 ) )
        = ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) )
     => ? [Bs2: list_P2281458131032066856st_a_b,As2: list_P2281458131032066856st_a_b] :
          ( ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Bs2 ) @ ( set_Pr6156699709417984823st_a_b @ Bs3 ) )
          & ( ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ R @ Bs2 ) )
              = Xs )
            | ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ R @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset
thf(fact_767_merge__split__supset,axiom,
    ! [As3: list_a,R: a,Bs3: list_a,Cmp: comparator_a,Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ As3 @ ( cons_a @ R @ Bs3 ) )
        = ( sorting_merge_a @ Cmp @ Xs @ Ys ) )
     => ? [Bs2: list_a,As2: list_a] :
          ( ( ord_less_eq_set_a @ ( set_a2 @ Bs2 ) @ ( set_a2 @ Bs3 ) )
          & ( ( ( append_a @ As2 @ ( cons_a @ R @ Bs2 ) )
              = Xs )
            | ( ( append_a @ As2 @ ( cons_a @ R @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset
thf(fact_768_merge__split__supset__fst,axiom,
    ! [As3: list_P2281458131032066856st_a_b,R: list_a,E: b,Bs3: list_P2281458131032066856st_a_b,Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ As3 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R @ E ) @ Bs3 ) )
        = ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) )
     => ? [As2: list_P2281458131032066856st_a_b,Bs2: list_P2281458131032066856st_a_b] :
          ( ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Bs2 ) @ ( set_Pr6156699709417984823st_a_b @ Bs3 ) )
          & ( ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R @ E ) @ Bs2 ) )
              = Xs )
            | ( ( append8668862642614166333st_a_b @ As2 @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ R @ E ) @ Bs2 ) )
              = Ys ) ) ) ) ).

% merge_split_supset_fst
thf(fact_769_merge__eq__appendI,axiom,
    ! [Xs: list_a,Ys: list_a,Cmp: comparator_a] :
      ( ! [X: a,Y3: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( ( member_a @ Y3 @ ( set_a2 @ Ys ) )
           => ( ( compare_a @ Cmp @ X @ Y3 )
             != greater ) ) )
     => ( ( sorting_merge_a @ Cmp @ Xs @ Ys )
        = ( append_a @ Xs @ Ys ) ) ) ).

% merge_eq_appendI
thf(fact_770_merge__eq__appendI,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Cmp: compar7542523840845723048st_a_b] :
      ( ! [X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
         => ( ( member8006451236149131979st_a_b @ Y3 @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
           => ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
             != greater ) ) )
     => ( ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys )
        = ( append8668862642614166333st_a_b @ Xs @ Ys ) ) ) ).

% merge_eq_appendI
thf(fact_771_pred__equals__eq2,axiom,
    ! [R2: set_Pr9033091269622839554st_a_b,S3: set_Pr9033091269622839554st_a_b] :
      ( ( ( ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_772_pred__subset__eq2,axiom,
    ! [R2: set_Pr9033091269622839554st_a_b,S3: set_Pr9033091269622839554st_a_b] :
      ( ( ord_le8305133717419367763_a_b_o
        @ ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ S3 ) )
      = ( ord_le2879471331418251426st_a_b @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_773_pre__digraph_Ocas_Ocases,axiom,
    ! [X2: produc9072104740861669386st_a_b] :
      ( ! [U: b,V3: b] :
          ( X2
         != ( produc2237782585831962490st_a_b @ U @ ( produc4781227316648555538st_a_b @ nil_a @ V3 ) ) )
     => ~ ! [U: b,E3: a,Es: list_a,V3: b] :
            ( X2
           != ( produc2237782585831962490st_a_b @ U @ ( produc4781227316648555538st_a_b @ ( cons_a @ E3 @ Es ) @ V3 ) ) ) ) ).

% pre_digraph.cas.cases
thf(fact_774_list__sel__aux_H_Ocases,axiom,
    ! [X2: produc2553673558697526164st_a_b] :
      ( ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Y3: produc2579390649552321826st_a_b] :
          ( X2
         != ( produc310556805594488006st_a_b @ Sel @ ( produc9021614133752304913st_a_b @ nil_Pr523822035851181096st_a_b @ Y3 ) ) )
     => ~ ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b] :
            ( X2
           != ( produc310556805594488006st_a_b @ Sel @ ( produc9021614133752304913st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Y3 ) ) ) ) ).

% list_sel_aux'.cases
thf(fact_775_wf__list__lverts_Ocases,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( X2 != nil_Pr523822035851181096st_a_b )
     => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) ) ) ).

% wf_list_lverts.cases
thf(fact_776_Succ__def,axiom,
    ( bNF_Gr6406994821186271933st_a_b
    = ( ^ [Kl: set_li7460651515946730888st_a_b,Kl2: list_P2281458131032066856st_a_b] :
          ( collec3957028476971440141st_a_b
          @ ^ [K: produc2579390649552321826st_a_b] : ( member1308482598881763409st_a_b @ ( append8668862642614166333st_a_b @ Kl2 @ ( cons_P2018802354021969880st_a_b @ K @ nil_Pr523822035851181096st_a_b ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_777_ldeep__s_Ocases,axiom,
    ! [X2: produc1825886450479747995st_a_b] :
      ( ! [F3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real] :
          ( X2
         != ( produc1145570397198343373st_a_b @ F3 @ nil_Pr523822035851181096st_a_b ) )
     => ~ ! [F3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc1145570397198343373st_a_b @ F3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ).

% ldeep_s.cases
thf(fact_778_list__sel_Ocases,axiom,
    ! [X2: produc1452261174440070554st_a_b] :
      ( ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Y3: list_P2281458131032066856st_a_b] :
          ( X2
         != ( produc8450139827066779468st_a_b @ Sel @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Y3 ) ) )
     => ~ ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc8450139827066779468st_a_b @ Sel @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Y3 ) ) ) ) ).

% list_sel.cases
thf(fact_779_list__sel_H_Ocases,axiom,
    ! [X2: produc1452261174440070554st_a_b] :
      ( ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: list_P2281458131032066856st_a_b] :
          ( X2
         != ( produc8450139827066779468st_a_b @ Sel @ ( produc717113118580727703st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) )
     => ~ ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc8450139827066779468st_a_b @ Sel @ ( produc717113118580727703st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ).

% list_sel'.cases
thf(fact_780_list__sel__aux_Ocases,axiom,
    ! [X2: produc4785139578647204768st_a_b] :
      ( ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b] :
          ( X2
         != ( produc6612496626342752978st_a_b @ Sel @ ( produc6648078606966411293st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) )
     => ~ ! [Sel: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc6612496626342752978st_a_b @ Sel @ ( produc6648078606966411293st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ).

% list_sel_aux.cases
thf(fact_781_SuccD,axiom,
    ! [K2: a,Kl3: set_list_a,Kl4: list_a] :
      ( ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl3 @ Kl4 ) )
     => ( member_list_a @ ( append_a @ Kl4 @ ( cons_a @ K2 @ nil_a ) ) @ Kl3 ) ) ).

% SuccD
thf(fact_782_SuccD,axiom,
    ! [K2: produc2579390649552321826st_a_b,Kl3: set_li7460651515946730888st_a_b,Kl4: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ K2 @ ( bNF_Gr6406994821186271933st_a_b @ Kl3 @ Kl4 ) )
     => ( member1308482598881763409st_a_b @ ( append8668862642614166333st_a_b @ Kl4 @ ( cons_P2018802354021969880st_a_b @ K2 @ nil_Pr523822035851181096st_a_b ) ) @ Kl3 ) ) ).

% SuccD
thf(fact_783_SuccI,axiom,
    ! [Kl4: list_a,K2: a,Kl3: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl4 @ ( cons_a @ K2 @ nil_a ) ) @ Kl3 )
     => ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl3 @ Kl4 ) ) ) ).

% SuccI
thf(fact_784_SuccI,axiom,
    ! [Kl4: list_P2281458131032066856st_a_b,K2: produc2579390649552321826st_a_b,Kl3: set_li7460651515946730888st_a_b] :
      ( ( member1308482598881763409st_a_b @ ( append8668862642614166333st_a_b @ Kl4 @ ( cons_P2018802354021969880st_a_b @ K2 @ nil_Pr523822035851181096st_a_b ) ) @ Kl3 )
     => ( member8006451236149131979st_a_b @ K2 @ ( bNF_Gr6406994821186271933st_a_b @ Kl3 @ Kl4 ) ) ) ).

% SuccI
thf(fact_785_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc2725289747333967015st_a_b] :
      ( ! [Ys2: list_P2281458131032066856st_a_b] :
          ( X2
         != ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
     => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Ys2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_786_longest__common__prefix_Ocases,axiom,
    ! [X2: produc2725289747333967015st_a_b] :
      ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
          ( X2
         != ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Uv ) )
       => ~ ! [Uu: list_P2281458131032066856st_a_b] :
              ( X2
             != ( produc717113118580727703st_a_b @ Uu @ nil_Pr523822035851181096st_a_b ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_787_empty__Shift,axiom,
    ! [Kl3: set_list_a,K2: a] :
      ( ( member_list_a @ nil_a @ Kl3 )
     => ( ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl3 @ nil_a ) )
       => ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl3 @ K2 ) ) ) ) ).

% empty_Shift
thf(fact_788_Succ__Shift,axiom,
    ! [Kl3: set_li7460651515946730888st_a_b,K2: produc2579390649552321826st_a_b,Kl4: list_P2281458131032066856st_a_b] :
      ( ( bNF_Gr6406994821186271933st_a_b @ ( bNF_Gr3010537980345091905st_a_b @ Kl3 @ K2 ) @ Kl4 )
      = ( bNF_Gr6406994821186271933st_a_b @ Kl3 @ ( cons_P2018802354021969880st_a_b @ K2 @ Kl4 ) ) ) ).

% Succ_Shift
thf(fact_789_ldeep__n_Ocases,axiom,
    ! [X2: produc1845366793593209155st_a_b] :
      ( ! [F3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Cf: produc2579390649552321826st_a_b > real] :
          ( X2
         != ( produc29907560316097525st_a_b @ F3 @ ( produc5167149376669553984st_a_b @ Cf @ nil_Pr523822035851181096st_a_b ) ) )
     => ~ ! [F3: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b > real,Cf: produc2579390649552321826st_a_b > real,R3: produc2579390649552321826st_a_b,Rs: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc29907560316097525st_a_b @ F3 @ ( produc5167149376669553984st_a_b @ Cf @ ( cons_P2018802354021969880st_a_b @ R3 @ Rs ) ) ) ) ) ).

% ldeep_n.cases
thf(fact_790_ShiftD,axiom,
    ! [Kl4: list_P2281458131032066856st_a_b,Kl3: set_li7460651515946730888st_a_b,K2: produc2579390649552321826st_a_b] :
      ( ( member1308482598881763409st_a_b @ Kl4 @ ( bNF_Gr3010537980345091905st_a_b @ Kl3 @ K2 ) )
     => ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ K2 @ Kl4 ) @ Kl3 ) ) ).

% ShiftD
thf(fact_791_Shift__def,axiom,
    ( bNF_Gr3010537980345091905st_a_b
    = ( ^ [Kl: set_li7460651515946730888st_a_b,K: produc2579390649552321826st_a_b] :
          ( collec3283141540447581331st_a_b
          @ ^ [Kl2: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ K @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_792_create__ldeep__rev_Ocases,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( X2 != nil_Pr523822035851181096st_a_b )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( X2
           != ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
       => ~ ! [X: produc2579390649552321826st_a_b,V3: produc2579390649552321826st_a_b,Va: list_P2281458131032066856st_a_b] :
              ( X2
             != ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ V3 @ Va ) ) ) ) ) ).

% create_ldeep_rev.cases
thf(fact_793_ldeep__T_H_Ocases,axiom,
    ! [X2: produc1958926909753010168st_a_b] :
      ( ! [F3: produc2579390649552321826st_a_b > real,Cf: produc2579390649552321826st_a_b > real] :
          ( X2
         != ( produc2842869207188926184st_a_b @ F3 @ ( produc5167149376669553984st_a_b @ Cf @ nil_Pr523822035851181096st_a_b ) ) )
     => ~ ! [F3: produc2579390649552321826st_a_b > real,Cf: produc2579390649552321826st_a_b > real,R3: produc2579390649552321826st_a_b,Rs: list_P2281458131032066856st_a_b] :
            ( X2
           != ( produc2842869207188926184st_a_b @ F3 @ ( produc5167149376669553984st_a_b @ Cf @ ( cons_P2018802354021969880st_a_b @ R3 @ Rs ) ) ) ) ) ).

% ldeep_T'.cases
thf(fact_794_prefixes__snoc,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( prefix1100606186622140975st_a_b @ ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) )
      = ( append2962543925774821315st_a_b @ ( prefix1100606186622140975st_a_b @ Xs ) @ ( cons_l7524743740370447070st_a_b @ ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) @ nil_li5528507983314571054st_a_b ) ) ) ).

% prefixes_snoc
thf(fact_795_list__exhaust__NSC,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( Xs
           != ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
       => ~ ! [X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
              ( Xs
             != ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ).

% list_exhaust_NSC
thf(fact_796_list__exhaust2,axiom,
    ! [Y2: list_P2281458131032066856st_a_b,Ya: list_P2281458131032066856st_a_b] :
      ( ( ( Y2 = nil_Pr523822035851181096st_a_b )
       => ( Ya != nil_Pr523822035851181096st_a_b ) )
     => ( ( ( Y2 = nil_Pr523822035851181096st_a_b )
         => ! [X212: produc2579390649552321826st_a_b,X222: list_P2281458131032066856st_a_b] :
              ( Ya
             != ( cons_P2018802354021969880st_a_b @ X212 @ X222 ) ) )
       => ( ( ? [X212: produc2579390649552321826st_a_b,X222: list_P2281458131032066856st_a_b] :
                ( Y2
                = ( cons_P2018802354021969880st_a_b @ X212 @ X222 ) )
           => ( Ya != nil_Pr523822035851181096st_a_b ) )
         => ~ ( ? [X212: produc2579390649552321826st_a_b,X222: list_P2281458131032066856st_a_b] :
                  ( Y2
                  = ( cons_P2018802354021969880st_a_b @ X212 @ X222 ) )
             => ! [X21a: produc2579390649552321826st_a_b,X22a: list_P2281458131032066856st_a_b] :
                  ( Ya
                 != ( cons_P2018802354021969880st_a_b @ X21a @ X22a ) ) ) ) ) ) ).

% list_exhaust2
thf(fact_797_vwalk__to__vpath_Ocases,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( X2 != nil_Pr523822035851181096st_a_b )
     => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
            ( X2
           != ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ).

% vwalk_to_vpath.cases
thf(fact_798_prefixes__eq__snoc,axiom,
    ! [Ys: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b,X2: list_P2281458131032066856st_a_b] :
      ( ( ( prefix1100606186622140975st_a_b @ Ys )
        = ( append2962543925774821315st_a_b @ Xs @ ( cons_l7524743740370447070st_a_b @ X2 @ nil_li5528507983314571054st_a_b ) ) )
      = ( ( ( ( Ys = nil_Pr523822035851181096st_a_b )
            & ( Xs = nil_li5528507983314571054st_a_b ) )
          | ? [Z6: produc2579390649552321826st_a_b,Zs3: list_P2281458131032066856st_a_b] :
              ( ( Ys
                = ( append8668862642614166333st_a_b @ Zs3 @ ( cons_P2018802354021969880st_a_b @ Z6 @ nil_Pr523822035851181096st_a_b ) ) )
              & ( Xs
                = ( prefix1100606186622140975st_a_b @ Zs3 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_799_suffixes__eq__snoc,axiom,
    ! [Ys: list_P2281458131032066856st_a_b,Xs: list_l3733004638376709294st_a_b,X2: list_P2281458131032066856st_a_b] :
      ( ( ( suffix3090462496574799984st_a_b @ Ys )
        = ( append2962543925774821315st_a_b @ Xs @ ( cons_l7524743740370447070st_a_b @ X2 @ nil_li5528507983314571054st_a_b ) ) )
      = ( ( ( ( Ys = nil_Pr523822035851181096st_a_b )
            & ( Xs = nil_li5528507983314571054st_a_b ) )
          | ? [Z6: produc2579390649552321826st_a_b,Zs3: list_P2281458131032066856st_a_b] :
              ( ( Ys
                = ( cons_P2018802354021969880st_a_b @ Z6 @ Zs3 ) )
              & ( Xs
                = ( suffix3090462496574799984st_a_b @ Zs3 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_800_suffixes_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( suffix3090462496574799984st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( append2962543925774821315st_a_b @ ( suffix3090462496574799984st_a_b @ Xs ) @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ nil_li5528507983314571054st_a_b ) ) ) ).

% suffixes.simps(2)
thf(fact_801_wf__list__verts_Opelims_I2_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( wf_lis5250910623959142362st_a_b @ X2 )
     => ( ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ X2 )
       => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
           => ~ ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ nil_Pr523822035851181096st_a_b ) )
         => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
               => ( ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
                 => ~ ( ~ ( member_list_a @ V3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
                      & ( wf_lis5250910623959142362st_a_b @ Xs2 ) ) ) ) ) ) ) ).

% wf_list_verts.pelims(2)
thf(fact_802_wf__list__verts_Opelims_I1_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Y2: $o] :
      ( ( ( wf_lis5250910623959142362st_a_b @ X2 )
        = Y2 )
     => ( ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ X2 )
       => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
           => ( Y2
             => ~ ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ nil_Pr523822035851181096st_a_b ) ) )
         => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
               => ( ( Y2
                    = ( ~ ( member_list_a @ V3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
                      & ( wf_lis5250910623959142362st_a_b @ Xs2 ) ) )
                 => ~ ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) ) ) ) ) ) ) ).

% wf_list_verts.pelims(1)
thf(fact_803_wf__list__verts_Opelims_I3_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ~ ( wf_lis5250910623959142362st_a_b @ X2 )
     => ( ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ X2 )
       => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( X2
                = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
             => ( ( accp_l493756998182423857st_a_b @ wf_lis494094236248123989st_a_b @ ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
               => ( ~ ( member_list_a @ V3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
                  & ( wf_lis5250910623959142362st_a_b @ Xs2 ) ) ) ) ) ) ).

% wf_list_verts.pelims(3)
thf(fact_804_merge_Opelims,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: list_P2281458131032066856st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( ( sortin5994461831780029029st_a_b @ Cmp @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ X2 @ Xa ) )
       => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
           => ( ( Y2 = Xa )
             => ~ ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Xa ) ) ) )
         => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
             => ( ( Y2 = X2 )
               => ~ ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ) )
           => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                  ( ( X2
                    = ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) )
                 => ! [Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                      ( ( Xa
                        = ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
                     => ( ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
                              = greater )
                           => ( Y2
                              = ( cons_P2018802354021969880st_a_b @ Y3 @ ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) )
                          & ( ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
                             != greater )
                           => ( Y2
                              = ( cons_P2018802354021969880st_a_b @ X @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) )
                       => ~ ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ).

% merge.pelims
thf(fact_805_merge_Opsimps_I3_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) )
     => ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ X2 @ Y2 )
            = greater )
         => ( ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
            = ( cons_P2018802354021969880st_a_b @ Y2 @ ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) ) ) )
        & ( ( ( compar8165832444956594340st_a_b @ Cmp @ X2 @ Y2 )
           != greater )
         => ( ( sortin5994461831780029029st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
            = ( cons_P2018802354021969880st_a_b @ X2 @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) ) ) ) ) ) ).

% merge.psimps(3)
thf(fact_806_merge_Opinduct,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,A0: list_P2281458131032066856st_a_b,A1: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > list_P2281458131032066856st_a_b > $o] :
      ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ A0 @ A1 ) )
     => ( ! [Ys2: list_P2281458131032066856st_a_b] :
            ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
           => ( P @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
       => ( ! [Xs2: list_P2281458131032066856st_a_b] :
              ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ Xs2 @ nil_Pr523822035851181096st_a_b ) )
             => ( P @ Xs2 @ nil_Pr523822035851181096st_a_b ) )
         => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) )
               => ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
                      = greater )
                   => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) )
                 => ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y3 )
                       != greater )
                     => ( P @ Xs2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) )
                   => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 ) ) ) ) ) ).

% merge.pinduct
thf(fact_807_merge_Opsimps_I1_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Ys ) )
     => ( ( sortin5994461831780029029st_a_b @ Cmp @ nil_Pr523822035851181096st_a_b @ Ys )
        = Ys ) ) ).

% merge.psimps(1)
thf(fact_808_merge_Opsimps_I2_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( accp_P4272997315293813680st_a_b @ ( sortin955834713828781028st_a_b @ Cmp ) @ ( produc717113118580727703st_a_b @ Xs @ nil_Pr523822035851181096st_a_b ) )
     => ( ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ nil_Pr523822035851181096st_a_b )
        = Xs ) ) ).

% merge.psimps(2)
thf(fact_809_splice_Opinduct,axiom,
    ! [A0: list_P2281458131032066856st_a_b,A1: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > list_P2281458131032066856st_a_b > $o] :
      ( ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ A0 @ A1 ) )
     => ( ! [Ys2: list_P2281458131032066856st_a_b] :
            ( ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
           => ( P @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
       => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Ys2: list_P2281458131032066856st_a_b] :
              ( ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) )
             => ( ( P @ Ys2 @ Xs2 )
               => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) ) )
         => ( P @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_810_shuffles_Opinduct,axiom,
    ! [A0: list_P2281458131032066856st_a_b,A1: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > list_P2281458131032066856st_a_b > $o] :
      ( ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ A0 @ A1 ) )
     => ( ! [Ys2: list_P2281458131032066856st_a_b] :
            ( ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
           => ( P @ nil_Pr523822035851181096st_a_b @ Ys2 ) )
       => ( ! [Xs2: list_P2281458131032066856st_a_b] :
              ( ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ Xs2 @ nil_Pr523822035851181096st_a_b ) )
             => ( P @ Xs2 @ nil_Pr523822035851181096st_a_b ) )
         => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                ( ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) )
               => ( ( P @ Xs2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
                 => ( ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 )
                   => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) )
           => ( P @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_811_suffixes__snoc,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( suffix3090462496574799984st_a_b @ ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) )
      = ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b
        @ ( map_li1268991901115354180st_a_b
          @ ^ [Ys4: list_P2281458131032066856st_a_b] : ( append8668862642614166333st_a_b @ Ys4 @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) )
          @ ( suffix3090462496574799984st_a_b @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_812_sorted__induct,axiom,
    ! [Cmp: comparator_a,Xs: list_a,P: list_a > $o] :
      ( ( sorting_sorted_a @ Cmp @ Xs )
     => ( ( P @ nil_a )
       => ( ! [X: a,Xs2: list_a] :
              ( ( sorting_sorted_a @ Cmp @ Xs2 )
             => ( ( P @ Xs2 )
               => ( ! [Y6: a] :
                      ( ( member_a @ Y6 @ ( set_a2 @ Xs2 ) )
                     => ( ( compare_a @ Cmp @ X @ Y6 )
                       != greater ) )
                 => ( P @ ( cons_a @ X @ Xs2 ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% sorted_induct
thf(fact_813_sorted__induct,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,P: list_P2281458131032066856st_a_b > $o] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs )
     => ( ( P @ nil_Pr523822035851181096st_a_b )
       => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs2 )
             => ( ( P @ Xs2 )
               => ( ! [Y6: produc2579390649552321826st_a_b] :
                      ( ( member8006451236149131979st_a_b @ Y6 @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) )
                     => ( ( compar8165832444956594340st_a_b @ Cmp @ X @ Y6 )
                       != greater ) )
                 => ( P @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% sorted_induct
thf(fact_814_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,R: set_Pr4988534092439063687st_a_b] :
      ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) @ ( append8668862642614166333st_a_b @ Ys @ ( cons_P2018802354021969880st_a_b @ Y2 @ nil_Pr523822035851181096st_a_b ) ) ) @ ( listre5635272895738803841st_a_b @ R ) )
      = ( ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) )
          & ( X2 = Y2 ) )
        | ( ( Xs = Ys )
          & ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y2 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_815_vwalk__arcs_Oelims,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Y2: list_P1351356133043840941st_a_b] :
      ( ( ( vertex2793823137597548186st_a_b @ X2 )
        = Y2 )
     => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
         => ( Y2 != nil_Pr6552475826380092333st_a_b ) )
       => ( ( ? [X: produc2579390649552321826st_a_b] :
                ( X2
                = ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
           => ( Y2 != nil_Pr6552475826380092333st_a_b ) )
         => ~ ! [X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Xs2 ) ) )
               => ( Y2
                 != ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ X @ Y3 ) @ ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ).

% vwalk_arcs.elims
thf(fact_816_list_Oset__map,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,V: list_P2281458131032066856st_a_b] :
      ( ( set_list_a2 @ ( map_Pr7332729320935904934list_a @ F @ V ) )
      = ( image_4065140400681997415list_a @ F @ ( set_Pr6156699709417984823st_a_b @ V ) ) ) ).

% list.set_map
thf(fact_817_list_Oset__map,axiom,
    ! [F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,V: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( map_Pr3595035732211294788st_a_b @ F @ V ) )
      = ( image_7587371051487426309st_a_b @ F @ ( set_Pr6156699709417984823st_a_b @ V ) ) ) ).

% list.set_map
thf(fact_818_list_Oset__map,axiom,
    ! [F: a > produc2579390649552321826st_a_b,V: list_a] :
      ( ( set_Pr6156699709417984823st_a_b @ ( map_a_7801393066950150376st_a_b @ F @ V ) )
      = ( image_1109531753604471593st_a_b @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_819_list_Oset__map,axiom,
    ! [F: produc2579390649552321826st_a_b > a,V: list_P2281458131032066856st_a_b] :
      ( ( set_a2 @ ( map_Pr1723537204529804064_a_b_a @ F @ V ) )
      = ( image_4255047928038901089_a_b_a @ F @ ( set_Pr6156699709417984823st_a_b @ V ) ) ) ).

% list.set_map
thf(fact_820_list_Oset__map,axiom,
    ! [F: a > a,V: list_a] :
      ( ( set_a2 @ ( map_a_a @ F @ V ) )
      = ( image_a_a @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_821_Cons__listrel1__Cons,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b,R: set_Pr4988534092439063687st_a_b] :
      ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) @ ( listre5635272895738803841st_a_b @ R ) )
      = ( ( ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y2 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X2 = Y2 )
          & ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_822_map__eq__Cons__conv,axiom,
    ! [F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( map_Pr3595035732211294788st_a_b @ F @ Xs )
        = ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
      = ( ? [Z6: produc2579390649552321826st_a_b,Zs3: list_P2281458131032066856st_a_b] :
            ( ( Xs
              = ( cons_P2018802354021969880st_a_b @ Z6 @ Zs3 ) )
            & ( ( F @ Z6 )
              = Y2 )
            & ( ( map_Pr3595035732211294788st_a_b @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_823_Cons__eq__map__conv,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
        = ( map_Pr3595035732211294788st_a_b @ F @ Ys ) )
      = ( ? [Z6: produc2579390649552321826st_a_b,Zs3: list_P2281458131032066856st_a_b] :
            ( ( Ys
              = ( cons_P2018802354021969880st_a_b @ Z6 @ Zs3 ) )
            & ( X2
              = ( F @ Z6 ) )
            & ( Xs
              = ( map_Pr3595035732211294788st_a_b @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_824_map__eq__Cons__D,axiom,
    ! [F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( map_Pr3595035732211294788st_a_b @ F @ Xs )
        = ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
     => ? [Z5: produc2579390649552321826st_a_b,Zs2: list_P2281458131032066856st_a_b] :
          ( ( Xs
            = ( cons_P2018802354021969880st_a_b @ Z5 @ Zs2 ) )
          & ( ( F @ Z5 )
            = Y2 )
          & ( ( map_Pr3595035732211294788st_a_b @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_825_Cons__eq__map__D,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
        = ( map_Pr3595035732211294788st_a_b @ F @ Ys ) )
     => ? [Z5: produc2579390649552321826st_a_b,Zs2: list_P2281458131032066856st_a_b] :
          ( ( Ys
            = ( cons_P2018802354021969880st_a_b @ Z5 @ Zs2 ) )
          & ( X2
            = ( F @ Z5 ) )
          & ( Xs
            = ( map_Pr3595035732211294788st_a_b @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_826_list_Osimps_I9_J,axiom,
    ! [F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] :
      ( ( map_Pr3595035732211294788st_a_b @ F @ ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) )
      = ( cons_P2018802354021969880st_a_b @ ( F @ X21 ) @ ( map_Pr3595035732211294788st_a_b @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_827_list_Omap__ident__strong,axiom,
    ! [T2: list_P2281458131032066856st_a_b,F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b] :
      ( ! [Z5: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ Z5 @ ( set_Pr6156699709417984823st_a_b @ T2 ) )
         => ( ( F @ Z5 )
            = Z5 ) )
     => ( ( map_Pr3595035732211294788st_a_b @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_828_list_Omap__ident__strong,axiom,
    ! [T2: list_a,F: a > a] :
      ( ! [Z5: a] :
          ( ( member_a @ Z5 @ ( set_a2 @ T2 ) )
         => ( ( F @ Z5 )
            = Z5 ) )
     => ( ( map_a_a @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_829_map__idI,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_Pr3595035732211294788st_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_830_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_831_sorted__Cons__imp__sorted,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
     => ( sortin3200817720516614760st_a_b @ Cmp @ Xs ) ) ).

% sorted_Cons_imp_sorted
thf(fact_832_sorted__merge,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs )
     => ( ( sortin3200817720516614760st_a_b @ Cmp @ Ys )
       => ( sortin3200817720516614760st_a_b @ Cmp @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) ) ) ) ).

% sorted_merge
thf(fact_833_image__set,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,Xs: list_P2281458131032066856st_a_b] :
      ( ( image_4065140400681997415list_a @ F @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( set_list_a2 @ ( map_Pr7332729320935904934list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_834_image__set,axiom,
    ! [F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( image_7587371051487426309st_a_b @ F @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( set_Pr6156699709417984823st_a_b @ ( map_Pr3595035732211294788st_a_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_835_image__set,axiom,
    ! [F: produc2579390649552321826st_a_b > a,Xs: list_P2281458131032066856st_a_b] :
      ( ( image_4255047928038901089_a_b_a @ F @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( set_a2 @ ( map_Pr1723537204529804064_a_b_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_836_image__set,axiom,
    ! [F: a > produc2579390649552321826st_a_b,Xs: list_a] :
      ( ( image_1109531753604471593st_a_b @ F @ ( set_a2 @ Xs ) )
      = ( set_Pr6156699709417984823st_a_b @ ( map_a_7801393066950150376st_a_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_837_image__set,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( image_a_a @ F @ ( set_a2 @ Xs ) )
      = ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_838_Sorting__Algorithms_Osorted__single,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: produc2579390649552321826st_a_b] : ( sortin3200817720516614760st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ).

% Sorting_Algorithms.sorted_single
thf(fact_839_listrel1I2,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,R: set_Pr4988534092439063687st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) )
     => ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ X2 @ Ys ) ) @ ( listre5635272895738803841st_a_b @ R ) ) ) ).

% listrel1I2
thf(fact_840_vwalk__arcs_Osimps_I3_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ Y2 @ Xs ) ) )
      = ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y2 ) @ ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 @ Xs ) ) ) ) ).

% vwalk_arcs.simps(3)
thf(fact_841_in__set__vwalk__arcsE,axiom,
    ! [U2: produc2579390649552321826st_a_b,V: produc2579390649552321826st_a_b,P2: list_P2281458131032066856st_a_b] :
      ( ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ U2 @ V ) @ ( set_Pr5950792022122277180st_a_b @ ( vertex2793823137597548186st_a_b @ P2 ) ) )
     => ~ ( ( member8006451236149131979st_a_b @ U2 @ ( set_Pr6156699709417984823st_a_b @ P2 ) )
         => ~ ( member8006451236149131979st_a_b @ V @ ( set_Pr6156699709417984823st_a_b @ P2 ) ) ) ) ).

% in_set_vwalk_arcsE
thf(fact_842_in__set__vwalk__arcsE,axiom,
    ! [U2: a,V: a,P2: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ V ) @ ( set_Product_prod_a_a2 @ ( vertex_vwalk_arcs_a @ P2 ) ) )
     => ~ ( ( member_a @ U2 @ ( set_a2 @ P2 ) )
         => ~ ( member_a @ V @ ( set_a2 @ P2 ) ) ) ) ).

% in_set_vwalk_arcsE
thf(fact_843_vwalk__arcs_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b] :
      ( ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) )
      = nil_Pr6552475826380092333st_a_b ) ).

% vwalk_arcs.simps(2)
thf(fact_844_set__vwalk__arcs__cons,axiom,
    ! [P2: list_P2281458131032066856st_a_b,U2: produc2579390649552321826st_a_b] : ( ord_le1374440848537019431st_a_b @ ( set_Pr5950792022122277180st_a_b @ ( vertex2793823137597548186st_a_b @ P2 ) ) @ ( set_Pr5950792022122277180st_a_b @ ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ U2 @ P2 ) ) ) ) ).

% set_vwalk_arcs_cons
thf(fact_845_listrel1I1,axiom,
    ! [X2: produc2579390649552321826st_a_b,Y2: produc2579390649552321826st_a_b,R: set_Pr4988534092439063687st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y2 ) @ R )
     => ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Xs ) ) @ ( listre5635272895738803841st_a_b @ R ) ) ) ).

% listrel1I1
thf(fact_846_Cons__listrel1E1,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,R: set_Pr4988534092439063687st_a_b] :
      ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) )
     => ( ! [Y3: produc2579390649552321826st_a_b] :
            ( ( Ys
              = ( cons_P2018802354021969880st_a_b @ Y3 @ Xs ) )
           => ~ ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_P2281458131032066856st_a_b] :
              ( ( Ys
                = ( cons_P2018802354021969880st_a_b @ X2 @ Zs2 ) )
             => ~ ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ Zs2 ) @ ( listre5635272895738803841st_a_b @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_847_Cons__listrel1E2,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b,R: set_Pr4988534092439063687st_a_b] :
      ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) @ ( listre5635272895738803841st_a_b @ R ) )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( ( Xs
              = ( cons_P2018802354021969880st_a_b @ X @ Ys ) )
           => ~ ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X @ Y2 ) @ R ) )
       => ~ ! [Zs2: list_P2281458131032066856st_a_b] :
              ( ( Xs
                = ( cons_P2018802354021969880st_a_b @ Y2 @ Zs2 ) )
             => ~ ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Zs2 @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_848_sorted_Oelims_I3_J,axiom,
    ! [X2: compar7542523840845723048st_a_b,Xa: list_P2281458131032066856st_a_b] :
      ( ~ ( sortin3200817720516614760st_a_b @ X2 @ Xa )
     => ~ ! [Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
            ( ( Xa
              = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) )
           => ( ( ( compar8165832444956594340st_a_b @ X2 @ Y3 @ X )
               != greater )
              & ( sortin3200817720516614760st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ).

% sorted.elims(3)
thf(fact_849_Sorting__Algorithms_Osorted__rec,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Y2: produc2579390649552321826st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ Y2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) )
      = ( ( ( compar8165832444956594340st_a_b @ Cmp @ Y2 @ X2 )
         != greater )
        & ( sortin3200817720516614760st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) ) ) ).

% Sorting_Algorithms.sorted_rec
thf(fact_850_sorted__ConsI,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs )
     => ( ! [Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
            ( ( Xs
              = ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
           => ( ( compar8165832444956594340st_a_b @ Cmp @ X2 @ Y3 )
             != greater ) )
       => ( sortin3200817720516614760st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) ) ) ).

% sorted_ConsI
thf(fact_851_prefixes_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( prefix1100606186622140975st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ ( map_li1268991901115354180st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ ( prefix1100606186622140975st_a_b @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_852_listrel1E,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,R: set_Pr4988534092439063687st_a_b] :
      ( ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) )
     => ~ ! [X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b] :
            ( ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X @ Y3 ) @ R )
           => ! [Us: list_P2281458131032066856st_a_b,Vs: list_P2281458131032066856st_a_b] :
                ( ( Xs
                  = ( append8668862642614166333st_a_b @ Us @ ( cons_P2018802354021969880st_a_b @ X @ Vs ) ) )
               => ( Ys
                 != ( append8668862642614166333st_a_b @ Us @ ( cons_P2018802354021969880st_a_b @ Y3 @ Vs ) ) ) ) ) ) ).

% listrel1E
thf(fact_853_listrel1I,axiom,
    ! [X2: produc2579390649552321826st_a_b,Y2: produc2579390649552321826st_a_b,R: set_Pr4988534092439063687st_a_b,Xs: list_P2281458131032066856st_a_b,Us2: list_P2281458131032066856st_a_b,Vs2: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( member7191766420993298128st_a_b @ ( produc7813282991251521687st_a_b @ X2 @ Y2 ) @ R )
     => ( ( Xs
          = ( append8668862642614166333st_a_b @ Us2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Vs2 ) ) )
       => ( ( Ys
            = ( append8668862642614166333st_a_b @ Us2 @ ( cons_P2018802354021969880st_a_b @ Y2 @ Vs2 ) ) )
         => ( member6028662021613063888st_a_b @ ( produc717113118580727703st_a_b @ Xs @ Ys ) @ ( listre5635272895738803841st_a_b @ R ) ) ) ) ) ).

% listrel1I
thf(fact_854_sublists_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( sublis6722123948842796986st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( append2962543925774821315st_a_b @ ( sublis6722123948842796986st_a_b @ Xs ) @ ( map_li1268991901115354180st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ ( prefix1100606186622140975st_a_b @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_855_sorted_Oelims_I2_J,axiom,
    ! [X2: compar7542523840845723048st_a_b,Xa: list_P2281458131032066856st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ X2 @ Xa )
     => ( ( Xa != nil_Pr523822035851181096st_a_b )
       => ( ! [X: produc2579390649552321826st_a_b] :
              ( Xa
             != ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
         => ~ ! [Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( Xa
                  = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) )
               => ~ ( ( ( compar8165832444956594340st_a_b @ X2 @ Y3 @ X )
                     != greater )
                    & ( sortin3200817720516614760st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ) ) ).

% sorted.elims(2)
thf(fact_856_sorted_Oelims_I1_J,axiom,
    ! [X2: compar7542523840845723048st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: $o] :
      ( ( ( sortin3200817720516614760st_a_b @ X2 @ Xa )
        = Y2 )
     => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
         => ~ Y2 )
       => ( ( ? [X: produc2579390649552321826st_a_b] :
                ( Xa
                = ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
           => ~ Y2 )
         => ~ ! [Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( Xa
                  = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) )
               => ( Y2
                  = ( ~ ( ( ( compar8165832444956594340st_a_b @ X2 @ Y3 @ X )
                         != greater )
                        & ( sortin3200817720516614760st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ) ) ) ) ).

% sorted.elims(1)
thf(fact_857_sorted__Cons__imp__not__less,axiom,
    ! [Cmp: comparator_a,Y2: a,Xs: list_a,X2: a] :
      ( ( sorting_sorted_a @ Cmp @ ( cons_a @ Y2 @ Xs ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( ( compare_a @ Cmp @ Y2 @ X2 )
         != greater ) ) ) ).

% sorted_Cons_imp_not_less
thf(fact_858_sorted__Cons__imp__not__less,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ ( cons_P2018802354021969880st_a_b @ Y2 @ Xs ) )
     => ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
       => ( ( compar8165832444956594340st_a_b @ Cmp @ Y2 @ X2 )
         != greater ) ) ) ).

% sorted_Cons_imp_not_less
thf(fact_859_sorted__append__iff,axiom,
    ! [Cmp: comparator_a,Xs: list_a,Ys: list_a] :
      ( ( sorting_sorted_a @ Cmp @ ( append_a @ Xs @ Ys ) )
      = ( ( sorting_sorted_a @ Cmp @ Xs )
        & ( sorting_sorted_a @ Cmp @ Ys )
        & ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ! [Y5: a] :
                ( ( member_a @ Y5 @ ( set_a2 @ Ys ) )
               => ( ( compare_a @ Cmp @ X4 @ Y5 )
                 != greater ) ) ) ) ) ).

% sorted_append_iff
thf(fact_860_sorted__append__iff,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ ( append8668862642614166333st_a_b @ Xs @ Ys ) )
      = ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs )
        & ( sortin3200817720516614760st_a_b @ Cmp @ Ys )
        & ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ! [Y5: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ Y5 @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
               => ( ( compar8165832444956594340st_a_b @ Cmp @ X4 @ Y5 )
                 != greater ) ) ) ) ) ).

% sorted_append_iff
thf(fact_861_vwalk__arcs_Opelims,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Y2: list_P1351356133043840941st_a_b] :
      ( ( ( vertex2793823137597548186st_a_b @ X2 )
        = Y2 )
     => ( ( accp_l493756998182423857st_a_b @ vertex806124994473458159st_a_b @ X2 )
       => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
           => ( ( Y2 = nil_Pr6552475826380092333st_a_b )
             => ~ ( accp_l493756998182423857st_a_b @ vertex806124994473458159st_a_b @ nil_Pr523822035851181096st_a_b ) ) )
         => ( ! [X: produc2579390649552321826st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
               => ( ( Y2 = nil_Pr6552475826380092333st_a_b )
                 => ~ ( accp_l493756998182423857st_a_b @ vertex806124994473458159st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) )
           => ~ ! [X: produc2579390649552321826st_a_b,Y3: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                  ( ( X2
                    = ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Xs2 ) ) )
                 => ( ( Y2
                      = ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ X @ Y3 ) @ ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ Y3 @ Xs2 ) ) ) )
                   => ~ ( accp_l493756998182423857st_a_b @ vertex806124994473458159st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ ( cons_P2018802354021969880st_a_b @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).

% vwalk_arcs.pelims
thf(fact_862_sorted_Opelims_I2_J,axiom,
    ! [X2: compar7542523840845723048st_a_b,Xa: list_P2281458131032066856st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ X2 @ Xa )
     => ( ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ Xa ) )
       => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
           => ~ ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) )
         => ( ! [X: produc2579390649552321826st_a_b] :
                ( ( Xa
                  = ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
               => ~ ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) )
           => ~ ! [Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                  ( ( Xa
                    = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) )
                 => ( ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) )
                   => ~ ( ( ( compar8165832444956594340st_a_b @ X2 @ Y3 @ X )
                         != greater )
                        & ( sortin3200817720516614760st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ) ) ) ) ).

% sorted.pelims(2)
thf(fact_863_sorted_Opelims_I1_J,axiom,
    ! [X2: compar7542523840845723048st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: $o] :
      ( ( ( sortin3200817720516614760st_a_b @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ Xa ) )
       => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
           => ( Y2
             => ~ ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ) )
         => ( ! [X: produc2579390649552321826st_a_b] :
                ( ( Xa
                  = ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) )
               => ( Y2
                 => ~ ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ nil_Pr523822035851181096st_a_b ) ) ) ) )
           => ~ ! [Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                  ( ( Xa
                    = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) )
                 => ( ( Y2
                      = ( ( ( compar8165832444956594340st_a_b @ X2 @ Y3 @ X )
                         != greater )
                        & ( sortin3200817720516614760st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) )
                   => ~ ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% sorted.pelims(1)
thf(fact_864_sorted_Opelims_I3_J,axiom,
    ! [X2: compar7542523840845723048st_a_b,Xa: list_P2281458131032066856st_a_b] :
      ( ~ ( sortin3200817720516614760st_a_b @ X2 @ Xa )
     => ( ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ Xa ) )
       => ~ ! [Y3: produc2579390649552321826st_a_b,X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( Xa
                = ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) )
             => ( ( accp_P8714544010401090352st_a_b @ sortin5914512992985630625st_a_b @ ( produc4166231589323681047st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) )
               => ( ( ( compar8165832444956594340st_a_b @ X2 @ Y3 @ X )
                   != greater )
                  & ( sortin3200817720516614760st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) ) ) ) ) ) ) ).

% sorted.pelims(3)
thf(fact_865_splice_Opelims,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( ( splice7289630126412227579st_a_b @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ X2 @ Xa ) )
       => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
           => ( ( Y2 = Xa )
             => ~ ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Xa ) ) ) )
         => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) )
               => ( ( Y2
                    = ( cons_P2018802354021969880st_a_b @ X @ ( splice7289630126412227579st_a_b @ Xa @ Xs2 ) ) )
                 => ~ ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_866_insort__eq__ConsI,axiom,
    ! [Cmp: comparator_a,Xs: list_a,X2: a] :
      ( ( sorting_sorted_a @ Cmp @ Xs )
     => ( ! [Y3: a] :
            ( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
           => ( ( compare_a @ Cmp @ X2 @ Y3 )
             != greater ) )
       => ( ( sorting_insort_a @ Cmp @ X2 @ Xs )
          = ( cons_a @ X2 @ Xs ) ) ) ) ).

% insort_eq_ConsI
thf(fact_867_insort__eq__ConsI,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs )
     => ( ! [Y3: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ Y3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ( ( compar8165832444956594340st_a_b @ Cmp @ X2 @ Y3 )
             != greater ) )
       => ( ( sortin1419078322245402082st_a_b @ Cmp @ X2 @ Xs )
          = ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) ) ) ).

% insort_eq_ConsI
thf(fact_868_splice_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( splice7289630126412227579st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys )
      = ( cons_P2018802354021969880st_a_b @ X2 @ ( splice7289630126412227579st_a_b @ Ys @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_869_insort_Osimps_I1_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Y2: produc2579390649552321826st_a_b] :
      ( ( sortin1419078322245402082st_a_b @ Cmp @ Y2 @ nil_Pr523822035851181096st_a_b )
      = ( cons_P2018802354021969880st_a_b @ Y2 @ nil_Pr523822035851181096st_a_b ) ) ).

% insort.simps(1)
thf(fact_870_splice_Oelims,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( ( splice7289630126412227579st_a_b @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
         => ( Y2 != Xa ) )
       => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( X2
                = ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) )
             => ( Y2
               != ( cons_P2018802354021969880st_a_b @ X @ ( splice7289630126412227579st_a_b @ Xa @ Xs2 ) ) ) ) ) ) ).

% splice.elims
thf(fact_871_insort_Osimps_I2_J,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Y2: produc2579390649552321826st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( ( compar8165832444956594340st_a_b @ Cmp @ Y2 @ X2 )
         != greater )
       => ( ( sortin1419078322245402082st_a_b @ Cmp @ Y2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = ( cons_P2018802354021969880st_a_b @ Y2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) ) ) )
      & ( ( ( compar8165832444956594340st_a_b @ Cmp @ Y2 @ X2 )
          = greater )
       => ( ( sortin1419078322245402082st_a_b @ Cmp @ Y2 @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = ( cons_P2018802354021969880st_a_b @ X2 @ ( sortin1419078322245402082st_a_b @ Cmp @ Y2 @ Xs ) ) ) ) ) ).

% insort.simps(2)
thf(fact_872_splice_Opsimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( accp_P4272997315293813680st_a_b @ splice4666273797428483534st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) )
     => ( ( splice7289630126412227579st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys )
        = ( cons_P2018802354021969880st_a_b @ X2 @ ( splice7289630126412227579st_a_b @ Ys @ Xs ) ) ) ) ).

% splice.psimps(2)
thf(fact_873_subseqs_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( subseq1577149208604677367st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( append2962543925774821315st_a_b @ ( map_li1268991901115354180st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ ( subseq1577149208604677367st_a_b @ Xs ) ) @ ( subseq1577149208604677367st_a_b @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_874_vwalk__arcs__Cons,axiom,
    ! [P2: list_P2281458131032066856st_a_b,U2: produc2579390649552321826st_a_b] :
      ( ( P2 != nil_Pr523822035851181096st_a_b )
     => ( ( vertex2793823137597548186st_a_b @ ( cons_P2018802354021969880st_a_b @ U2 @ P2 ) )
        = ( cons_P5669161040283791197st_a_b @ ( produc7813282991251521687st_a_b @ U2 @ ( hd_Pro4070172631639903629st_a_b @ P2 ) ) @ ( vertex2793823137597548186st_a_b @ P2 ) ) ) ) ).

% vwalk_arcs_Cons
thf(fact_875_product_Osimps_I2_J,axiom,
    ! [X2: list_a,Xs: list_list_a,Ys: list_b] :
      ( ( product_list_a_b @ ( cons_list_a @ X2 @ Xs ) @ Ys )
      = ( append8668862642614166333st_a_b @ ( map_b_2284495172895197351st_a_b @ ( produc4781227316648555538st_a_b @ X2 ) @ Ys ) @ ( product_list_a_b @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_876_list_Osel_I1_J,axiom,
    ! [X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] :
      ( ( hd_Pro4070172631639903629st_a_b @ ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_877_Cons__in__subseqsD,axiom,
    ! [Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) @ ( set_li5246279850479030973st_a_b @ ( subseq1577149208604677367st_a_b @ Xs ) ) )
     => ( member1308482598881763409st_a_b @ Ys @ ( set_li5246279850479030973st_a_b @ ( subseq1577149208604677367st_a_b @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_878_hd__in__set,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( member8006451236149131979st_a_b @ ( hd_Pro4070172631639903629st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ).

% hd_in_set
thf(fact_879_hd__in__set,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_880_list_Oset__sel_I1_J,axiom,
    ! [A2: list_P2281458131032066856st_a_b] :
      ( ( A2 != nil_Pr523822035851181096st_a_b )
     => ( member8006451236149131979st_a_b @ ( hd_Pro4070172631639903629st_a_b @ A2 ) @ ( set_Pr6156699709417984823st_a_b @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_881_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_882_merge__eq__Cons__imp,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Z: produc2579390649552321826st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys )
        = ( cons_P2018802354021969880st_a_b @ Z @ Zs ) )
     => ( ( ( Xs != nil_Pr523822035851181096st_a_b )
          & ( Z
            = ( hd_Pro4070172631639903629st_a_b @ Xs ) ) )
        | ( ( Ys != nil_Pr523822035851181096st_a_b )
          & ( Z
            = ( hd_Pro4070172631639903629st_a_b @ Ys ) ) ) ) ) ).

% merge_eq_Cons_imp
thf(fact_883_subset__subseqs,axiom,
    ! [X5: set_Pr9033091269622839554st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ord_le2879471331418251426st_a_b @ X5 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( member358942219636986539st_a_b @ X5 @ ( image_3249384991236752479st_a_b @ set_Pr6156699709417984823st_a_b @ ( set_li5246279850479030973st_a_b @ ( subseq1577149208604677367st_a_b @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_884_subset__subseqs,axiom,
    ! [X5: set_a,Xs: list_a] :
      ( ( ord_less_eq_set_a @ X5 @ ( set_a2 @ Xs ) )
     => ( member_set_a @ X5 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_885_Id__on__set,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( id_on_5984594033129004095st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( set_Pr5950792022122277180st_a_b
        @ ( map_Pr3613949492719739849st_a_b
          @ ^ [X4: produc2579390649552321826st_a_b] : ( produc7813282991251521687st_a_b @ X4 @ X4 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_886_Id__on__set,axiom,
    ! [Xs: list_a] :
      ( ( id_on_a @ ( set_a2 @ Xs ) )
      = ( set_Product_prod_a_a2
        @ ( map_a_7860052162900579309od_a_a
          @ ^ [X4: a] : ( product_Pair_a_a @ X4 @ X4 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_887_map__tailrec__rev_Opelims,axiom,
    ! [X2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Xa: list_P2281458131032066856st_a_b,Xb: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( ( map_ta462313747816757493st_a_b @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp_P1297149347635948462st_a_b @ map_ta86837710956154436st_a_b @ ( produc6221001472763327657st_a_b @ X2 @ ( produc717113118580727703st_a_b @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
           => ( ( Y2 = Xb )
             => ~ ( accp_P1297149347635948462st_a_b @ map_ta86837710956154436st_a_b @ ( produc6221001472763327657st_a_b @ X2 @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Xb ) ) ) ) )
         => ~ ! [A6: produc2579390649552321826st_a_b,As: list_P2281458131032066856st_a_b] :
                ( ( Xa
                  = ( cons_P2018802354021969880st_a_b @ A6 @ As ) )
               => ( ( Y2
                    = ( map_ta462313747816757493st_a_b @ X2 @ As @ ( cons_P2018802354021969880st_a_b @ ( X2 @ A6 ) @ Xb ) ) )
                 => ~ ( accp_P1297149347635948462st_a_b @ map_ta86837710956154436st_a_b @ ( produc6221001472763327657st_a_b @ X2 @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ A6 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_888_Id__onI,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ ( id_on_a @ A ) ) ) ).

% Id_onI
thf(fact_889_last__snoc,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( last_P2607716600779573313st_a_b @ ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) )
      = X2 ) ).

% last_snoc
thf(fact_890_Id__on__iff,axiom,
    ! [X2: a,Y2: a,A: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( id_on_a @ A ) )
      = ( ( X2 = Y2 )
        & ( member_a @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_891_Id__on__eqI,axiom,
    ! [A2: a,B2: a,A: set_a] :
      ( ( A2 = B2 )
     => ( ( member_a @ A2 @ A )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( id_on_a @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_892_Id__onE,axiom,
    ! [C2: product_prod_a_a,A: set_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( id_on_a @ A ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A )
           => ( C2
             != ( product_Pair_a_a @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_893_last_Osimps,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( ( Xs = nil_Pr523822035851181096st_a_b )
       => ( ( last_P2607716600779573313st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = X2 ) )
      & ( ( Xs != nil_Pr523822035851181096st_a_b )
       => ( ( last_P2607716600779573313st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = ( last_P2607716600779573313st_a_b @ Xs ) ) ) ) ).

% last.simps
thf(fact_894_last__ConsL,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( Xs = nil_Pr523822035851181096st_a_b )
     => ( ( last_P2607716600779573313st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
        = X2 ) ) ).

% last_ConsL
thf(fact_895_last__ConsR,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ( last_P2607716600779573313st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
        = ( last_P2607716600779573313st_a_b @ Xs ) ) ) ).

% last_ConsR
thf(fact_896_last__in__set,axiom,
    ! [As3: list_P2281458131032066856st_a_b] :
      ( ( As3 != nil_Pr523822035851181096st_a_b )
     => ( member8006451236149131979st_a_b @ ( last_P2607716600779573313st_a_b @ As3 ) @ ( set_Pr6156699709417984823st_a_b @ As3 ) ) ) ).

% last_in_set
thf(fact_897_last__in__set,axiom,
    ! [As3: list_a] :
      ( ( As3 != nil_a )
     => ( member_a @ ( last_a @ As3 ) @ ( set_a2 @ As3 ) ) ) ).

% last_in_set
thf(fact_898_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,A2: produc2579390649552321826st_a_b,As3: list_P2281458131032066856st_a_b,Bs3: list_P2281458131032066856st_a_b] :
      ( ( map_ta462313747816757493st_a_b @ F @ ( cons_P2018802354021969880st_a_b @ A2 @ As3 ) @ Bs3 )
      = ( map_ta462313747816757493st_a_b @ F @ As3 @ ( cons_P2018802354021969880st_a_b @ ( F @ A2 ) @ Bs3 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_899_map__tailrec__rev_Oelims,axiom,
    ! [X2: produc2579390649552321826st_a_b > produc2579390649552321826st_a_b,Xa: list_P2281458131032066856st_a_b,Xb: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( ( map_ta462313747816757493st_a_b @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
         => ( Y2 != Xb ) )
       => ~ ! [A6: produc2579390649552321826st_a_b,As: list_P2281458131032066856st_a_b] :
              ( ( Xa
                = ( cons_P2018802354021969880st_a_b @ A6 @ As ) )
             => ( Y2
               != ( map_ta462313747816757493st_a_b @ X2 @ As @ ( cons_P2018802354021969880st_a_b @ ( X2 @ A6 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_900_split__list__last__sep,axiom,
    ! [Y2: a,Xs: list_a] :
      ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
     => ( ( Y2
         != ( last_a @ Xs ) )
       => ? [As: list_a,Bs: list_a] :
            ( ( append_a @ As @ ( cons_a @ Y2 @ ( append_a @ Bs @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) ) ) )
            = Xs ) ) ) ).

% split_list_last_sep
thf(fact_901_split__list__last__sep,axiom,
    ! [Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( ( Y2
         != ( last_P2607716600779573313st_a_b @ Xs ) )
       => ? [As: list_P2281458131032066856st_a_b,Bs: list_P2281458131032066856st_a_b] :
            ( ( append8668862642614166333st_a_b @ As @ ( cons_P2018802354021969880st_a_b @ Y2 @ ( append8668862642614166333st_a_b @ Bs @ ( cons_P2018802354021969880st_a_b @ ( last_P2607716600779573313st_a_b @ Xs ) @ nil_Pr523822035851181096st_a_b ) ) ) )
            = Xs ) ) ) ).

% split_list_last_sep
thf(fact_902_split__list__not__last,axiom,
    ! [Y2: a,Xs: list_a] :
      ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
     => ( ( Y2
         != ( last_a @ Xs ) )
       => ? [As: list_a,Bs: list_a] :
            ( ( ( append_a @ As @ ( cons_a @ Y2 @ Bs ) )
              = Xs )
            & ( Bs != nil_a ) ) ) ) ).

% split_list_not_last
thf(fact_903_split__list__not__last,axiom,
    ! [Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ Y2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
     => ( ( Y2
         != ( last_P2607716600779573313st_a_b @ Xs ) )
       => ? [As: list_P2281458131032066856st_a_b,Bs: list_P2281458131032066856st_a_b] :
            ( ( ( append8668862642614166333st_a_b @ As @ ( cons_P2018802354021969880st_a_b @ Y2 @ Bs ) )
              = Xs )
            & ( Bs != nil_Pr523822035851181096st_a_b ) ) ) ) ).

% split_list_not_last
thf(fact_904_split__last__eq,axiom,
    ! [As3: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Bs3: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ As3 @ ( cons_P2018802354021969880st_a_b @ Y2 @ Bs3 ) )
        = Xs )
     => ( ( Bs3 != nil_Pr523822035851181096st_a_b )
       => ( ( last_P2607716600779573313st_a_b @ Bs3 )
          = ( last_P2607716600779573313st_a_b @ Xs ) ) ) ) ).

% split_last_eq
thf(fact_905_set__vwalk__arcs__snoc,axiom,
    ! [P2: list_P2281458131032066856st_a_b,A2: produc2579390649552321826st_a_b] :
      ( ( P2 != nil_Pr523822035851181096st_a_b )
     => ( ( set_Pr5950792022122277180st_a_b @ ( vertex2793823137597548186st_a_b @ ( append8668862642614166333st_a_b @ P2 @ ( cons_P2018802354021969880st_a_b @ A2 @ nil_Pr523822035851181096st_a_b ) ) ) )
        = ( insert7502441213859096951st_a_b @ ( produc7813282991251521687st_a_b @ ( last_P2607716600779573313st_a_b @ P2 ) @ A2 ) @ ( set_Pr5950792022122277180st_a_b @ ( vertex2793823137597548186st_a_b @ P2 ) ) ) ) ) ).

% set_vwalk_arcs_snoc
thf(fact_906_insert__iff,axiom,
    ! [A2: a,B2: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a2 @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_907_insertCI,axiom,
    ! [A2: a,B: set_a,B2: a] :
      ( ( ~ ( member_a @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_a @ A2 @ ( insert_a2 @ B2 @ B ) ) ) ).

% insertCI
thf(fact_908_image__insert,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A2: produc2579390649552321826st_a_b,B: set_Pr9033091269622839554st_a_b] :
      ( ( image_4065140400681997415list_a @ F @ ( insert7903963420998390514st_a_b @ A2 @ B ) )
      = ( insert_list_a2 @ ( F @ A2 ) @ ( image_4065140400681997415list_a @ F @ B ) ) ) ).

% image_insert
thf(fact_909_insert__image,axiom,
    ! [X2: produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b,F: produc2579390649552321826st_a_b > list_a] :
      ( ( member8006451236149131979st_a_b @ X2 @ A )
     => ( ( insert_list_a2 @ ( F @ X2 ) @ ( image_4065140400681997415list_a @ F @ A ) )
        = ( image_4065140400681997415list_a @ F @ A ) ) ) ).

% insert_image
thf(fact_910_insert__subset,axiom,
    ! [X2: a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ A ) @ B )
      = ( ( member_a @ X2 @ B )
        & ( ord_less_eq_set_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_911_list_Osimps_I15_J,axiom,
    ! [X21: a,X22: list_a] :
      ( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
      = ( insert_a2 @ X21 @ ( set_a2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_912_list_Osimps_I15_J,axiom,
    ! [X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) )
      = ( insert7903963420998390514st_a_b @ X21 @ ( set_Pr6156699709417984823st_a_b @ X22 ) ) ) ).

% list.simps(15)
thf(fact_913_set__insort,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( sortin1419078322245402082st_a_b @ Cmp @ X2 @ Xs ) )
      = ( insert7903963420998390514st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ).

% set_insort
thf(fact_914_set__insort,axiom,
    ! [Cmp: comparator_a,X2: a,Xs: list_a] :
      ( ( set_a2 @ ( sorting_insort_a @ Cmp @ X2 @ Xs ) )
      = ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_insort
thf(fact_915_List_Oset__insert,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( insert6036452702741353534st_a_b @ X2 @ Xs ) )
      = ( insert7903963420998390514st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ).

% List.set_insert
thf(fact_916_List_Oset__insert,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( set_a2 @ ( insert_a @ X2 @ Xs ) )
      = ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_917_insert__subsetI,axiom,
    ! [X2: a,A: set_a,X5: set_a] :
      ( ( member_a @ X2 @ A )
     => ( ( ord_less_eq_set_a @ X5 @ A )
       => ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ X5 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_918_insert__mono,axiom,
    ! [C3: set_a,D: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ C3 @ D )
     => ( ord_less_eq_set_a @ ( insert_a2 @ A2 @ C3 ) @ ( insert_a2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_919_subset__insert,axiom,
    ! [X2: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A )
     => ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X2 @ B ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% subset_insert
thf(fact_920_subset__insertI,axiom,
    ! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a2 @ A2 @ B ) ) ).

% subset_insertI
thf(fact_921_subset__insertI2,axiom,
    ! [A: set_a,B: set_a,B2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_less_eq_set_a @ A @ ( insert_a2 @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_922_insert__compr,axiom,
    ( insert_a2
    = ( ^ [A4: a,B3: set_a] :
          ( collect_a
          @ ^ [X4: a] :
              ( ( X4 = A4 )
              | ( member_a @ X4 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_923_mk__disjoint__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ? [B7: set_a] :
          ( ( A
            = ( insert_a2 @ A2 @ B7 ) )
          & ~ ( member_a @ A2 @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_924_insert__eq__iff,axiom,
    ! [A2: a,A: set_a,B2: a,B: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ~ ( member_a @ B2 @ B )
       => ( ( ( insert_a2 @ A2 @ A )
            = ( insert_a2 @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C5: set_a] :
                  ( ( A
                    = ( insert_a2 @ B2 @ C5 ) )
                  & ~ ( member_a @ B2 @ C5 )
                  & ( B
                    = ( insert_a2 @ A2 @ C5 ) )
                  & ~ ( member_a @ A2 @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_925_insert__absorb,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( insert_a2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_926_insert__ident,axiom,
    ! [X2: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A )
     => ( ~ ( member_a @ X2 @ B )
       => ( ( ( insert_a2 @ X2 @ A )
            = ( insert_a2 @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_927_Set_Oset__insert,axiom,
    ! [X2: a,A: set_a] :
      ( ( member_a @ X2 @ A )
     => ~ ! [B7: set_a] :
            ( ( A
              = ( insert_a2 @ X2 @ B7 ) )
           => ( member_a @ X2 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_928_insertI2,axiom,
    ! [A2: a,B: set_a,B2: a] :
      ( ( member_a @ A2 @ B )
     => ( member_a @ A2 @ ( insert_a2 @ B2 @ B ) ) ) ).

% insertI2
thf(fact_929_insertI1,axiom,
    ! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a2 @ A2 @ B ) ) ).

% insertI1
thf(fact_930_insertE,axiom,
    ! [A2: a,B2: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a2 @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_931_append__butlast__last__id,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ( append8668862642614166333st_a_b @ ( butlas3702971404721225588st_a_b @ Xs ) @ ( cons_P2018802354021969880st_a_b @ ( last_P2607716600779573313st_a_b @ Xs ) @ nil_Pr523822035851181096st_a_b ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_932_the__elem__set,axiom,
    ! [X2: a] :
      ( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_933_the__elem__set,axiom,
    ! [X2: produc2579390649552321826st_a_b] :
      ( ( the_el5479306861426336663st_a_b @ ( set_Pr6156699709417984823st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_934_vwalk__join__split,axiom,
    ! [U2: produc2579390649552321826st_a_b,P2: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ U2 @ ( set_Pr6156699709417984823st_a_b @ P2 ) )
     => ? [Q2: list_P2281458131032066856st_a_b,R3: list_P2281458131032066856st_a_b] :
          ( ( P2
            = ( vertex3498345769568195249st_a_b @ Q2 @ R3 ) )
          & ( ( last_P2607716600779573313st_a_b @ Q2 )
            = U2 )
          & ( ( hd_Pro4070172631639903629st_a_b @ R3 )
            = U2 )
          & ( Q2 != nil_Pr523822035851181096st_a_b )
          & ( R3 != nil_Pr523822035851181096st_a_b ) ) ) ).

% vwalk_join_split
thf(fact_935_vwalk__join__split,axiom,
    ! [U2: a,P2: list_a] :
      ( ( member_a @ U2 @ ( set_a2 @ P2 ) )
     => ? [Q2: list_a,R3: list_a] :
          ( ( P2
            = ( vertex_vwalk_join_a @ Q2 @ R3 ) )
          & ( ( last_a @ Q2 )
            = U2 )
          & ( ( hd_a @ R3 )
            = U2 )
          & ( Q2 != nil_a )
          & ( R3 != nil_a ) ) ) ).

% vwalk_join_split
thf(fact_936_distinct__adj__Cons__Cons,axiom,
    ! [X2: produc2579390649552321826st_a_b,Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( distin8513967041251542528st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ ( cons_P2018802354021969880st_a_b @ Y2 @ Xs ) ) )
      = ( ( X2 != Y2 )
        & ( distin8513967041251542528st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 @ Xs ) ) ) ) ).

% distinct_adj_Cons_Cons
thf(fact_937_vwalk__join__Cons,axiom,
    ! [P2: list_P2281458131032066856st_a_b,U2: produc2579390649552321826st_a_b,Q3: list_P2281458131032066856st_a_b] :
      ( ( P2 != nil_Pr523822035851181096st_a_b )
     => ( ( vertex3498345769568195249st_a_b @ ( cons_P2018802354021969880st_a_b @ U2 @ P2 ) @ Q3 )
        = ( cons_P2018802354021969880st_a_b @ U2 @ ( vertex3498345769568195249st_a_b @ P2 @ Q3 ) ) ) ) ).

% vwalk_join_Cons
thf(fact_938_butlast__snoc,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( butlas3702971404721225588st_a_b @ ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_939_in__set__butlastD,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( butlas3702971404721225588st_a_b @ Xs ) ) )
     => ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_940_in__set__butlastD,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_941_distinct__adj__ConsD,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( distin8513967041251542528st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
     => ( distin8513967041251542528st_a_b @ Xs ) ) ).

% distinct_adj_ConsD
thf(fact_942_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( ( Xs = nil_Pr523822035851181096st_a_b )
       => ( ( butlas3702971404721225588st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = nil_Pr523822035851181096st_a_b ) )
      & ( ( Xs != nil_Pr523822035851181096st_a_b )
       => ( ( butlas3702971404721225588st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = ( cons_P2018802354021969880st_a_b @ X2 @ ( butlas3702971404721225588st_a_b @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_943_in__set__butlast__appendI,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( butlas3702971404721225588st_a_b @ Xs ) ) )
        | ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( butlas3702971404721225588st_a_b @ Ys ) ) ) )
     => ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( butlas3702971404721225588st_a_b @ ( append8668862642614166333st_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_944_in__set__butlast__appendI,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a] :
      ( ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
        | ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
     => ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_945_distinct__adj__singleton,axiom,
    ! [X2: produc2579390649552321826st_a_b] : ( distin8513967041251542528st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ).

% distinct_adj_singleton
thf(fact_946_distinct__adj__Cons,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( distin8513967041251542528st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( ( Xs = nil_Pr523822035851181096st_a_b )
        | ( ( X2
           != ( hd_Pro4070172631639903629st_a_b @ Xs ) )
          & ( distin8513967041251542528st_a_b @ Xs ) ) ) ) ).

% distinct_adj_Cons
thf(fact_947_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) )
        = Ys )
      = ( ( Ys != nil_Pr523822035851181096st_a_b )
        & ( ( butlas3702971404721225588st_a_b @ Ys )
          = Xs )
        & ( ( last_P2607716600779573313st_a_b @ Ys )
          = X2 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_948_separate__P__fst__acc,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Acc: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,As3: list_P2281458131032066856st_a_b,Bs3: list_P2281458131032066856st_a_b] :
      ( ( ( iKKBZ_2269102172772517653st_a_b @ P @ Acc @ Xs )
        = ( produc717113118580727703st_a_b @ As3 @ Bs3 ) )
     => ? [As2: list_P2281458131032066856st_a_b] :
          ( ( As3
            = ( append8668862642614166333st_a_b @ As2 @ Acc ) )
          & ! [X3: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ As2 ) )
             => ( P @ X3 ) ) ) ) ).

% separate_P_fst_acc
thf(fact_949_separate__P__fst__acc,axiom,
    ! [P: a > $o,Acc: list_a,Xs: list_a,As3: list_a,Bs3: list_a] :
      ( ( ( iKKBZ_separate_P_a @ P @ Acc @ Xs )
        = ( produc6837034575241423639list_a @ As3 @ Bs3 ) )
     => ? [As2: list_a] :
          ( ( As3
            = ( append_a @ As2 @ Acc ) )
          & ! [X3: a] :
              ( ( member_a @ X3 @ ( set_a2 @ As2 ) )
             => ( P @ X3 ) ) ) ) ).

% separate_P_fst_acc
thf(fact_950_separate__P__fst,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Xs: list_P2281458131032066856st_a_b,As3: list_P2281458131032066856st_a_b,Bs3: list_P2281458131032066856st_a_b] :
      ( ( ( iKKBZ_2269102172772517653st_a_b @ P @ nil_Pr523822035851181096st_a_b @ Xs )
        = ( produc717113118580727703st_a_b @ As3 @ Bs3 ) )
     => ! [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ As3 ) )
         => ( P @ X3 ) ) ) ).

% separate_P_fst
thf(fact_951_separate__P__fst,axiom,
    ! [P: a > $o,Xs: list_a,As3: list_a,Bs3: list_a] :
      ( ( ( iKKBZ_separate_P_a @ P @ nil_a @ Xs )
        = ( produc6837034575241423639list_a @ As3 @ Bs3 ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ As3 ) )
         => ( P @ X3 ) ) ) ).

% separate_P_fst
thf(fact_952_joinable__Cons__l__iff,axiom,
    ! [P2: list_P2281458131032066856st_a_b,V: produc2579390649552321826st_a_b,Q3: list_P2281458131032066856st_a_b] :
      ( ( P2 != nil_Pr523822035851181096st_a_b )
     => ( ( vertex6651755781183652644st_a_b @ ( cons_P2018802354021969880st_a_b @ V @ P2 ) @ Q3 )
        = ( vertex6651755781183652644st_a_b @ P2 @ Q3 ) ) ) ).

% joinable_Cons_l_iff
thf(fact_953_joinable__Snoc__r__iff,axiom,
    ! [Q3: list_P2281458131032066856st_a_b,P2: list_P2281458131032066856st_a_b,V: produc2579390649552321826st_a_b] :
      ( ( Q3 != nil_Pr523822035851181096st_a_b )
     => ( ( vertex6651755781183652644st_a_b @ P2 @ ( append8668862642614166333st_a_b @ Q3 @ ( cons_P2018802354021969880st_a_b @ V @ nil_Pr523822035851181096st_a_b ) ) )
        = ( vertex6651755781183652644st_a_b @ P2 @ Q3 ) ) ) ).

% joinable_Snoc_r_iff
thf(fact_954_separate__not__P__snd,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Acc: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,As3: list_P2281458131032066856st_a_b,Bs3: list_P2281458131032066856st_a_b] :
      ( ( ( iKKBZ_2269102172772517653st_a_b @ P @ Acc @ Xs )
        = ( produc717113118580727703st_a_b @ As3 @ Bs3 ) )
     => ! [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Bs3 ) )
         => ~ ( P @ X3 ) ) ) ).

% separate_not_P_snd
thf(fact_955_separate__not__P__snd,axiom,
    ! [P: a > $o,Acc: list_a,Xs: list_a,As3: list_a,Bs3: list_a] :
      ( ( ( iKKBZ_separate_P_a @ P @ Acc @ Xs )
        = ( produc6837034575241423639list_a @ As3 @ Bs3 ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Bs3 ) )
         => ~ ( P @ X3 ) ) ) ).

% separate_not_P_snd
thf(fact_956_permutation__of__list__id,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P1351356133043840941st_a_b] :
      ( ~ ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( map_Pr2735670493911151103st_a_b @ produc6544331879426417283st_a_b @ Xs ) ) )
     => ( ( permut7041589675599641919st_a_b @ Xs @ X2 )
        = X2 ) ) ).

% permutation_of_list_id
thf(fact_957_permutation__of__list__id,axiom,
    ! [X2: a,Xs: list_P1396940483166286381od_a_a] :
      ( ~ ( member_a @ X2 @ ( set_a2 @ ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a @ Xs ) ) )
     => ( ( permut8467757748254915227list_a @ Xs @ X2 )
        = X2 ) ) ).

% permutation_of_list_id
thf(fact_958_product__concat__map,axiom,
    ( product_list_a_b
    = ( ^ [Xs4: list_list_a,Ys4: list_b] :
          ( concat588675724576574691st_a_b
          @ ( map_li3503115258579264104st_a_b
            @ ^ [X4: list_a] : ( map_b_2284495172895197351st_a_b @ ( produc4781227316648555538st_a_b @ X4 ) @ Ys4 )
            @ Xs4 ) ) ) ) ).

% product_concat_map
thf(fact_959_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Xss2: list_l3733004638376709294st_a_b] :
      ( ( produc5480500085755353745st_a_b @ ( cons_l7524743740370447070st_a_b @ Xs @ Xss2 ) )
      = ( concat4126514578872141417st_a_b
        @ ( map_Pr7268644361359556560st_a_b
          @ ^ [X4: produc2579390649552321826st_a_b] : ( map_li1268991901115354180st_a_b @ ( cons_P2018802354021969880st_a_b @ X4 ) @ ( produc5480500085755353745st_a_b @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_960_product__code,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( produc9145407795026516050st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
      = ( set_Pr5950792022122277180st_a_b
        @ ( concat1772608278777562344st_a_b
          @ ( map_Pr2389323529323935439st_a_b
            @ ^ [X4: produc2579390649552321826st_a_b] : ( map_Pr3613949492719739849st_a_b @ ( produc7813282991251521687st_a_b @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_961_product__code,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_a] :
      ( ( produc669184731872937518_a_b_a @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_a2 @ Ys ) )
      = ( set_Pr896729820931812248_a_b_a
        @ ( concat105229137197504836_a_b_a
          @ ( map_Pr7256514589255010091_a_b_a
            @ ^ [X4: produc2579390649552321826st_a_b] : ( map_a_5937377274483198025_a_b_a @ ( produc1676824617466569203_a_b_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_962_product__code,axiom,
    ! [Xs: list_a,Ys: list_P2281458131032066856st_a_b] :
      ( ( produc6747040594293283830st_a_b @ ( set_a2 @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
      = ( set_Pr2913479409760856416st_a_b
        @ ( concat2121978726026549004st_a_b
          @ ( map_a_512126053687075223st_a_b
            @ ^ [X4: a] : ( map_Pr8901270321044693869st_a_b @ ( produc7754680479886915515st_a_b @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_963_product__code,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( product_product_a_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
      = ( set_Product_prod_a_a2
        @ ( concat6071998529708183912od_a_a
          @ ( map_a_6618942900564114931od_a_a
            @ ^ [X4: a] : ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_964_product__code,axiom,
    ! [Xs: list_list_a,Ys: list_b] :
      ( ( produc3268770062311508813st_a_b @ ( set_list_a2 @ Xs ) @ ( set_b2 @ Ys ) )
      = ( set_Pr6156699709417984823st_a_b
        @ ( concat588675724576574691st_a_b
          @ ( map_li3503115258579264104st_a_b
            @ ^ [X4: list_a] : ( map_b_2284495172895197351st_a_b @ ( produc4781227316648555538st_a_b @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_965_set__concat,axiom,
    ! [Xs: list_l3733004638376709294st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( concat588675724576574691st_a_b @ Xs ) )
      = ( comple3847176067670185711st_a_b @ ( image_3249384991236752479st_a_b @ set_Pr6156699709417984823st_a_b @ ( set_li5246279850479030973st_a_b @ Xs ) ) ) ) ).

% set_concat
thf(fact_966_set__concat,axiom,
    ! [Xs: list_list_a] :
      ( ( set_a2 @ ( concat_a @ Xs ) )
      = ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_concat
thf(fact_967_set__list__bind,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,F: produc2579390649552321826st_a_b > list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( bind_P4816016919889662599st_a_b @ Xs @ F ) )
      = ( comple3847176067670185711st_a_b
        @ ( image_3062191654419162853st_a_b
          @ ^ [X4: produc2579390649552321826st_a_b] : ( set_Pr6156699709417984823st_a_b @ ( F @ X4 ) )
          @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_968_set__list__bind,axiom,
    ! [Xs: list_a,F: a > list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( bind_a6226939647125828011st_a_b @ Xs @ F ) )
      = ( comple3847176067670185711st_a_b
        @ ( image_8858725927978154761st_a_b
          @ ^ [X4: a] : ( set_Pr6156699709417984823st_a_b @ ( F @ X4 ) )
          @ ( set_a2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_969_set__list__bind,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,F: produc2579390649552321826st_a_b > list_a] :
      ( ( set_a2 @ ( bind_P149083784705481699_a_b_a @ Xs @ F ) )
      = ( comple2307003609928055243_set_a
        @ ( image_4720221803530543809_set_a
          @ ^ [X4: produc2579390649552321826st_a_b] : ( set_a2 @ ( F @ X4 ) )
          @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_970_set__list__bind,axiom,
    ! [Xs: list_a,F: a > list_a] :
      ( ( set_a2 @ ( bind_a_a @ Xs @ F ) )
      = ( comple2307003609928055243_set_a
        @ ( image_a_set_a
          @ ^ [X4: a] : ( set_a2 @ ( F @ X4 ) )
          @ ( set_a2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_971_UN__I,axiom,
    ! [A2: a,A: set_a,B2: a,B: a > set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a @ B2 @ ( B @ A2 ) )
       => ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_972_SUP__Sup__eq2,axiom,
    ! [S3: set_se1825328058903261410st_a_b] :
      ( ( comple3674869584817201568_a_b_o
        @ ( image_3185165532000840118_a_b_o
          @ ^ [I: set_Pr9033091269622839554st_a_b,X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ I )
          @ S3 ) )
      = ( ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ ( comple3847176067670185711st_a_b @ S3 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_973_Sup__set__def,axiom,
    ( comple2307003609928055243_set_a
    = ( ^ [A3: set_set_a] :
          ( collect_a
          @ ^ [X4: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X4 ) @ A3 ) ) ) ) ) ).

% Sup_set_def
thf(fact_974_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_a_o
    = ( ^ [S4: set_a_o,X4: a] : ( member_a @ X4 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_975_SUP__Sup__eq,axiom,
    ! [S3: set_set_a] :
      ( ( complete_Sup_Sup_a_o
        @ ( image_set_a_a_o
          @ ^ [I: set_a,X4: a] : ( member_a @ X4 @ I )
          @ S3 ) )
      = ( ^ [X4: a] : ( member_a @ X4 @ ( comple2307003609928055243_set_a @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_976_Inf_OINF__cong,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,B: set_Pr9033091269622839554st_a_b,C3: produc2579390649552321826st_a_b > list_a,D: produc2579390649552321826st_a_b > list_a,Inf: set_list_a > list_a] :
      ( ( A = B )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X @ B )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Inf @ ( image_4065140400681997415list_a @ C3 @ A ) )
          = ( Inf @ ( image_4065140400681997415list_a @ D @ B ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_977_Sup_OSUP__cong,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,B: set_Pr9033091269622839554st_a_b,C3: produc2579390649552321826st_a_b > list_a,D: produc2579390649552321826st_a_b > list_a,Sup: set_list_a > list_a] :
      ( ( A = B )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X @ B )
           => ( ( C3 @ X )
              = ( D @ X ) ) )
       => ( ( Sup @ ( image_4065140400681997415list_a @ C3 @ A ) )
          = ( Sup @ ( image_4065140400681997415list_a @ D @ B ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_978_Sup__eqI,axiom,
    ! [A: set_set_a,X2: set_a] :
      ( ! [Y3: set_a] :
          ( ( member_set_a @ Y3 @ A )
         => ( ord_less_eq_set_a @ Y3 @ X2 ) )
     => ( ! [Y3: set_a] :
            ( ! [Z7: set_a] :
                ( ( member_set_a @ Z7 @ A )
               => ( ord_less_eq_set_a @ Z7 @ Y3 ) )
           => ( ord_less_eq_set_a @ X2 @ Y3 ) )
       => ( ( comple2307003609928055243_set_a @ A )
          = X2 ) ) ) ).

% Sup_eqI
thf(fact_979_Sup__mono,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [A6: set_a] :
          ( ( member_set_a @ A6 @ A )
         => ? [X3: set_a] :
              ( ( member_set_a @ X3 @ B )
              & ( ord_less_eq_set_a @ A6 @ X3 ) ) )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).

% Sup_mono
thf(fact_980_Sup__least,axiom,
    ! [A: set_set_a,Z: set_a] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ A )
         => ( ord_less_eq_set_a @ X @ Z ) )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ Z ) ) ).

% Sup_least
thf(fact_981_Sup__upper,axiom,
    ! [X2: set_a,A: set_set_a] :
      ( ( member_set_a @ X2 @ A )
     => ( ord_less_eq_set_a @ X2 @ ( comple2307003609928055243_set_a @ A ) ) ) ).

% Sup_upper
thf(fact_982_Sup__le__iff,axiom,
    ! [A: set_set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ B2 )
      = ( ! [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
           => ( ord_less_eq_set_a @ X4 @ B2 ) ) ) ) ).

% Sup_le_iff
thf(fact_983_Sup__upper2,axiom,
    ! [U2: set_a,A: set_set_a,V: set_a] :
      ( ( member_set_a @ U2 @ A )
     => ( ( ord_less_eq_set_a @ V @ U2 )
       => ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).

% Sup_upper2
thf(fact_984_Union__mono,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).

% Union_mono
thf(fact_985_Union__least,axiom,
    ! [A: set_set_a,C3: set_a] :
      ( ! [X6: set_a] :
          ( ( member_set_a @ X6 @ A )
         => ( ord_less_eq_set_a @ X6 @ C3 ) )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ C3 ) ) ).

% Union_least
thf(fact_986_Union__upper,axiom,
    ! [B: set_a,A: set_set_a] :
      ( ( member_set_a @ B @ A )
     => ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ).

% Union_upper
thf(fact_987_Union__subsetI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ A )
         => ? [Y6: set_a] :
              ( ( member_set_a @ Y6 @ B )
              & ( ord_less_eq_set_a @ X @ Y6 ) ) )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).

% Union_subsetI
thf(fact_988_image__Union,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,S3: set_se1825328058903261410st_a_b] :
      ( ( image_4065140400681997415list_a @ F @ ( comple3847176067670185711st_a_b @ S3 ) )
      = ( comple6928918032620976721list_a @ ( image_8488307704784700519list_a @ ( image_4065140400681997415list_a @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_989_UN__E,axiom,
    ! [B2: a,B: a > set_a,A: set_a] :
      ( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A )
           => ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).

% UN_E
thf(fact_990_SUP__eq,axiom,
    ! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
      ( ! [I2: a] :
          ( ( member_a @ I2 @ A )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
     => ( ! [J: a] :
            ( ( member_a @ J @ B )
           => ? [X3: a] :
                ( ( member_a @ X3 @ A )
                & ( ord_less_eq_set_a @ ( G @ J ) @ ( F @ X3 ) ) ) )
       => ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) )
          = ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_991_Sup__subset__mono,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).

% Sup_subset_mono
thf(fact_992_SUP__upper2,axiom,
    ! [I3: a,A: set_a,U2: set_a,F: a > set_a] :
      ( ( member_a @ I3 @ A )
     => ( ( ord_less_eq_set_a @ U2 @ ( F @ I3 ) )
       => ( ord_less_eq_set_a @ U2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).

% SUP_upper2
thf(fact_993_SUP__upper,axiom,
    ! [I3: a,A: set_a,F: a > set_a] :
      ( ( member_a @ I3 @ A )
     => ( ord_less_eq_set_a @ ( F @ I3 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ).

% SUP_upper
thf(fact_994_SUP__least,axiom,
    ! [A: set_a,F: a > set_a,U2: set_a] :
      ( ! [I2: a] :
          ( ( member_a @ I2 @ A )
         => ( ord_less_eq_set_a @ ( F @ I2 ) @ U2 ) )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ U2 ) ) ).

% SUP_least
thf(fact_995_SUP__eqI,axiom,
    ! [A: set_a,F: a > set_a,X2: set_a] :
      ( ! [I2: a] :
          ( ( member_a @ I2 @ A )
         => ( ord_less_eq_set_a @ ( F @ I2 ) @ X2 ) )
     => ( ! [Y3: set_a] :
            ( ! [I4: a] :
                ( ( member_a @ I4 @ A )
               => ( ord_less_eq_set_a @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_a @ X2 @ Y3 ) )
       => ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) )
          = X2 ) ) ) ).

% SUP_eqI
thf(fact_996_UN__upper,axiom,
    ! [A2: a,A: set_a,B: a > set_a] :
      ( ( member_a @ A2 @ A )
     => ( ord_less_eq_set_a @ ( B @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ).

% UN_upper
thf(fact_997_UN__least,axiom,
    ! [A: set_a,B: a > set_a,C3: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( ord_less_eq_set_a @ ( B @ X ) @ C3 ) )
     => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) @ C3 ) ) ).

% UN_least
thf(fact_998_UN__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ! [X: a] :
            ( ( member_a @ X @ A )
           => ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).

% UN_mono
thf(fact_999_SUP__subset__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ! [X: a] :
            ( ( member_a @ X @ A )
           => ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_1000_transpose_Opinduct,axiom,
    ! [A0: list_l3733004638376709294st_a_b,P: list_l3733004638376709294st_a_b > $o] :
      ( ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ A0 )
     => ( ( ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ nil_li5528507983314571054st_a_b )
         => ( P @ nil_li5528507983314571054st_a_b ) )
       => ( ! [Xss: list_l3733004638376709294st_a_b] :
              ( ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ Xss ) ) ) )
         => ( ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Xss: list_l3733004638376709294st_a_b] :
                ( ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_l7524743740370447070st_a_b @ Xs2
                      @ ( concat4126514578872141417st_a_b
                        @ ( map_li2046058465311281098st_a_b
                          @ ( case_l2363047217959649370st_a_b @ nil_li5528507983314571054st_a_b
                            @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_l7524743740370447070st_a_b @ T @ nil_li5528507983314571054st_a_b ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_1001_transpose_Oelims,axiom,
    ! [X2: list_l3733004638376709294st_a_b,Y2: list_l3733004638376709294st_a_b] :
      ( ( ( transp26204301798245554st_a_b @ X2 )
        = Y2 )
     => ( ( ( X2 = nil_li5528507983314571054st_a_b )
         => ( Y2 != nil_li5528507983314571054st_a_b ) )
       => ( ! [Xss: list_l3733004638376709294st_a_b] :
              ( ( X2
                = ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ Xss ) )
             => ( Y2
               != ( transp26204301798245554st_a_b @ Xss ) ) )
         => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Xss: list_l3733004638376709294st_a_b] :
                ( ( X2
                  = ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xss ) )
               => ( Y2
                 != ( cons_l7524743740370447070st_a_b
                    @ ( cons_P2018802354021969880st_a_b @ X
                      @ ( concat588675724576574691st_a_b
                        @ ( map_li1268991901115354180st_a_b
                          @ ( case_l2687411144252764896st_a_b @ nil_Pr523822035851181096st_a_b
                            @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_P2018802354021969880st_a_b @ H2 @ nil_Pr523822035851181096st_a_b ) )
                          @ Xss ) ) )
                    @ ( transp26204301798245554st_a_b
                      @ ( cons_l7524743740370447070st_a_b @ Xs2
                        @ ( concat4126514578872141417st_a_b
                          @ ( map_li2046058465311281098st_a_b
                            @ ( case_l2363047217959649370st_a_b @ nil_li5528507983314571054st_a_b
                              @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_l7524743740370447070st_a_b @ T @ nil_li5528507983314571054st_a_b ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_1002_transpose_Osimps_I3_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Xss2: list_l3733004638376709294st_a_b] :
      ( ( transp26204301798245554st_a_b @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Xss2 ) )
      = ( cons_l7524743740370447070st_a_b
        @ ( cons_P2018802354021969880st_a_b @ X2
          @ ( concat588675724576574691st_a_b
            @ ( map_li1268991901115354180st_a_b
              @ ( case_l2687411144252764896st_a_b @ nil_Pr523822035851181096st_a_b
                @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_P2018802354021969880st_a_b @ H2 @ nil_Pr523822035851181096st_a_b ) )
              @ Xss2 ) ) )
        @ ( transp26204301798245554st_a_b
          @ ( cons_l7524743740370447070st_a_b @ Xs
            @ ( concat4126514578872141417st_a_b
              @ ( map_li2046058465311281098st_a_b
                @ ( case_l2363047217959649370st_a_b @ nil_li5528507983314571054st_a_b
                  @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_l7524743740370447070st_a_b @ T @ nil_li5528507983314571054st_a_b ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_1003_conj__subset__def,axiom,
    ! [A: set_a,P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ A
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
        & ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1004_transpose_Opelims,axiom,
    ! [X2: list_l3733004638376709294st_a_b,Y2: list_l3733004638376709294st_a_b] :
      ( ( ( transp26204301798245554st_a_b @ X2 )
        = Y2 )
     => ( ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ X2 )
       => ( ( ( X2 = nil_li5528507983314571054st_a_b )
           => ( ( Y2 = nil_li5528507983314571054st_a_b )
             => ~ ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ nil_li5528507983314571054st_a_b ) ) )
         => ( ! [Xss: list_l3733004638376709294st_a_b] :
                ( ( X2
                  = ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ Xss ) )
               => ( ( Y2
                    = ( transp26204301798245554st_a_b @ Xss ) )
                 => ~ ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ ( cons_l7524743740370447070st_a_b @ nil_Pr523822035851181096st_a_b @ Xss ) ) ) )
           => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b,Xss: list_l3733004638376709294st_a_b] :
                  ( ( X2
                    = ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xss ) )
                 => ( ( Y2
                      = ( cons_l7524743740370447070st_a_b
                        @ ( cons_P2018802354021969880st_a_b @ X
                          @ ( concat588675724576574691st_a_b
                            @ ( map_li1268991901115354180st_a_b
                              @ ( case_l2687411144252764896st_a_b @ nil_Pr523822035851181096st_a_b
                                @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_P2018802354021969880st_a_b @ H2 @ nil_Pr523822035851181096st_a_b ) )
                              @ Xss ) ) )
                        @ ( transp26204301798245554st_a_b
                          @ ( cons_l7524743740370447070st_a_b @ Xs2
                            @ ( concat4126514578872141417st_a_b
                              @ ( map_li2046058465311281098st_a_b
                                @ ( case_l2363047217959649370st_a_b @ nil_li5528507983314571054st_a_b
                                  @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_l7524743740370447070st_a_b @ T @ nil_li5528507983314571054st_a_b ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_1005_transpose_Opsimps_I3_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Xss2: list_l3733004638376709294st_a_b] :
      ( ( accp_l7938954131099498039st_a_b @ transp8025728776616114647st_a_b @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Xss2 ) )
     => ( ( transp26204301798245554st_a_b @ ( cons_l7524743740370447070st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Xss2 ) )
        = ( cons_l7524743740370447070st_a_b
          @ ( cons_P2018802354021969880st_a_b @ X2
            @ ( concat588675724576574691st_a_b
              @ ( map_li1268991901115354180st_a_b
                @ ( case_l2687411144252764896st_a_b @ nil_Pr523822035851181096st_a_b
                  @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_P2018802354021969880st_a_b @ H2 @ nil_Pr523822035851181096st_a_b ) )
                @ Xss2 ) ) )
          @ ( transp26204301798245554st_a_b
            @ ( cons_l7524743740370447070st_a_b @ Xs
              @ ( concat4126514578872141417st_a_b
                @ ( map_li2046058465311281098st_a_b
                  @ ( case_l2363047217959649370st_a_b @ nil_li5528507983314571054st_a_b
                    @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_l7524743740370447070st_a_b @ T @ nil_li5528507983314571054st_a_b ) )
                  @ Xss2 ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_1006_image__Fpow__mono,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( image_4065140400681997415list_a @ F @ A ) @ B )
     => ( ord_le8877086941679407844list_a @ ( image_8488307704784700519list_a @ ( image_4065140400681997415list_a @ F ) @ ( finite4368465842446244488st_a_b @ A ) ) @ ( finite_Fpow_list_a @ B ) ) ) ).

% image_Fpow_mono
thf(fact_1007_transpose__aux__filter__head,axiom,
    ! [Xss2: list_l3733004638376709294st_a_b] :
      ( ( concat588675724576574691st_a_b
        @ ( map_li1268991901115354180st_a_b
          @ ( case_l2687411144252764896st_a_b @ nil_Pr523822035851181096st_a_b
            @ ^ [H2: produc2579390649552321826st_a_b,T: list_P2281458131032066856st_a_b] : ( cons_P2018802354021969880st_a_b @ H2 @ nil_Pr523822035851181096st_a_b ) )
          @ Xss2 ) )
      = ( map_li6330388269032553534st_a_b @ hd_Pro4070172631639903629st_a_b
        @ ( filter5139274660221957797st_a_b
          @ ^ [Ys4: list_P2281458131032066856st_a_b] : ( Ys4 != nil_Pr523822035851181096st_a_b )
          @ Xss2 ) ) ) ).

% transpose_aux_filter_head
thf(fact_1008_filter__True,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
         => ( P @ X ) )
     => ( ( filter4355500224637553439st_a_b @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_1009_filter__True,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( P @ X ) )
     => ( ( filter_a @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_1010_set__filter,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Xs: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( filter4355500224637553439st_a_b @ P @ Xs ) )
      = ( collec3957028476971440141st_a_b
        @ ^ [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
            & ( P @ X4 ) ) ) ) ).

% set_filter
thf(fact_1011_set__filter,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( set_a2 @ ( filter_a @ P @ Xs ) )
      = ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
            & ( P @ X4 ) ) ) ) ).

% set_filter
thf(fact_1012_filter__False,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
         => ~ ( P @ X ) )
     => ( ( filter4355500224637553439st_a_b @ P @ Xs )
        = nil_Pr523822035851181096st_a_b ) ) ).

% filter_False
thf(fact_1013_filter__False,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ~ ( P @ X ) )
     => ( ( filter_a @ P @ Xs )
        = nil_a ) ) ).

% filter_False
thf(fact_1014_filter__cong,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o,Q: produc2579390649552321826st_a_b > $o] :
      ( ( Xs = Ys )
     => ( ! [X: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
           => ( ( P @ X )
              = ( Q @ X ) ) )
       => ( ( filter4355500224637553439st_a_b @ P @ Xs )
          = ( filter4355500224637553439st_a_b @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_1015_filter__cong,axiom,
    ! [Xs: list_a,Ys: list_a,P: a > $o,Q: a > $o] :
      ( ( Xs = Ys )
     => ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Ys ) )
           => ( ( P @ X )
              = ( Q @ X ) ) )
       => ( ( filter_a @ P @ Xs )
          = ( filter_a @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_1016_filter__id__conv,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( filter4355500224637553439st_a_b @ P @ Xs )
        = Xs )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ( P @ X4 ) ) ) ) ).

% filter_id_conv
thf(fact_1017_filter__id__conv,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( ( filter_a @ P @ Xs )
        = Xs )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( P @ X4 ) ) ) ) ).

% filter_id_conv
thf(fact_1018_filter_Osimps_I2_J,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( P @ X2 )
       => ( ( filter4355500224637553439st_a_b @ P @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = ( cons_P2018802354021969880st_a_b @ X2 @ ( filter4355500224637553439st_a_b @ P @ Xs ) ) ) )
      & ( ~ ( P @ X2 )
       => ( ( filter4355500224637553439st_a_b @ P @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
          = ( filter4355500224637553439st_a_b @ P @ Xs ) ) ) ) ).

% filter.simps(2)
thf(fact_1019_filter__empty__conv,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( filter4355500224637553439st_a_b @ P @ Xs )
        = nil_Pr523822035851181096st_a_b )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ~ ( P @ X4 ) ) ) ) ).

% filter_empty_conv
thf(fact_1020_filter__empty__conv,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( ( filter_a @ P @ Xs )
        = nil_a )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ~ ( P @ X4 ) ) ) ) ).

% filter_empty_conv
thf(fact_1021_empty__filter__conv,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Xs: list_P2281458131032066856st_a_b] :
      ( ( nil_Pr523822035851181096st_a_b
        = ( filter4355500224637553439st_a_b @ P @ Xs ) )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ~ ( P @ X4 ) ) ) ) ).

% empty_filter_conv
thf(fact_1022_empty__filter__conv,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( nil_a
        = ( filter_a @ P @ Xs ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ~ ( P @ X4 ) ) ) ) ).

% empty_filter_conv
thf(fact_1023_filter__is__subset,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Xs: list_P2281458131032066856st_a_b] : ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ ( filter4355500224637553439st_a_b @ P @ Xs ) ) @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ).

% filter_is_subset
thf(fact_1024_filter__is__subset,axiom,
    ! [P: a > $o,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( filter_a @ P @ Xs ) ) @ ( set_a2 @ Xs ) ) ).

% filter_is_subset
thf(fact_1025_filter__merge,axiom,
    ! [Cmp: compar7542523840845723048st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o] :
      ( ( sortin3200817720516614760st_a_b @ Cmp @ Xs )
     => ( ( sortin3200817720516614760st_a_b @ Cmp @ Ys )
       => ( ( filter4355500224637553439st_a_b @ P @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) )
          = ( sortin5994461831780029029st_a_b @ Cmp @ ( filter4355500224637553439st_a_b @ P @ Xs ) @ ( filter4355500224637553439st_a_b @ P @ Ys ) ) ) ) ) ).

% filter_merge
thf(fact_1026_Cons__eq__filterD,axiom,
    ! [X2: a,Xs: list_a,P: a > $o,Ys: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( filter_a @ P @ Ys ) )
     => ? [Us: list_a,Vs: list_a] :
          ( ( Ys
            = ( append_a @ Us @ ( cons_a @ X2 @ Vs ) ) )
          & ! [X3: a] :
              ( ( member_a @ X3 @ ( set_a2 @ Us ) )
             => ~ ( P @ X3 ) )
          & ( P @ X2 )
          & ( Xs
            = ( filter_a @ P @ Vs ) ) ) ) ).

% Cons_eq_filterD
thf(fact_1027_Cons__eq__filterD,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
        = ( filter4355500224637553439st_a_b @ P @ Ys ) )
     => ? [Us: list_P2281458131032066856st_a_b,Vs: list_P2281458131032066856st_a_b] :
          ( ( Ys
            = ( append8668862642614166333st_a_b @ Us @ ( cons_P2018802354021969880st_a_b @ X2 @ Vs ) ) )
          & ! [X3: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Us ) )
             => ~ ( P @ X3 ) )
          & ( P @ X2 )
          & ( Xs
            = ( filter4355500224637553439st_a_b @ P @ Vs ) ) ) ) ).

% Cons_eq_filterD
thf(fact_1028_filter__eq__ConsD,axiom,
    ! [P: a > $o,Ys: list_a,X2: a,Xs: list_a] :
      ( ( ( filter_a @ P @ Ys )
        = ( cons_a @ X2 @ Xs ) )
     => ? [Us: list_a,Vs: list_a] :
          ( ( Ys
            = ( append_a @ Us @ ( cons_a @ X2 @ Vs ) ) )
          & ! [X3: a] :
              ( ( member_a @ X3 @ ( set_a2 @ Us ) )
             => ~ ( P @ X3 ) )
          & ( P @ X2 )
          & ( Xs
            = ( filter_a @ P @ Vs ) ) ) ) ).

% filter_eq_ConsD
thf(fact_1029_filter__eq__ConsD,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Ys: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( filter4355500224637553439st_a_b @ P @ Ys )
        = ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
     => ? [Us: list_P2281458131032066856st_a_b,Vs: list_P2281458131032066856st_a_b] :
          ( ( Ys
            = ( append8668862642614166333st_a_b @ Us @ ( cons_P2018802354021969880st_a_b @ X2 @ Vs ) ) )
          & ! [X3: produc2579390649552321826st_a_b] :
              ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Us ) )
             => ~ ( P @ X3 ) )
          & ( P @ X2 )
          & ( Xs
            = ( filter4355500224637553439st_a_b @ P @ Vs ) ) ) ) ).

% filter_eq_ConsD
thf(fact_1030_Cons__eq__filter__iff,axiom,
    ! [X2: a,Xs: list_a,P: a > $o,Ys: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( filter_a @ P @ Ys ) )
      = ( ? [Us3: list_a,Vs3: list_a] :
            ( ( Ys
              = ( append_a @ Us3 @ ( cons_a @ X2 @ Vs3 ) ) )
            & ! [X4: a] :
                ( ( member_a @ X4 @ ( set_a2 @ Us3 ) )
               => ~ ( P @ X4 ) )
            & ( P @ X2 )
            & ( Xs
              = ( filter_a @ P @ Vs3 ) ) ) ) ) ).

% Cons_eq_filter_iff
thf(fact_1031_Cons__eq__filter__iff,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,P: produc2579390649552321826st_a_b > $o,Ys: list_P2281458131032066856st_a_b] :
      ( ( ( cons_P2018802354021969880st_a_b @ X2 @ Xs )
        = ( filter4355500224637553439st_a_b @ P @ Ys ) )
      = ( ? [Us3: list_P2281458131032066856st_a_b,Vs3: list_P2281458131032066856st_a_b] :
            ( ( Ys
              = ( append8668862642614166333st_a_b @ Us3 @ ( cons_P2018802354021969880st_a_b @ X2 @ Vs3 ) ) )
            & ! [X4: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Us3 ) )
               => ~ ( P @ X4 ) )
            & ( P @ X2 )
            & ( Xs
              = ( filter4355500224637553439st_a_b @ P @ Vs3 ) ) ) ) ) ).

% Cons_eq_filter_iff
thf(fact_1032_filter__eq__Cons__iff,axiom,
    ! [P: a > $o,Ys: list_a,X2: a,Xs: list_a] :
      ( ( ( filter_a @ P @ Ys )
        = ( cons_a @ X2 @ Xs ) )
      = ( ? [Us3: list_a,Vs3: list_a] :
            ( ( Ys
              = ( append_a @ Us3 @ ( cons_a @ X2 @ Vs3 ) ) )
            & ! [X4: a] :
                ( ( member_a @ X4 @ ( set_a2 @ Us3 ) )
               => ~ ( P @ X4 ) )
            & ( P @ X2 )
            & ( Xs
              = ( filter_a @ P @ Vs3 ) ) ) ) ) ).

% filter_eq_Cons_iff
thf(fact_1033_filter__eq__Cons__iff,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,Ys: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( ( filter4355500224637553439st_a_b @ P @ Ys )
        = ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( ? [Us3: list_P2281458131032066856st_a_b,Vs3: list_P2281458131032066856st_a_b] :
            ( ( Ys
              = ( append8668862642614166333st_a_b @ Us3 @ ( cons_P2018802354021969880st_a_b @ X2 @ Vs3 ) ) )
            & ! [X4: produc2579390649552321826st_a_b] :
                ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Us3 ) )
               => ~ ( P @ X4 ) )
            & ( P @ X2 )
            & ( Xs
              = ( filter4355500224637553439st_a_b @ P @ Vs3 ) ) ) ) ) ).

% filter_eq_Cons_iff
thf(fact_1034_Fpow__mono,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).

% Fpow_mono
thf(fact_1035_cSup__eq__maximum,axiom,
    ! [Z: set_a,X5: set_set_a] :
      ( ( member_set_a @ Z @ X5 )
     => ( ! [X: set_a] :
            ( ( member_set_a @ X @ X5 )
           => ( ord_less_eq_set_a @ X @ Z ) )
       => ( ( comple2307003609928055243_set_a @ X5 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1036_hd__Cons__tl,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ( cons_P2018802354021969880st_a_b @ ( hd_Pro4070172631639903629st_a_b @ Xs ) @ ( tl_Pro4179349441156287377st_a_b @ Xs ) )
        = Xs ) ) ).

% hd_Cons_tl
thf(fact_1037_list_Ocollapse,axiom,
    ! [List: list_P2281458131032066856st_a_b] :
      ( ( List != nil_Pr523822035851181096st_a_b )
     => ( ( cons_P2018802354021969880st_a_b @ ( hd_Pro4070172631639903629st_a_b @ List ) @ ( tl_Pro4179349441156287377st_a_b @ List ) )
        = List ) ) ).

% list.collapse
thf(fact_1038_list_Osel_I3_J,axiom,
    ! [X21: produc2579390649552321826st_a_b,X22: list_P2281458131032066856st_a_b] :
      ( ( tl_Pro4179349441156287377st_a_b @ ( cons_P2018802354021969880st_a_b @ X21 @ X22 ) )
      = X22 ) ).

% list.sel(3)
thf(fact_1039_list__set__tl,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( tl_Pro4179349441156287377st_a_b @ Xs ) ) )
     => ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) ) ) ).

% list_set_tl
thf(fact_1040_list__set__tl,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( tl_a @ Xs ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% list_set_tl
thf(fact_1041_Nil__tl,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( nil_Pr523822035851181096st_a_b
        = ( tl_Pro4179349441156287377st_a_b @ Xs ) )
      = ( ( Xs = nil_Pr523822035851181096st_a_b )
        | ? [X4: produc2579390649552321826st_a_b] :
            ( Xs
            = ( cons_P2018802354021969880st_a_b @ X4 @ nil_Pr523822035851181096st_a_b ) ) ) ) ).

% Nil_tl
thf(fact_1042_tl__Nil,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( ( tl_Pro4179349441156287377st_a_b @ Xs )
        = nil_Pr523822035851181096st_a_b )
      = ( ( Xs = nil_Pr523822035851181096st_a_b )
        | ? [X4: produc2579390649552321826st_a_b] :
            ( Xs
            = ( cons_P2018802354021969880st_a_b @ X4 @ nil_Pr523822035851181096st_a_b ) ) ) ) ).

% tl_Nil
thf(fact_1043_list_Oset__sel_I2_J,axiom,
    ! [A2: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( A2 != nil_Pr523822035851181096st_a_b )
     => ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ ( tl_Pro4179349441156287377st_a_b @ A2 ) ) )
       => ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ A2 ) ) ) ) ).

% list.set_sel(2)
thf(fact_1044_list_Oset__sel_I2_J,axiom,
    ! [A2: list_a,X2: a] :
      ( ( A2 != nil_a )
     => ( ( member_a @ X2 @ ( set_a2 @ ( tl_a @ A2 ) ) )
       => ( member_a @ X2 @ ( set_a2 @ A2 ) ) ) ) ).

% list.set_sel(2)
thf(fact_1045_list_Oexhaust__sel,axiom,
    ! [List: list_P2281458131032066856st_a_b] :
      ( ( List != nil_Pr523822035851181096st_a_b )
     => ( List
        = ( cons_P2018802354021969880st_a_b @ ( hd_Pro4070172631639903629st_a_b @ List ) @ ( tl_Pro4179349441156287377st_a_b @ List ) ) ) ) ).

% list.exhaust_sel
thf(fact_1046_rotate1__hd__tl,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( Xs != nil_Pr523822035851181096st_a_b )
     => ( ( rotate4984794991180899445st_a_b @ Xs )
        = ( append8668862642614166333st_a_b @ ( tl_Pro4179349441156287377st_a_b @ Xs ) @ ( cons_P2018802354021969880st_a_b @ ( hd_Pro4070172631639903629st_a_b @ Xs ) @ nil_Pr523822035851181096st_a_b ) ) ) ) ).

% rotate1_hd_tl
thf(fact_1047_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_1048_empty__iff,axiom,
    ! [C2: a] :
      ~ ( member_a @ C2 @ bot_bot_set_a ) ).

% empty_iff
thf(fact_1049_image__is__empty,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( ( image_4065140400681997415list_a @ F @ A )
        = bot_bot_set_list_a )
      = ( A = bot_bo6763464233164786030st_a_b ) ) ).

% image_is_empty
thf(fact_1050_empty__is__image,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( bot_bot_set_list_a
        = ( image_4065140400681997415list_a @ F @ A ) )
      = ( A = bot_bo6763464233164786030st_a_b ) ) ).

% empty_is_image
thf(fact_1051_image__empty,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a] :
      ( ( image_4065140400681997415list_a @ F @ bot_bo6763464233164786030st_a_b )
      = bot_bot_set_list_a ) ).

% image_empty
thf(fact_1052_empty__subsetI,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% empty_subsetI
thf(fact_1053_subset__empty,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_1054_singletonI,axiom,
    ! [A2: a] : ( member_a @ A2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_1055_set__rotate1,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( rotate4984794991180899445st_a_b @ Xs ) )
      = ( set_Pr6156699709417984823st_a_b @ Xs ) ) ).

% set_rotate1
thf(fact_1056_set__rotate1,axiom,
    ! [Xs: list_a] :
      ( ( set_a2 @ ( rotate1_a @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% set_rotate1
thf(fact_1057_set__empty,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( ( set_Pr6156699709417984823st_a_b @ Xs )
        = bot_bo6763464233164786030st_a_b )
      = ( Xs = nil_Pr523822035851181096st_a_b ) ) ).

% set_empty
thf(fact_1058_set__empty,axiom,
    ! [Xs: list_a] :
      ( ( ( set_a2 @ Xs )
        = bot_bot_set_a )
      = ( Xs = nil_a ) ) ).

% set_empty
thf(fact_1059_set__empty2,axiom,
    ! [Xs: list_P2281458131032066856st_a_b] :
      ( ( bot_bo6763464233164786030st_a_b
        = ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( Xs = nil_Pr523822035851181096st_a_b ) ) ).

% set_empty2
thf(fact_1060_set__empty2,axiom,
    ! [Xs: list_a] :
      ( ( bot_bot_set_a
        = ( set_a2 @ Xs ) )
      = ( Xs = nil_a ) ) ).

% set_empty2
thf(fact_1061_singleton__insert__inj__eq,axiom,
    ! [B2: a,A2: a,A: set_a] :
      ( ( ( insert_a2 @ B2 @ bot_bot_set_a )
        = ( insert_a2 @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_a @ A @ ( insert_a2 @ B2 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1062_singleton__insert__inj__eq_H,axiom,
    ! [A2: a,A: set_a,B2: a] :
      ( ( ( insert_a2 @ A2 @ A )
        = ( insert_a2 @ B2 @ bot_bot_set_a ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_a @ A @ ( insert_a2 @ B2 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1063_subset__emptyI,axiom,
    ! [A: set_a] :
      ( ! [X: a] :
          ~ ( member_a @ X @ A )
     => ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).

% subset_emptyI
thf(fact_1064_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
     => ( A2 = bot_bot_set_a ) ) ).

% bot.extremum_uniqueI
thf(fact_1065_bot_Oextremum__unique,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
      = ( A2 = bot_bot_set_a ) ) ).

% bot.extremum_unique
thf(fact_1066_bot_Oextremum,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).

% bot.extremum
thf(fact_1067_singleton__iff,axiom,
    ! [B2: a,A2: a] :
      ( ( member_a @ B2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1068_singletonD,axiom,
    ! [B2: a,A2: a] :
      ( ( member_a @ B2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1069_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X4: a] : ( member_a @ X4 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_1070_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y3: a] :
          ~ ( member_a @ Y3 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_1071_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_1072_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_1073_less__eq__Sup,axiom,
    ! [A: set_set_a,U2: set_a] :
      ( ! [V3: set_a] :
          ( ( member_set_a @ V3 @ A )
         => ( ord_less_eq_set_a @ U2 @ V3 ) )
     => ( ( A != bot_bot_set_set_a )
       => ( ord_less_eq_set_a @ U2 @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).

% less_eq_Sup
thf(fact_1074_cSup__least,axiom,
    ! [X5: set_set_a,Z: set_a] :
      ( ( X5 != bot_bot_set_set_a )
     => ( ! [X: set_a] :
            ( ( member_set_a @ X @ X5 )
           => ( ord_less_eq_set_a @ X @ Z ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X5 ) @ Z ) ) ) ).

% cSup_least
thf(fact_1075_cSup__eq__non__empty,axiom,
    ! [X5: set_set_a,A2: set_a] :
      ( ( X5 != bot_bot_set_set_a )
     => ( ! [X: set_a] :
            ( ( member_set_a @ X @ X5 )
           => ( ord_less_eq_set_a @ X @ A2 ) )
       => ( ! [Y3: set_a] :
              ( ! [X3: set_a] :
                  ( ( member_set_a @ X3 @ X5 )
                 => ( ord_less_eq_set_a @ X3 @ Y3 ) )
             => ( ord_less_eq_set_a @ A2 @ Y3 ) )
         => ( ( comple2307003609928055243_set_a @ X5 )
            = A2 ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1076_empty__set,axiom,
    ( bot_bo6763464233164786030st_a_b
    = ( set_Pr6156699709417984823st_a_b @ nil_Pr523822035851181096st_a_b ) ) ).

% empty_set
thf(fact_1077_empty__set,axiom,
    ( bot_bot_set_a
    = ( set_a2 @ nil_a ) ) ).

% empty_set
thf(fact_1078_subset__singletonD,axiom,
    ! [A: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
     => ( ( A = bot_bot_set_a )
        | ( A
          = ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_1079_subset__singleton__iff,axiom,
    ! [X5: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ X5 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
      = ( ( X5 = bot_bot_set_a )
        | ( X5
          = ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_1080_image__constant,axiom,
    ! [X2: produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b,C2: list_a] :
      ( ( member8006451236149131979st_a_b @ X2 @ A )
     => ( ( image_4065140400681997415list_a
          @ ^ [X4: produc2579390649552321826st_a_b] : C2
          @ A )
        = ( insert_list_a2 @ C2 @ bot_bot_set_list_a ) ) ) ).

% image_constant
thf(fact_1081_image__constant__conv,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,C2: list_a] :
      ( ( ( A = bot_bo6763464233164786030st_a_b )
       => ( ( image_4065140400681997415list_a
            @ ^ [X4: produc2579390649552321826st_a_b] : C2
            @ A )
          = bot_bot_set_list_a ) )
      & ( ( A != bot_bo6763464233164786030st_a_b )
       => ( ( image_4065140400681997415list_a
            @ ^ [X4: produc2579390649552321826st_a_b] : C2
            @ A )
          = ( insert_list_a2 @ C2 @ bot_bot_set_list_a ) ) ) ) ).

% image_constant_conv
thf(fact_1082_set__rec,axiom,
    ( set_Pr6156699709417984823st_a_b
    = ( rec_li446784655536826070st_a_b @ bot_bo6763464233164786030st_a_b
      @ ^ [X4: produc2579390649552321826st_a_b,Uu2: list_P2281458131032066856st_a_b] : ( insert7903963420998390514st_a_b @ X4 ) ) ) ).

% set_rec
thf(fact_1083_set__rec,axiom,
    ( set_a2
    = ( rec_list_set_a_a @ bot_bot_set_a
      @ ^ [X4: a,Uu2: list_a] : ( insert_a2 @ X4 ) ) ) ).

% set_rec
thf(fact_1084_the__elem__image__unique,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,F: produc2579390649552321826st_a_b > list_a,X2: produc2579390649552321826st_a_b] :
      ( ( A != bot_bo6763464233164786030st_a_b )
     => ( ! [Y3: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ Y3 @ A )
           => ( ( F @ Y3 )
              = ( F @ X2 ) ) )
       => ( ( the_elem_list_a @ ( image_4065140400681997415list_a @ F @ A ) )
          = ( F @ X2 ) ) ) ) ).

% the_elem_image_unique
thf(fact_1085_SUP__eq__iff,axiom,
    ! [I5: set_a,C2: set_a,F: a > set_a] :
      ( ( I5 != bot_bot_set_a )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I5 )
           => ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
       => ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I5 ) )
            = C2 )
          = ( ! [X4: a] :
                ( ( member_a @ X4 @ I5 )
               => ( ( F @ X4 )
                  = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1086_cSUP__least,axiom,
    ! [A: set_a,F: a > set_a,M: set_a] :
      ( ( A != bot_bot_set_a )
     => ( ! [X: a] :
            ( ( member_a @ X @ A )
           => ( ord_less_eq_set_a @ ( F @ X ) @ M ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ M ) ) ) ).

% cSUP_least
thf(fact_1087_rotate1_Osimps_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( rotate4984794991180899445st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) )
      = ( append8668862642614166333st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ) ).

% rotate1.simps(2)
thf(fact_1088_UNION__singleton__eq__range,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( comple6928918032620976721list_a
        @ ( image_1176824672273543239list_a
          @ ^ [X4: produc2579390649552321826st_a_b] : ( insert_list_a2 @ ( F @ X4 ) @ bot_bot_set_list_a )
          @ A ) )
      = ( image_4065140400681997415list_a @ F @ A ) ) ).

% UNION_singleton_eq_range
thf(fact_1089_bot__empty__eq,axiom,
    ( bot_bot_a_o
    = ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).

% bot_empty_eq
thf(fact_1090_bot__empty__eq2,axiom,
    ( bot_bot_list_a_b_o
    = ( ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ bot_bo6763464233164786030st_a_b ) ) ) ).

% bot_empty_eq2
thf(fact_1091_set__Cons__sing__Nil,axiom,
    ! [A: set_Pr9033091269622839554st_a_b] :
      ( ( set_Co6673716987944833240st_a_b @ A @ ( insert4196886921343744248st_a_b @ nil_Pr523822035851181096st_a_b @ bot_bo6367405917657264116st_a_b ) )
      = ( image_6196736372919613323st_a_b
        @ ^ [X4: produc2579390649552321826st_a_b] : ( cons_P2018802354021969880st_a_b @ X4 @ nil_Pr523822035851181096st_a_b )
        @ A ) ) ).

% set_Cons_sing_Nil
thf(fact_1092_is__singletonI_H,axiom,
    ! [A: set_a] :
      ( ( A != bot_bot_set_a )
     => ( ! [X: a,Y3: a] :
            ( ( member_a @ X @ A )
           => ( ( member_a @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_a @ A ) ) ) ).

% is_singletonI'
thf(fact_1093_Longest__common__prefix__eq__Cons,axiom,
    ! [L: set_li7460651515946730888st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( L != bot_bo6367405917657264116st_a_b )
     => ( ~ ( member1308482598881763409st_a_b @ nil_Pr523822035851181096st_a_b @ L )
       => ( ! [X: list_P2281458131032066856st_a_b] :
              ( ( member1308482598881763409st_a_b @ X @ L )
             => ( ( hd_Pro4070172631639903629st_a_b @ X )
                = X2 ) )
         => ( ( longes6057516002205906636st_a_b @ L )
            = ( cons_P2018802354021969880st_a_b @ X2
              @ ( longes6057516002205906636st_a_b
                @ ( collec3283141540447581331st_a_b
                  @ ^ [Ys4: list_P2281458131032066856st_a_b] : ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Ys4 ) @ L ) ) ) ) ) ) ) ) ).

% Longest_common_prefix_eq_Cons
thf(fact_1094_Longest__common__prefix__eq__Nil,axiom,
    ! [X2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b,L: set_li7460651515946730888st_a_b,Y2: produc2579390649552321826st_a_b,Zs: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Ys ) @ L )
     => ( ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 @ Zs ) @ L )
       => ( ( X2 != Y2 )
         => ( ( longes6057516002205906636st_a_b @ L )
            = nil_Pr523822035851181096st_a_b ) ) ) ) ).

% Longest_common_prefix_eq_Nil
thf(fact_1095_Longest__common__prefix__image__Cons,axiom,
    ! [L: set_li7460651515946730888st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( L != bot_bo6367405917657264116st_a_b )
     => ( ( longes6057516002205906636st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ L ) )
        = ( cons_P2018802354021969880st_a_b @ X2 @ ( longes6057516002205906636st_a_b @ L ) ) ) ) ).

% Longest_common_prefix_image_Cons
thf(fact_1096_shufflesE,axiom,
    ! [Zs: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ Zs @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) )
     => ( ( ( Zs = Xs )
         => ( Ys != nil_Pr523822035851181096st_a_b ) )
       => ( ( ( Zs = Ys )
           => ( Xs != nil_Pr523822035851181096st_a_b ) )
         => ( ! [X: produc2579390649552321826st_a_b,Xs5: list_P2281458131032066856st_a_b] :
                ( ( Xs
                  = ( cons_P2018802354021969880st_a_b @ X @ Xs5 ) )
               => ! [Z5: produc2579390649552321826st_a_b,Zs4: list_P2281458131032066856st_a_b] :
                    ( ( Zs
                      = ( cons_P2018802354021969880st_a_b @ Z5 @ Zs4 ) )
                   => ( ( X = Z5 )
                     => ~ ( member1308482598881763409st_a_b @ Zs4 @ ( shuffl8224214619213514045st_a_b @ Xs5 @ Ys ) ) ) ) )
           => ~ ! [Y3: produc2579390649552321826st_a_b,Ys6: list_P2281458131032066856st_a_b] :
                  ( ( Ys
                    = ( cons_P2018802354021969880st_a_b @ Y3 @ Ys6 ) )
                 => ! [Z5: produc2579390649552321826st_a_b,Zs4: list_P2281458131032066856st_a_b] :
                      ( ( Zs
                        = ( cons_P2018802354021969880st_a_b @ Z5 @ Zs4 ) )
                     => ( ( Y3 = Z5 )
                       => ~ ( member1308482598881763409st_a_b @ Zs4 @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).

% shufflesE
thf(fact_1097_Cons__in__shuffles__leftI,axiom,
    ! [Zs: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Z: produc2579390649552321826st_a_b] :
      ( ( member1308482598881763409st_a_b @ Zs @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) )
     => ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ Z @ Zs ) @ ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ Z @ Xs ) @ Ys ) ) ) ).

% Cons_in_shuffles_leftI
thf(fact_1098_Cons__in__shuffles__rightI,axiom,
    ! [Zs: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Z: produc2579390649552321826st_a_b] :
      ( ( member1308482598881763409st_a_b @ Zs @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) )
     => ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ Z @ Zs ) @ ( shuffl8224214619213514045st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ Z @ Ys ) ) ) ) ).

% Cons_in_shuffles_rightI
thf(fact_1099_Cons__shuffles__subset2,axiom,
    ! [Y2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] : ( ord_le2045451444129209640st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 ) @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) ) @ ( shuffl8224214619213514045st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) ) ).

% Cons_shuffles_subset2
thf(fact_1100_Cons__shuffles__subset1,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] : ( ord_le2045451444129209640st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) ) @ ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) ) ).

% Cons_shuffles_subset1
thf(fact_1101_Cons__in__shuffles__iff,axiom,
    ! [Z: produc2579390649552321826st_a_b,Zs: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ Z @ Zs ) @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) )
      = ( ( ( Xs != nil_Pr523822035851181096st_a_b )
          & ( ( hd_Pro4070172631639903629st_a_b @ Xs )
            = Z )
          & ( member1308482598881763409st_a_b @ Zs @ ( shuffl8224214619213514045st_a_b @ ( tl_Pro4179349441156287377st_a_b @ Xs ) @ Ys ) ) )
        | ( ( Ys != nil_Pr523822035851181096st_a_b )
          & ( ( hd_Pro4070172631639903629st_a_b @ Ys )
            = Z )
          & ( member1308482598881763409st_a_b @ Zs @ ( shuffl8224214619213514045st_a_b @ Xs @ ( tl_Pro4179349441156287377st_a_b @ Ys ) ) ) ) ) ) ).

% Cons_in_shuffles_iff
thf(fact_1102_shuffles_Opelims,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: set_li7460651515946730888st_a_b] :
      ( ( ( shuffl8224214619213514045st_a_b @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ X2 @ Xa ) )
       => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
           => ( ( Y2
                = ( insert4196886921343744248st_a_b @ Xa @ bot_bo6367405917657264116st_a_b ) )
             => ~ ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ nil_Pr523822035851181096st_a_b @ Xa ) ) ) )
         => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
             => ( ( Y2
                  = ( insert4196886921343744248st_a_b @ X2 @ bot_bo6367405917657264116st_a_b ) )
               => ~ ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ X2 @ nil_Pr523822035851181096st_a_b ) ) ) )
           => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                  ( ( X2
                    = ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) )
                 => ! [Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                      ( ( Xa
                        = ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
                     => ( ( Y2
                          = ( sup_su3629596151668714332st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ X ) @ ( shuffl8224214619213514045st_a_b @ Xs2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ Y3 ) @ ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) )
                       => ~ ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) ) ) ) ) ) ) ) ).

% shuffles.pelims
thf(fact_1103_shuffles_Opsimps_I3_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( accp_P4272997315293813680st_a_b @ shuffl5911578961019143948st_a_b @ ( produc717113118580727703st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) )
     => ( ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
        = ( sup_su3629596151668714332st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ ( shuffl8224214619213514045st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) ) @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 ) @ ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) ) ) ) ) ).

% shuffles.psimps(3)
thf(fact_1104_UnCI,axiom,
    ! [C2: a,B: set_a,A: set_a] :
      ( ( ~ ( member_a @ C2 @ B )
       => ( member_a @ C2 @ A ) )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnCI
thf(fact_1105_Un__iff,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
      = ( ( member_a @ C2 @ A )
        | ( member_a @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1106_Un__subset__iff,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 )
      = ( ( ord_less_eq_set_a @ A @ C3 )
        & ( ord_less_eq_set_a @ B @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1107_set__append,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( append8668862642614166333st_a_b @ Xs @ Ys ) )
      = ( sup_su6420029347046086870st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) ) ) ).

% set_append
thf(fact_1108_set__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( append_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_append
thf(fact_1109_set__shuffles,axiom,
    ! [Zs: list_P2281458131032066856st_a_b,Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( member1308482598881763409st_a_b @ Zs @ ( shuffl8224214619213514045st_a_b @ Xs @ Ys ) )
     => ( ( set_Pr6156699709417984823st_a_b @ Zs )
        = ( sup_su6420029347046086870st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) ) ) ) ).

% set_shuffles
thf(fact_1110_set__shuffles,axiom,
    ! [Zs: list_a,Xs: list_a,Ys: list_a] :
      ( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
     => ( ( set_a2 @ Zs )
        = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ) ).

% set_shuffles
thf(fact_1111_image__Un,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,B: set_Pr9033091269622839554st_a_b] :
      ( ( image_4065140400681997415list_a @ F @ ( sup_su6420029347046086870st_a_b @ A @ B ) )
      = ( sup_sup_set_list_a @ ( image_4065140400681997415list_a @ F @ A ) @ ( image_4065140400681997415list_a @ F @ B ) ) ) ).

% image_Un
thf(fact_1112_UnE,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
     => ( ~ ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% UnE
thf(fact_1113_UnI1,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ A )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1114_UnI2,axiom,
    ! [C2: a,B: set_a,A: set_a] :
      ( ( member_a @ C2 @ B )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1115_Un__mono,axiom,
    ! [A: set_a,C3: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A @ C3 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C3 @ D ) ) ) ) ).

% Un_mono
thf(fact_1116_Un__least,axiom,
    ! [A: set_a,C3: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ C3 )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 ) ) ) ).

% Un_least
thf(fact_1117_Un__upper1,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).

% Un_upper1
thf(fact_1118_Un__upper2,axiom,
    ! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).

% Un_upper2
thf(fact_1119_Un__absorb1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( sup_sup_set_a @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_1120_Un__absorb2,axiom,
    ! [B: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( sup_sup_set_a @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_1121_subset__UnE,axiom,
    ! [C3: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A @ B ) )
     => ~ ! [A7: set_a] :
            ( ( ord_less_eq_set_a @ A7 @ A )
           => ! [B8: set_a] :
                ( ( ord_less_eq_set_a @ B8 @ B )
               => ( C3
                 != ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_1122_subset__Un__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( sup_sup_set_a @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_1123_Un__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A3 )
              | ( member_a @ X4 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_1124_set__merge,axiom,
    ! [Xs: list_a,Ys: list_a,Cmp: comparator_a] :
      ( ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
      = ( set_a2 @ ( sorting_merge_a @ Cmp @ Xs @ Ys ) ) ) ).

% set_merge
thf(fact_1125_set__merge,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b,Cmp: compar7542523840845723048st_a_b] :
      ( ( sup_su6420029347046086870st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) )
      = ( set_Pr6156699709417984823st_a_b @ ( sortin5994461831780029029st_a_b @ Cmp @ Xs @ Ys ) ) ) ).

% set_merge
thf(fact_1126_shuffles_Osimps_I3_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,Y2: produc2579390649552321826st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) )
      = ( sup_su3629596151668714332st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 ) @ ( shuffl8224214619213514045st_a_b @ Xs @ ( cons_P2018802354021969880st_a_b @ Y2 @ Ys ) ) ) @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ Y2 ) @ ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ Ys ) ) ) ) ).

% shuffles.simps(3)
thf(fact_1127_shuffles_Oelims,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Xa: list_P2281458131032066856st_a_b,Y2: set_li7460651515946730888st_a_b] :
      ( ( ( shuffl8224214619213514045st_a_b @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = nil_Pr523822035851181096st_a_b )
         => ( Y2
           != ( insert4196886921343744248st_a_b @ Xa @ bot_bo6367405917657264116st_a_b ) ) )
       => ( ( ( Xa = nil_Pr523822035851181096st_a_b )
           => ( Y2
             != ( insert4196886921343744248st_a_b @ X2 @ bot_bo6367405917657264116st_a_b ) ) )
         => ~ ! [X: produc2579390649552321826st_a_b,Xs2: list_P2281458131032066856st_a_b] :
                ( ( X2
                  = ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) )
               => ! [Y3: produc2579390649552321826st_a_b,Ys2: list_P2281458131032066856st_a_b] :
                    ( ( Xa
                      = ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) )
                   => ( Y2
                     != ( sup_su3629596151668714332st_a_b @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ X ) @ ( shuffl8224214619213514045st_a_b @ Xs2 @ ( cons_P2018802354021969880st_a_b @ Y3 @ Ys2 ) ) ) @ ( image_5474055693525301253st_a_b @ ( cons_P2018802354021969880st_a_b @ Y3 ) @ ( shuffl8224214619213514045st_a_b @ ( cons_P2018802354021969880st_a_b @ X @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).

% shuffles.elims
thf(fact_1128_sup_Obounded__iff,axiom,
    ! [B2: set_a,C2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_set_a @ B2 @ A2 )
        & ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1129_le__sup__iff,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_eq_set_a @ X2 @ Z )
        & ( ord_less_eq_set_a @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_1130_sup__Un__eq2,axiom,
    ! [R2: set_Pr9033091269622839554st_a_b,S3: set_Pr9033091269622839554st_a_b] :
      ( ( sup_sup_list_a_b_o
        @ ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: list_a,Y5: b] : ( member8006451236149131979st_a_b @ ( produc4781227316648555538st_a_b @ X4 @ Y5 ) @ ( sup_su6420029347046086870st_a_b @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_1131_sup__set__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ( sup_sup_a_o
            @ ^ [X4: a] : ( member_a @ X4 @ A3 )
            @ ^ [X4: a] : ( member_a @ X4 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_1132_sup__Un__eq,axiom,
    ! [R2: set_a,S3: set_a] :
      ( ( sup_sup_a_o
        @ ^ [X4: a] : ( member_a @ X4 @ R2 )
        @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
      = ( ^ [X4: a] : ( member_a @ X4 @ ( sup_sup_set_a @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_1133_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_1134_inf__sup__ord_I3_J,axiom,
    ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_1135_le__supE,axiom,
    ! [A2: set_a,B2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 )
     => ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
         => ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_1136_le__supI,axiom,
    ! [A2: set_a,X2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X2 )
     => ( ( ord_less_eq_set_a @ B2 @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_1137_sup__ge1,axiom,
    ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).

% sup_ge1
thf(fact_1138_sup__ge2,axiom,
    ! [Y2: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X2 @ Y2 ) ) ).

% sup_ge2
thf(fact_1139_le__supI1,axiom,
    ! [X2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A2 )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1140_le__supI2,axiom,
    ! [X2: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ B2 )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1141_sup_Omono,axiom,
    ! [C2: set_a,A2: set_a,D2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ A2 )
     => ( ( ord_less_eq_set_a @ D2 @ B2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1142_sup__mono,axiom,
    ! [A2: set_a,C2: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1143_sup__least,axiom,
    ! [Y2: set_a,X2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ Y2 @ X2 )
     => ( ( ord_less_eq_set_a @ Z @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_1144_le__iff__sup,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( sup_sup_set_a @ X4 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_1145_sup_OorderE,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1146_sup_OorderI,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2
        = ( sup_sup_set_a @ A2 @ B2 ) )
     => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1147_sup__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y2: set_a] :
      ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X @ ( F @ X @ Y3 ) )
     => ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X @ Y3 ) )
       => ( ! [X: set_a,Y3: set_a,Z5: set_a] :
              ( ( ord_less_eq_set_a @ Y3 @ X )
             => ( ( ord_less_eq_set_a @ Z5 @ X )
               => ( ord_less_eq_set_a @ ( F @ Y3 @ Z5 ) @ X ) ) )
         => ( ( sup_sup_set_a @ X2 @ Y2 )
            = ( F @ X2 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_1148_sup_Oabsorb1,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1149_sup_Oabsorb2,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1150_sup__absorb1,axiom,
    ! [Y2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y2 @ X2 )
     => ( ( sup_sup_set_a @ X2 @ Y2 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_1151_sup__absorb2,axiom,
    ! [X2: set_a,Y2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( sup_sup_set_a @ X2 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_1152_sup_OboundedE,axiom,
    ! [B2: set_a,C2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
         => ~ ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1153_sup_OboundedI,axiom,
    ! [B2: set_a,A2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C2 @ A2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1154_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( A4
          = ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1155_sup_Ocobounded1,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1156_sup_Ocobounded2,axiom,
    ! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1157_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_1158_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1159_sup_OcoboundedI1,axiom,
    ! [C2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ A2 )
     => ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1160_sup_OcoboundedI2,axiom,
    ! [C2: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ B2 )
     => ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1161_set__union,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( set_Pr6156699709417984823st_a_b @ ( union_92921776482251292st_a_b @ Xs @ Ys ) )
      = ( sup_su6420029347046086870st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) ) ) ).

% set_union
thf(fact_1162_set__union,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( union_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_union
thf(fact_1163_FieldI2,axiom,
    ! [I3: a,J2: a,R2: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I3 @ J2 ) @ R2 )
     => ( member_a @ J2 @ ( field_a @ R2 ) ) ) ).

% FieldI2
thf(fact_1164_FieldI1,axiom,
    ! [I3: a,J2: a,R2: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I3 @ J2 ) @ R2 )
     => ( member_a @ I3 @ ( field_a @ R2 ) ) ) ).

% FieldI1
thf(fact_1165_mono__Field,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ R @ S )
     => ( ord_less_eq_set_a @ ( field_a @ R ) @ ( field_a @ S ) ) ) ).

% mono_Field
thf(fact_1166_Cons__in__lists__iff,axiom,
    ! [X2: a,Xs: list_a,A: set_a] :
      ( ( member_list_a @ ( cons_a @ X2 @ Xs ) @ ( lists_a @ A ) )
      = ( ( member_a @ X2 @ A )
        & ( member_list_a @ Xs @ ( lists_a @ A ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1167_Cons__in__lists__iff,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ ( lists_6060421456370130710st_a_b @ A ) )
      = ( ( member8006451236149131979st_a_b @ X2 @ A )
        & ( member1308482598881763409st_a_b @ Xs @ ( lists_6060421456370130710st_a_b @ A ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_1168_in__listsI,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ! [X: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
         => ( member8006451236149131979st_a_b @ X @ A ) )
     => ( member1308482598881763409st_a_b @ Xs @ ( lists_6060421456370130710st_a_b @ A ) ) ) ).

% in_listsI
thf(fact_1169_in__listsI,axiom,
    ! [Xs: list_a,A: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( member_a @ X @ A ) )
     => ( member_list_a @ Xs @ ( lists_a @ A ) ) ) ).

% in_listsI
thf(fact_1170_sublist__code_I2_J,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ~ ( sublis640185979109914773st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ nil_Pr523822035851181096st_a_b ) ).

% sublist_code(2)
thf(fact_1171_lists__eq__set,axiom,
    ( lists_6060421456370130710st_a_b
    = ( ^ [A3: set_Pr9033091269622839554st_a_b] :
          ( collec3283141540447581331st_a_b
          @ ^ [Xs4: list_P2281458131032066856st_a_b] : ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs4 ) @ A3 ) ) ) ) ).

% lists_eq_set
thf(fact_1172_lists__eq__set,axiom,
    ( lists_a
    = ( ^ [A3: set_a] :
          ( collect_list_a
          @ ^ [Xs4: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs4 ) @ A3 ) ) ) ) ).

% lists_eq_set
thf(fact_1173_set__mono__sublist,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,Ys: list_P2281458131032066856st_a_b] :
      ( ( sublis640185979109914773st_a_b @ Xs @ Ys )
     => ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs ) @ ( set_Pr6156699709417984823st_a_b @ Ys ) ) ) ).

% set_mono_sublist
thf(fact_1174_set__mono__sublist,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( sublist_a @ Xs @ Ys )
     => ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_mono_sublist
thf(fact_1175_sublist__set__elem,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,A: list_P2281458131032066856st_a_b,B: list_P2281458131032066856st_a_b,X2: produc2579390649552321826st_a_b] :
      ( ( sublis640185979109914773st_a_b @ Xs @ ( append8668862642614166333st_a_b @ A @ B ) )
     => ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
       => ( ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ A ) )
          | ( member8006451236149131979st_a_b @ X2 @ ( set_Pr6156699709417984823st_a_b @ B ) ) ) ) ) ).

% sublist_set_elem
thf(fact_1176_sublist__set__elem,axiom,
    ! [Xs: list_a,A: list_a,B: list_a,X2: a] :
      ( ( sublist_a @ Xs @ ( append_a @ A @ B ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( ( member_a @ X2 @ ( set_a2 @ A ) )
          | ( member_a @ X2 @ ( set_a2 @ B ) ) ) ) ) ).

% sublist_set_elem
thf(fact_1177_in__lists__conv__set,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( member1308482598881763409st_a_b @ Xs @ ( lists_6060421456370130710st_a_b @ A ) )
      = ( ! [X4: produc2579390649552321826st_a_b] :
            ( ( member8006451236149131979st_a_b @ X4 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
           => ( member8006451236149131979st_a_b @ X4 @ A ) ) ) ) ).

% in_lists_conv_set
thf(fact_1178_in__lists__conv__set,axiom,
    ! [Xs: list_a,A: set_a] :
      ( ( member_list_a @ Xs @ ( lists_a @ A ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( member_a @ X4 @ A ) ) ) ) ).

% in_lists_conv_set
thf(fact_1179_in__listsD,axiom,
    ! [Xs: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( member1308482598881763409st_a_b @ Xs @ ( lists_6060421456370130710st_a_b @ A ) )
     => ! [X3: produc2579390649552321826st_a_b] :
          ( ( member8006451236149131979st_a_b @ X3 @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
         => ( member8006451236149131979st_a_b @ X3 @ A ) ) ) ).

% in_listsD
thf(fact_1180_in__listsD,axiom,
    ! [Xs: list_a,A: set_a] :
      ( ( member_list_a @ Xs @ ( lists_a @ A ) )
     => ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( member_a @ X3 @ A ) ) ) ).

% in_listsD
thf(fact_1181_listsE,axiom,
    ! [X2: a,L2: list_a,A: set_a] :
      ( ( member_list_a @ ( cons_a @ X2 @ L2 ) @ ( lists_a @ A ) )
     => ~ ( ( member_a @ X2 @ A )
         => ~ ( member_list_a @ L2 @ ( lists_a @ A ) ) ) ) ).

% listsE
thf(fact_1182_listsE,axiom,
    ! [X2: produc2579390649552321826st_a_b,L2: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ L2 ) @ ( lists_6060421456370130710st_a_b @ A ) )
     => ~ ( ( member8006451236149131979st_a_b @ X2 @ A )
         => ~ ( member1308482598881763409st_a_b @ L2 @ ( lists_6060421456370130710st_a_b @ A ) ) ) ) ).

% listsE
thf(fact_1183_lists_OCons,axiom,
    ! [A2: a,A: set_a,L2: list_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_list_a @ L2 @ ( lists_a @ A ) )
       => ( member_list_a @ ( cons_a @ A2 @ L2 ) @ ( lists_a @ A ) ) ) ) ).

% lists.Cons
thf(fact_1184_lists_OCons,axiom,
    ! [A2: produc2579390649552321826st_a_b,A: set_Pr9033091269622839554st_a_b,L2: list_P2281458131032066856st_a_b] :
      ( ( member8006451236149131979st_a_b @ A2 @ A )
     => ( ( member1308482598881763409st_a_b @ L2 @ ( lists_6060421456370130710st_a_b @ A ) )
       => ( member1308482598881763409st_a_b @ ( cons_P2018802354021969880st_a_b @ A2 @ L2 ) @ ( lists_6060421456370130710st_a_b @ A ) ) ) ) ).

% lists.Cons
thf(fact_1185_sublist__Cons,axiom,
    ! [A: produc2579390649552321826st_a_b,B: list_P2281458131032066856st_a_b,C3: list_P2281458131032066856st_a_b] :
      ( ( sublis640185979109914773st_a_b @ ( cons_P2018802354021969880st_a_b @ A @ B ) @ C3 )
     => ( ( sublis640185979109914773st_a_b @ ( cons_P2018802354021969880st_a_b @ A @ nil_Pr523822035851181096st_a_b ) @ C3 )
        & ( sublis640185979109914773st_a_b @ B @ C3 ) ) ) ).

% sublist_Cons
thf(fact_1186_sublist__Cons__Nil,axiom,
    ! [X2: produc2579390649552321826st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ~ ( sublis640185979109914773st_a_b @ ( cons_P2018802354021969880st_a_b @ X2 @ Xs ) @ nil_Pr523822035851181096st_a_b ) ).

% sublist_Cons_Nil
thf(fact_1187_lists_Osimps,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.simps
thf(fact_1188_lists_Osimps,axiom,
    ! [A2: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( member1308482598881763409st_a_b @ A2 @ ( lists_6060421456370130710st_a_b @ A ) )
      = ( ( A2 = nil_Pr523822035851181096st_a_b )
        | ? [A4: produc2579390649552321826st_a_b,L3: list_P2281458131032066856st_a_b] :
            ( ( A2
              = ( cons_P2018802354021969880st_a_b @ A4 @ L3 ) )
            & ( member8006451236149131979st_a_b @ A4 @ A )
            & ( member1308482598881763409st_a_b @ L3 @ ( lists_6060421456370130710st_a_b @ A ) ) ) ) ) ).

% lists.simps
thf(fact_1189_lists_Ocases,axiom,
    ! [A2: list_a,A: set_a] :
      ( ( member_list_a @ A2 @ ( lists_a @ A ) )
     => ( ( A2 != nil_a )
       => ~ ! [A6: a,L4: list_a] :
              ( ( A2
                = ( cons_a @ A6 @ L4 ) )
             => ( ( member_a @ A6 @ A )
               => ~ ( member_list_a @ L4 @ ( lists_a @ A ) ) ) ) ) ) ).

% lists.cases
thf(fact_1190_lists_Ocases,axiom,
    ! [A2: list_P2281458131032066856st_a_b,A: set_Pr9033091269622839554st_a_b] :
      ( ( member1308482598881763409st_a_b @ A2 @ ( lists_6060421456370130710st_a_b @ A ) )
     => ( ( A2 != nil_Pr523822035851181096st_a_b )
       => ~ ! [A6: produc2579390649552321826st_a_b,L4: list_P2281458131032066856st_a_b] :
              ( ( A2
                = ( cons_P2018802354021969880st_a_b @ A6 @ L4 ) )
             => ( ( member8006451236149131979st_a_b @ A6 @ A )
               => ~ ( member1308482598881763409st_a_b @ L4 @ ( lists_6060421456370130710st_a_b @ A ) ) ) ) ) ) ).

% lists.cases
thf(fact_1191_lists__mono,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le8861187494160871172list_a @ ( lists_a @ A ) @ ( lists_a @ B ) ) ) ).

% lists_mono
thf(fact_1192_lists__image,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b] :
      ( ( lists_list_a @ ( image_4065140400681997415list_a @ F @ A ) )
      = ( image_1297384162028396903list_a @ ( map_Pr7332729320935904934list_a @ F ) @ ( lists_6060421456370130710st_a_b @ A ) ) ) ).

% lists_image
thf(fact_1193_subset__snd__if__hd__notin__fst,axiom,
    ! [Ys: list_P2281458131032066856st_a_b,V4: list_P2281458131032066856st_a_b,B: list_P2281458131032066856st_a_b] :
      ( ( sublis640185979109914773st_a_b @ Ys @ ( append8668862642614166333st_a_b @ V4 @ B ) )
     => ( ~ ( member8006451236149131979st_a_b @ ( hd_Pro4070172631639903629st_a_b @ Ys ) @ ( set_Pr6156699709417984823st_a_b @ V4 ) )
       => ( ( Ys != nil_Pr523822035851181096st_a_b )
         => ( ord_le2879471331418251426st_a_b @ ( set_Pr6156699709417984823st_a_b @ Ys ) @ ( set_Pr6156699709417984823st_a_b @ B ) ) ) ) ) ).

% subset_snd_if_hd_notin_fst
thf(fact_1194_subset__snd__if__hd__notin__fst,axiom,
    ! [Ys: list_a,V4: list_a,B: list_a] :
      ( ( sublist_a @ Ys @ ( append_a @ V4 @ B ) )
     => ( ~ ( member_a @ ( hd_a @ Ys ) @ ( set_a2 @ V4 ) )
       => ( ( Ys != nil_a )
         => ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ ( set_a2 @ B ) ) ) ) ) ).

% subset_snd_if_hd_notin_fst
thf(fact_1195_wf__list__lverts_Oelims_I2_J,axiom,
    ! [X2: list_P2281458131032066856st_a_b] :
      ( ( list_w2225310435451993436ts_a_b @ X2 )
     => ( ( X2 != nil_Pr523822035851181096st_a_b )
       => ~ ! [V3: list_a,E3: b,Xs2: list_P2281458131032066856st_a_b] :
              ( ( X2
                = ( cons_P2018802354021969880st_a_b @ ( produc4781227316648555538st_a_b @ V3 @ E3 ) @ Xs2 ) )
             => ~ ( ( V3 != nil_a )
                  & ! [X3: list_a] :
                      ( ( member_list_a @ X3 @ ( image_4065140400681997415list_a @ product_fst_list_a_b @ ( set_Pr6156699709417984823st_a_b @ Xs2 ) ) )
                     => ( ( inf_inf_set_a @ ( set_a2 @ V3 ) @ ( set_a2 @ X3 ) )
                        = bot_bot_set_a ) )
                  & ( list_w2225310435451993436ts_a_b @ Xs2 ) ) ) ) ) ).

% wf_list_lverts.elims(2)
thf(fact_1196_IntI,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ A )
     => ( ( member_a @ C2 @ B )
       => ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_1197_Int__iff,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
      = ( ( member_a @ C2 @ A )
        & ( member_a @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_1198_inf_Obounded__iff,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
      = ( ( ord_less_eq_set_a @ A2 @ B2 )
        & ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1199_le__inf__iff,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
      = ( ( ord_less_eq_set_a @ X2 @ Y2 )
        & ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_1200_Int__subset__iff,axiom,
    ! [C3: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B ) )
      = ( ( ord_less_eq_set_a @ C3 @ A )
        & ( ord_less_eq_set_a @ C3 @ B ) ) ) ).

% Int_subset_iff
thf(fact_1201_Int__insert__left__if0,axiom,
    ! [A2: a,C3: set_a,B: set_a] :
      ( ~ ( member_a @ A2 @ C3 )
     => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B ) @ C3 )
        = ( inf_inf_set_a @ B @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1202_Int__insert__left__if1,axiom,
    ! [A2: a,C3: set_a,B: set_a] :
      ( ( member_a @ A2 @ C3 )
     => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B ) @ C3 )
        = ( insert_a2 @ A2 @ ( inf_inf_set_a @ B @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1203_Int__insert__right__if0,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B ) )
        = ( inf_inf_set_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1204_Int__insert__right__if1,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B ) )
        = ( insert_a2 @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1205_insert__disjoint_I1_J,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ A ) @ B )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B )
        & ( ( inf_inf_set_a @ A @ B )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1206_insert__disjoint_I2_J,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a2 @ A2 @ A ) @ B ) )
      = ( ~ ( member_a @ A2 @ B )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1207_disjoint__insert_I1_J,axiom,
    ! [B: set_a,A2: a,A: set_a] :
      ( ( ( inf_inf_set_a @ B @ ( insert_a2 @ A2 @ A ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B )
        & ( ( inf_inf_set_a @ B @ A )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1208_disjoint__insert_I2_J,axiom,
    ! [A: set_a,B2: a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A @ ( insert_a2 @ B2 @ B ) ) )
      = ( ~ ( member_a @ B2 @ A )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1209_if__image__distrib,axiom,
    ! [P: produc2579390649552321826st_a_b > $o,F: produc2579390649552321826st_a_b > list_a,G: produc2579390649552321826st_a_b > list_a,S3: set_Pr9033091269622839554st_a_b] :
      ( ( image_4065140400681997415list_a
        @ ^ [X4: produc2579390649552321826st_a_b] : ( if_list_a @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S3 )
      = ( sup_sup_set_list_a @ ( image_4065140400681997415list_a @ F @ ( inf_in7575054441957001712st_a_b @ S3 @ ( collec3957028476971440141st_a_b @ P ) ) )
        @ ( image_4065140400681997415list_a @ G
          @ ( inf_in7575054441957001712st_a_b @ S3
            @ ( collec3957028476971440141st_a_b
              @ ^ [X4: produc2579390649552321826st_a_b] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1210_image__Int__subset,axiom,
    ! [F: produc2579390649552321826st_a_b > list_a,A: set_Pr9033091269622839554st_a_b,B: set_Pr9033091269622839554st_a_b] : ( ord_le8861187494160871172list_a @ ( image_4065140400681997415list_a @ F @ ( inf_in7575054441957001712st_a_b @ A @ B ) ) @ ( inf_inf_set_list_a @ ( image_4065140400681997415list_a @ F @ A ) @ ( image_4065140400681997415list_a @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1211_inf_OcoboundedI2,axiom,
    ! [B2: set_a,C2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ C2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1212_inf_OcoboundedI1,axiom,
    ! [A2: set_a,C2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1213_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( inf_inf_set_a @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_1214_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( inf_inf_set_a @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_1215_inf_Ocobounded2,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1216_inf_Ocobounded1,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1217_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( A4
          = ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_1218_inf__greatest,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( ord_less_eq_set_a @ X2 @ Z )
       => ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1219_inf_OboundedI,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ A2 @ C2 )
       => ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1220_inf_OboundedE,axiom,
    ! [A2: set_a,B2: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ~ ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1221_inf__absorb2,axiom,
    ! [Y2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y2 @ X2 )
     => ( ( inf_inf_set_a @ X2 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_1222_inf__absorb1,axiom,
    ! [X2: set_a,Y2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( inf_inf_set_a @ X2 @ Y2 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_1223_inf_Oabsorb2,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1224_inf_Oabsorb1,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1225_le__iff__inf,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( inf_inf_set_a @ X4 @ Y5 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_1226_inf__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y2: set_a] :
      ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ X )
     => ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ Y3 )
       => ( ! [X: set_a,Y3: set_a,Z5: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ( ord_less_eq_set_a @ X @ Z5 )
               => ( ord_less_eq_set_a @ X @ ( F @ Y3 @ Z5 ) ) ) )
         => ( ( inf_inf_set_a @ X2 @ Y2 )
            = ( F @ X2 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_1227_inf_OorderI,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2
        = ( inf_inf_set_a @ A2 @ B2 ) )
     => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1228_inf_OorderE,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_a @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1229_le__infI2,axiom,
    ! [B2: set_a,X2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ X2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).

% le_infI2
thf(fact_1230_le__infI1,axiom,
    ! [A2: set_a,X2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).

% le_infI1
thf(fact_1231_inf__mono,axiom,
    ! [A2: set_a,C2: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1232_le__infI,axiom,
    ! [X2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ X2 @ B2 )
       => ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1233_le__infE,axiom,
    ! [X2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_a @ X2 @ A2 )
         => ~ ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ).

% le_infE
thf(fact_1234_inf__le2,axiom,
    ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_1235_inf__le1,axiom,
    ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).

% inf_le1
thf(fact_1236_inf__sup__ord_I1_J,axiom,
    ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_1237_inf__sup__ord_I2_J,axiom,
    ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_1238_inter__set__filter,axiom,
    ! [A: set_Pr9033091269622839554st_a_b,Xs: list_P2281458131032066856st_a_b] :
      ( ( inf_in7575054441957001712st_a_b @ A @ ( set_Pr6156699709417984823st_a_b @ Xs ) )
      = ( set_Pr6156699709417984823st_a_b
        @ ( filter4355500224637553439st_a_b
          @ ^ [X4: produc2579390649552321826st_a_b] : ( member8006451236149131979st_a_b @ X4 @ A )
          @ Xs ) ) ) ).

% inter_set_filter
thf(fact_1239_inter__set__filter,axiom,
    ! [A: set_a,Xs: list_a] :
      ( ( inf_inf_set_a @ A @ ( set_a2 @ Xs ) )
      = ( set_a2
        @ ( filter_a
          @ ^ [X4: a] : ( member_a @ X4 @ A )
          @ Xs ) ) ) ).

% inter_set_filter
thf(fact_1240_disjoint__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A @ B )
        = bot_bot_set_a )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ~ ( member_a @ X4 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_1241_Int__emptyI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ~ ( member_a @ X @ B ) )
     => ( ( inf_inf_set_a @ A @ B )
        = bot_bot_set_a ) ) ).

% Int_emptyI
thf(fact_1242_Int__insert__right,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B ) )
          = ( insert_a2 @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
      & ( ~ ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B ) )
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1243_Int__insert__left,axiom,
    ! [A2: a,C3: set_a,B: set_a] :
      ( ( ( member_a @ A2 @ C3 )
       => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B ) @ C3 )
          = ( insert_a2 @ A2 @ ( inf_inf_set_a @ B @ C3 ) ) ) )
      & ( ~ ( member_a @ A2 @ C3 )
       => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B ) @ C3 )
          = ( inf_inf_set_a @ B @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_1244_IntE,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
     => ~ ( ( member_a @ C2 @ A )
         => ~ ( member_a @ C2 @ B ) ) ) ).

% IntE
thf(fact_1245_IntD1,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
     => ( member_a @ C2 @ A ) ) ).

% IntD1
thf(fact_1246_IntD2,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
     => ( member_a @ C2 @ B ) ) ).

% IntD2
thf(fact_1247_Int__Collect,axiom,
    ! [X2: a,A: set_a,P: a > $o] :
      ( ( member_a @ X2 @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
      = ( ( member_a @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_1248_Int__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A3 )
              & ( member_a @ X4 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_1249_Union__Int__subset,axiom,
    ! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).

% Union_Int_subset
thf(fact_1250_Sup__inter__less__eq,axiom,
    ! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).

% Sup_inter_less_eq
thf(fact_1251_Int__Collect__mono,axiom,
    ! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ! [X: a] :
            ( ( member_a @ X @ A )
           => ( ( P @ X )
             => ( Q @ X ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1252_Int__greatest,axiom,
    ! [C3: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C3 @ A )
     => ( ( ord_less_eq_set_a @ C3 @ B )
       => ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_1253_Int__absorb2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( inf_inf_set_a @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_1254_Int__absorb1,axiom,
    ! [B: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( inf_inf_set_a @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_1255_Int__lower2,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_1256_Int__lower1,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_1257_Int__mono,axiom,
    ! [A: set_a,C3: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A @ C3 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C3 @ D ) ) ) ) ).

% Int_mono
thf(fact_1258_distrib__inf__le,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ ( inf_inf_set_a @ X2 @ Z ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1259_distrib__sup__le,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y2 ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1260_Un__Int__assoc__eq,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C3 )
        = ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C3 ) ) )
      = ( ord_less_eq_set_a @ C3 @ A ) ) ).

% Un_Int_assoc_eq

% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X2: list_a,Y2: list_a] :
      ( ( if_list_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X2: list_a,Y2: list_a] :
      ( ( if_list_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_T,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( if_lis3463779252948870126st_a_b @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__b_J_J_T,axiom,
    ! [X2: list_P2281458131032066856st_a_b,Y2: list_P2281458131032066856st_a_b] :
      ( ( if_lis3463779252948870126st_a_b @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ord_less_eq_set_a @ ( iKKBZ_8930711501659780123st_a_b @ ( cons_P2018802354021969880st_a_b @ y @ ysa ) @ a2 ) @ ( iKKBZ_8930711501659780123st_a_b @ ( sortin5994461831780029029st_a_b @ cmp @ ( cons_P2018802354021969880st_a_b @ x @ xsa ) @ ( cons_P2018802354021969880st_a_b @ y @ ysa ) ) @ a2 ) ).

%------------------------------------------------------------------------------