TPTP Problem File: SLH0579^1.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    : VYDRA_MDL/0009_Window/prob_01147_049828__16421958_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1717 ( 543 unt; 448 typ;   0 def)
%            Number of atoms       : 3734 (1383 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11539 ( 509   ~;  66   |; 294   &;8946   @)
%                                         (   0 <=>;1724  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :   94 (  93 usr)
%            Number of type conns  : 1554 (1554   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  358 ( 355 usr;  53 con; 0-9 aty)
%            Number of variables   : 4300 ( 545   ^;3563   !; 192   ?;4300   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:53:09.463
%------------------------------------------------------------------------------
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    sup_leadsto_b_a_c: b > ( b > a > b ) > list_P125642481956313003od_c_a > nat > nat > b > option_c ).

thf(sy_c_Window_Ots__at_001tf__c_001tf__a,type,
    ts_at_c_a: list_P125642481956313003od_c_a > nat > c ).

thf(sy_c_Window_Ovalid__s_001tf__b_001tf__a_001tf__c,type,
    valid_s_b_a_c: b > ( b > a > b ) > mappin8597647756751374250_b_a_b > ( b > $o ) > list_P125642481956313003od_c_a > nat > nat > nat > list_P7417839048565863355_c_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    member5552703068553123156od_c_a: list_P125642481956313003od_c_a > set_li4905775889721270027od_c_a > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member5855424355840516880at_nat: produc6487378988399798503at_nat > set_Pr5297940549829899463at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_Itf__b_J_J,type,
    member318967379524898064_set_b: produc3262564261791608551_set_b > set_Pr7275202699945397959_set_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__c_J_Mt__Set__Oset_Itf__c_J_J,type,
    member1877963456866042576_set_c: produc4821560339132753063_set_c > set_Pr8704909817274950791_set_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J,type,
    member7562873241046315796_c_nat: produc4862256710654508797_c_nat > set_Pr8806432033423503795_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    member7862447932407534991od_b_a: product_prod_b_a > set_Product_prod_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    member7862447941013992593od_b_c: product_prod_b_c > set_Product_prod_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    member8195077246299207702_c_nat: product_prod_c_nat > set_Pr6903500605879609269_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__a_J,type,
    member5074992350434858958od_c_a: product_prod_c_a > set_Product_prod_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    member5074992359041316560od_c_c: product_prod_c_c > set_Product_prod_c_c > $o ).

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

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

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_v_L____,type,
    l: set_nat ).

thf(sy_v_Z____,type,
    z: set_b ).

thf(sy_v_ac_H____,type,
    ac: mapping_b_o ).

thf(sy_v_ac____,type,
    ac2: mapping_b_o ).

thf(sy_v_accept____,type,
    accept: b > $o ).

thf(sy_v_bs,type,
    bs: a ).

thf(sy_v_e_H____,type,
    e: list_P903359562653991662od_b_c ).

thf(sy_v_e____,type,
    e2: list_P903359562653991662od_b_c ).

thf(sy_v_i____,type,
    i: nat ).

thf(sy_v_init____,type,
    init: b ).

thf(sy_v_inv____,type,
    inv: mappin8597647756751374250_b_a_b > $o ).

thf(sy_v_j____,type,
    j: nat ).

thf(sy_v_q____,type,
    q: b ).

thf(sy_v_rho,type,
    rho: list_P125642481956313003od_c_a ).

thf(sy_v_rho_H____,type,
    rho2: list_P125642481956313003od_c_a ).

thf(sy_v_s_H____,type,
    s: list_P7417839048565863355_c_nat ).

thf(sy_v_s____,type,
    s2: list_P7417839048565863355_c_nat ).

thf(sy_v_st_H_H____,type,
    st: mappin8597647756751374250_b_a_b ).

thf(sy_v_st_H____,type,
    st2: mappin8597647756751374250_b_a_b ).

thf(sy_v_st____,type,
    st3: mappin8597647756751374250_b_a_b ).

thf(sy_v_step____,type,
    step: b > a > b ).

thf(sy_v_t,type,
    t: c ).

% Relevant facts (1261)
thf(fact_0_False,axiom,
    z != bot_bot_set_b ).

% False
thf(fact_1_steps__refl,axiom,
    ! [Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b,I: nat] :
      ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ I ) )
      = Q ) ).

% steps_refl
thf(fact_2_bs__at__rho_H__j,axiom,
    ( ( bs_at_c_a @ rho2 @ j )
    = bs ) ).

% bs_at_rho'_j
thf(fact_3_L__def,axiom,
    ( l
    = ( collect_nat
      @ ^ [L: nat] :
          ( ( ord_less_nat @ L @ i )
          & ( ( steps_b_a_c @ step @ rho2 @ init @ ( product_Pair_nat_nat @ L @ ( suc @ j ) ) )
            = q ) ) ) ) ).

% L_def
thf(fact_4_keys__e__alt,axiom,
    ( ( mmap_keys_b_c @ e2 )
    = ( collect_b
      @ ^ [Q2: b] :
        ? [L: nat] :
          ( ( ord_less_nat @ L @ i )
          & ( ( steps_b_a_c @ step @ rho2 @ init @ ( product_Pair_nat_nat @ L @ j ) )
            = Q2 ) ) ) ) ).

% keys_e_alt
thf(fact_5_i__j,axiom,
    ord_less_eq_nat @ i @ j ).

% i_j
thf(fact_6_Z__def,axiom,
    ( z
    = ( collect_b
      @ ^ [X: b] :
          ( ( member_b @ X @ ( mmap_keys_b_c @ e2 ) )
          & ( ( step @ X @ bs )
            = q ) ) ) ) ).

% Z_def
thf(fact_7_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: c,X2: a,Y1: c,Y2: a] :
      ( ( ( product_Pair_c_a @ X1 @ X2 )
        = ( product_Pair_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_prod_Oinject,axiom,
    ! [X1: b,X2: option7520157102916957007_c_nat,Y1: b,Y2: option7520157102916957007_c_nat] :
      ( ( ( produc5716802255202478839_c_nat @ X1 @ X2 )
        = ( produc5716802255202478839_c_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: b,X2: a,Y1: b,Y2: a] :
      ( ( ( product_Pair_b_a @ X1 @ X2 )
        = ( product_Pair_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: c,B: a,A2: c,B2: a] :
      ( ( ( product_Pair_c_a @ A @ B )
        = ( product_Pair_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A2: b,B2: option7520157102916957007_c_nat] :
      ( ( ( produc5716802255202478839_c_nat @ A @ B )
        = ( produc5716802255202478839_c_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
    ! [A: b,B: c,A2: b,B2: c] :
      ( ( ( product_Pair_b_c @ A @ B )
        = ( product_Pair_b_c @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: b,B: a,A2: b,B2: a] :
      ( ( ( product_Pair_b_a @ A @ B )
        = ( product_Pair_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

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

% old.prod.inject
thf(fact_19_lookup__e,axiom,
    ! [Q: b] :
      ( ( mmap_lookup_b_c @ e2 @ Q )
      = ( sup_leadsto_b_a_c @ init @ step @ rho2 @ i @ j @ Q ) ) ).

% lookup_e
thf(fact_20_ts__at__rho_H__j,axiom,
    ( ( ts_at_c_a @ rho2 @ j )
    = t ) ).

% ts_at_rho'_j
thf(fact_21_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R ) )
        = ( ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_22_pred__equals__eq2,axiom,
    ! [R: set_Pr6903500605879609269_c_nat,S: set_Pr6903500605879609269_c_nat] :
      ( ( ( ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R ) )
        = ( ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_23_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_c_a,S: set_Product_prod_c_a] :
      ( ( ( ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ R ) )
        = ( ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_24_pred__equals__eq2,axiom,
    ! [R: set_Pr8806432033423503795_c_nat,S: set_Pr8806432033423503795_c_nat] :
      ( ( ( ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R ) )
        = ( ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_25_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( ( ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ R ) )
        = ( ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_26_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_a,S: set_Product_prod_b_a] :
      ( ( ( ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ R ) )
        = ( ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_27_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_28_in__measure,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur8038558561449204169at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_29_in__measure,axiom,
    ! [X3: nat,Y3: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_30_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] :
          ( X3
         != ( product_Pair_nat_nat @ K @ M ) ) ).

% prod_decode_aux.cases
thf(fact_31_steps__appE,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b,Q3: b] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) )
          = Q3 )
       => ? [Q4: b] :
            ( ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ J ) )
              = Q4 )
            & ( Q3
              = ( Step @ Q4 @ ( bs_at_c_a @ Rho @ J ) ) ) ) ) ) ).

% steps_appE
thf(fact_32_steps__split,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b] :
      ( ( ord_less_nat @ I @ J )
     => ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ J ) )
        = ( steps_b_a_c @ Step @ Rho @ ( Step @ Q @ ( bs_at_c_a @ Rho @ I ) ) @ ( product_Pair_nat_nat @ ( suc @ I ) @ J ) ) ) ) ).

% steps_split
thf(fact_33_steps__comp,axiom,
    ! [I: nat,L2: nat,J: nat,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b,Q3: b,Q5: b] :
      ( ( ord_less_eq_nat @ I @ L2 )
     => ( ( ord_less_eq_nat @ L2 @ J )
       => ( ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ L2 ) )
            = Q3 )
         => ( ( ( steps_b_a_c @ Step @ Rho @ Q3 @ ( product_Pair_nat_nat @ L2 @ J ) )
              = Q5 )
           => ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ J ) )
              = Q5 ) ) ) ) ) ).

% steps_comp
thf(fact_34_Pair__inject,axiom,
    ! [A: c,B: nat,A2: c,B2: nat] :
      ( ( ( product_Pair_c_nat @ A @ B )
        = ( product_Pair_c_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_35_Pair__inject,axiom,
    ! [A: c,B: a,A2: c,B2: a] :
      ( ( ( product_Pair_c_a @ A @ B )
        = ( product_Pair_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_36_Pair__inject,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A2: b,B2: option7520157102916957007_c_nat] :
      ( ( ( produc5716802255202478839_c_nat @ A @ B )
        = ( produc5716802255202478839_c_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_37_Pair__inject,axiom,
    ! [A: b,B: c,A2: b,B2: c] :
      ( ( ( product_Pair_b_c @ A @ B )
        = ( product_Pair_b_c @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_38_Pair__inject,axiom,
    ! [A: b,B: a,A2: b,B2: a] :
      ( ( ( product_Pair_b_a @ A @ B )
        = ( product_Pair_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_39_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_40_prod__cases,axiom,
    ! [P: product_prod_c_nat > $o,P2: product_prod_c_nat] :
      ( ! [A3: c,B3: nat] : ( P @ ( product_Pair_c_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_41_prod__cases,axiom,
    ! [P: product_prod_c_a > $o,P2: product_prod_c_a] :
      ( ! [A3: c,B3: a] : ( P @ ( product_Pair_c_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_42_prod__cases,axiom,
    ! [P: produc4862256710654508797_c_nat > $o,P2: produc4862256710654508797_c_nat] :
      ( ! [A3: b,B3: option7520157102916957007_c_nat] : ( P @ ( produc5716802255202478839_c_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_43_prod__cases,axiom,
    ! [P: product_prod_b_c > $o,P2: product_prod_b_c] :
      ( ! [A3: b,B3: c] : ( P @ ( product_Pair_b_c @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_44_prod__cases,axiom,
    ! [P: product_prod_b_a > $o,P2: product_prod_b_a] :
      ( ! [A3: b,B3: a] : ( P @ ( product_Pair_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_45_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_46_surj__pair,axiom,
    ! [P2: product_prod_c_nat] :
    ? [X4: c,Y4: nat] :
      ( P2
      = ( product_Pair_c_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_47_surj__pair,axiom,
    ! [P2: product_prod_c_a] :
    ? [X4: c,Y4: a] :
      ( P2
      = ( product_Pair_c_a @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_48_surj__pair,axiom,
    ! [P2: produc4862256710654508797_c_nat] :
    ? [X4: b,Y4: option7520157102916957007_c_nat] :
      ( P2
      = ( produc5716802255202478839_c_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_49_surj__pair,axiom,
    ! [P2: product_prod_b_c] :
    ? [X4: b,Y4: c] :
      ( P2
      = ( product_Pair_b_c @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_50_surj__pair,axiom,
    ! [P2: product_prod_b_a] :
    ? [X4: b,Y4: a] :
      ( P2
      = ( product_Pair_b_a @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_51_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y4: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_c_nat] :
      ~ ! [A3: c,B3: nat] :
          ( Y3
         != ( product_Pair_c_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_c_a] :
      ~ ! [A3: c,B3: a] :
          ( Y3
         != ( product_Pair_c_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_54_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4862256710654508797_c_nat] :
      ~ ! [A3: b,B3: option7520157102916957007_c_nat] :
          ( Y3
         != ( produc5716802255202478839_c_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_55_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_b_c] :
      ~ ! [A3: b,B3: c] :
          ( Y3
         != ( product_Pair_b_c @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_b_a] :
      ~ ! [A3: b,B3: a] :
          ( Y3
         != ( product_Pair_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_57_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_58_valid__before_I5_J,axiom,
    ! [Q6: b] :
      ( ( mmap_lookup_b_c @ e2 @ Q6 )
      = ( sup_leadsto_b_a_c @ init @ step @ rho @ i @ j @ Q6 ) ) ).

% valid_before(5)
thf(fact_59_e_H__fold__sup__st_H_H_I1_J,axiom,
    ( e
    = ( fold_sup_b_c @ e2
      @ ^ [Q2: b] : ( step @ Q2 @ bs ) ) ) ).

% e'_fold_sup_st''(1)
thf(fact_60_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_61_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_62_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_63_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_64_finite__keys__e,axiom,
    finite_finite_b @ ( mmap_keys_b_c @ e2 ) ).

% finite_keys_e
thf(fact_65_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: c,P: c > $o] :
      ( ( member_c @ A @ ( collect_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A: b,P: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_69_Collect__mem__eq,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A4: set_c] :
      ( ( collect_c
        @ ^ [X: c] : ( member_c @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A4: set_b] :
      ( ( collect_b
        @ ^ [X: b] : ( member_b @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_73_Collect__cong,axiom,
    ! [P: b > $o,Q7: b > $o] :
      ( ! [X4: b] :
          ( ( P @ X4 )
          = ( Q7 @ X4 ) )
     => ( ( collect_b @ P )
        = ( collect_b @ Q7 ) ) ) ).

% Collect_cong
thf(fact_74_Collect__cong,axiom,
    ! [P: nat > $o,Q7: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q7 @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q7 ) ) ) ).

% Collect_cong
thf(fact_75_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_76_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_77_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_78_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_79_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_80_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_81_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_82_Suc__inject,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Suc_inject
thf(fact_83_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_84_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_85_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_86_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_87_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_88_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_89_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_90_linorder__neqE__nat,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_91_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_92_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_93_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_94_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_95_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_96_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_97_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ I @ K2 )
     => ( ( K2
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K2
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_98_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_99_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K2 )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K2
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_100_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_101_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_102_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_103_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_104_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_105_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_106_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ I2 ) ) ) ) ).

% All_less_Suc
thf(fact_107_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M4: nat] :
            ( ( M2
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_108_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_109_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_110_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K2 )
       => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).

% less_trans_Suc
thf(fact_111_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K )
                   => ( P @ I3 @ K ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_112_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_113_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_114_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y4: nat,Z: nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_115_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_116_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_117_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_118_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_119_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_120_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
     => ? [M: nat] :
          ( M5
          = ( suc @ M ) ) ) ).

% Suc_le_D
thf(fact_121_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_122_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_123_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_124_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M6 @ N3 )
          & ( M6 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_125_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_126_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N3: nat] :
          ( ( ord_less_nat @ M6 @ N3 )
          | ( M6 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_127_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_128_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_129_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_130_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_131_lift__Suc__antimono__le,axiom,
    ! [F: nat > c,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_c @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_c @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_132_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_133_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_134_lift__Suc__mono__le,axiom,
    ! [F: nat > c,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_c @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_c @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_135_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_136_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_137_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_138_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_139_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_140_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_141_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_142_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_143_finite__Collect__bounded__ex,axiom,
    ! [P: b > $o,Q7: b > b > $o] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_b
          @ ( collect_b
            @ ^ [X: b] :
              ? [Y: b] :
                ( ( P @ Y )
                & ( Q7 @ X @ Y ) ) ) )
        = ( ! [Y: b] :
              ( ( P @ Y )
             => ( finite_finite_b
                @ ( collect_b
                  @ ^ [X: b] : ( Q7 @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_144_finite__Collect__bounded__ex,axiom,
    ! [P: b > $o,Q7: nat > b > $o] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: b] :
                ( ( P @ Y )
                & ( Q7 @ X @ Y ) ) ) )
        = ( ! [Y: b] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q7 @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_145_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q7: b > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_b
          @ ( collect_b
            @ ^ [X: b] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q7 @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_b
                @ ( collect_b
                  @ ^ [X: b] : ( Q7 @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_146_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q7: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q7 @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q7 @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_147_finite__Collect__disjI,axiom,
    ! [P: b > $o,Q7: b > $o] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( P @ X )
              | ( Q7 @ X ) ) ) )
      = ( ( finite_finite_b @ ( collect_b @ P ) )
        & ( finite_finite_b @ ( collect_b @ Q7 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_148_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q7: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q7 @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q7 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_149_finite__Collect__conjI,axiom,
    ! [P: b > $o,Q7: b > $o] :
      ( ( ( finite_finite_b @ ( collect_b @ P ) )
        | ( finite_finite_b @ ( collect_b @ Q7 ) ) )
     => ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( P @ X )
              & ( Q7 @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_150_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q7: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q7 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q7 @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_151_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S )
                  & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_152_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X5 )
                & ( ord_less_nat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_153_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_154_finite__has__minimal,axiom,
    ! [A4: set_c] :
      ( ( finite_finite_c @ A4 )
     => ( ( A4 != bot_bot_set_c )
       => ? [X4: c] :
            ( ( member_c @ X4 @ A4 )
            & ! [Xa: c] :
                ( ( member_c @ Xa @ A4 )
               => ( ( ord_less_eq_c @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_155_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_156_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_157_finite__has__maximal,axiom,
    ! [A4: set_c] :
      ( ( finite_finite_c @ A4 )
     => ( ( A4 != bot_bot_set_c )
       => ? [X4: c] :
            ( ( member_c @ X4 @ A4 )
            & ! [Xa: c] :
                ( ( member_c @ Xa @ A4 )
               => ( ( ord_less_eq_c @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_158_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_159_fin__L,axiom,
    finite_finite_nat @ l ).

% fin_L
thf(fact_160_empty__Collect__eq,axiom,
    ! [P: b > $o] :
      ( ( bot_bot_set_b
        = ( collect_b @ P ) )
      = ( ! [X: b] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_161_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_162_Collect__empty__eq,axiom,
    ! [P: b > $o] :
      ( ( ( collect_b @ P )
        = bot_bot_set_b )
      = ( ! [X: b] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_163_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_164_subset__empty,axiom,
    ! [A4: set_b] :
      ( ( ord_less_eq_set_b @ A4 @ bot_bot_set_b )
      = ( A4 = bot_bot_set_b ) ) ).

% subset_empty
thf(fact_165_subset__empty,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_166_empty__subsetI,axiom,
    ! [A4: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A4 ) ).

% empty_subsetI
thf(fact_167_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_168_empty__iff,axiom,
    ! [C: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ C @ bot_bo5327735625951526323at_nat ) ).

% empty_iff
thf(fact_169_empty__iff,axiom,
    ! [C: c] :
      ~ ( member_c @ C @ bot_bot_set_c ) ).

% empty_iff
thf(fact_170_empty__iff,axiom,
    ! [C: b] :
      ~ ( member_b @ C @ bot_bot_set_b ) ).

% empty_iff
thf(fact_171_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_172_all__not__in__conv,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( ! [X: produc859450856879609959at_nat] :
            ~ ( member8206827879206165904at_nat @ X @ A4 ) )
      = ( A4 = bot_bo5327735625951526323at_nat ) ) ).

% all_not_in_conv
thf(fact_173_all__not__in__conv,axiom,
    ! [A4: set_c] :
      ( ( ! [X: c] :
            ~ ( member_c @ X @ A4 ) )
      = ( A4 = bot_bot_set_c ) ) ).

% all_not_in_conv
thf(fact_174_all__not__in__conv,axiom,
    ! [A4: set_b] :
      ( ( ! [X: b] :
            ~ ( member_b @ X @ A4 ) )
      = ( A4 = bot_bot_set_b ) ) ).

% all_not_in_conv
thf(fact_175_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_176_finite__Collect__subsets,axiom,
    ! [A4: set_b] :
      ( ( finite_finite_b @ A4 )
     => ( finite_finite_set_b
        @ ( collect_set_b
          @ ^ [B4: set_b] : ( ord_less_eq_set_b @ B4 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_177_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_178_finite__Collect__less__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_less_nat
thf(fact_179_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_180_not__psubset__empty,axiom,
    ! [A4: set_b] :
      ~ ( ord_less_set_b @ A4 @ bot_bot_set_b ) ).

% not_psubset_empty
thf(fact_181_not__psubset__empty,axiom,
    ! [A4: set_nat] :
      ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_182_finite__psubset__induct,axiom,
    ! [A4: set_b,P: set_b > $o] :
      ( ( finite_finite_b @ A4 )
     => ( ! [A5: set_b] :
            ( ( finite_finite_b @ A5 )
           => ( ! [B5: set_b] :
                  ( ( ord_less_set_b @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_183_finite__psubset__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A5: set_nat] :
            ( ( finite_finite_nat @ A5 )
           => ( ! [B5: set_nat] :
                  ( ( ord_less_set_nat @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_184_rev__finite__subset,axiom,
    ! [B6: set_b,A4: set_b] :
      ( ( finite_finite_b @ B6 )
     => ( ( ord_less_eq_set_b @ A4 @ B6 )
       => ( finite_finite_b @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_185_rev__finite__subset,axiom,
    ! [B6: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B6 )
     => ( ( ord_less_eq_set_nat @ A4 @ B6 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_186_infinite__super,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( ord_less_eq_set_b @ S @ T2 )
     => ( ~ ( finite_finite_b @ S )
       => ~ ( finite_finite_b @ T2 ) ) ) ).

% infinite_super
thf(fact_187_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_188_finite__subset,axiom,
    ! [A4: set_b,B6: set_b] :
      ( ( ord_less_eq_set_b @ A4 @ B6 )
     => ( ( finite_finite_b @ B6 )
       => ( finite_finite_b @ A4 ) ) ) ).

% finite_subset
thf(fact_189_finite__subset,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( finite_finite_nat @ B6 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_190_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
        @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ S ) )
      = ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_191_pred__subset__eq2,axiom,
    ! [R: set_Pr6903500605879609269_c_nat,S: set_Pr6903500605879609269_c_nat] :
      ( ( ord_less_eq_c_nat_o
        @ ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R )
        @ ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ S ) )
      = ( ord_le1411700432677832725_c_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_192_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_c_a,S: set_Product_prod_c_a] :
      ( ( ord_less_eq_c_a_o
        @ ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ R )
        @ ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ S ) )
      = ( ord_le8698776994054418981od_c_a @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_193_pred__subset__eq2,axiom,
    ! [R: set_Pr8806432033423503795_c_nat,S: set_Pr8806432033423503795_c_nat] :
      ( ( ord_le4248069829227848518_nat_o
        @ ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R )
        @ ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ S ) )
      = ( ord_le7634486248364433939_c_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_194_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( ord_less_eq_b_c_o
        @ ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ R )
        @ ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ S ) )
      = ( ord_le253122037897321832od_b_c @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_195_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_b_a,S: set_Product_prod_b_a] :
      ( ( ord_less_eq_b_a_o
        @ ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ R )
        @ ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ S ) )
      = ( ord_le111053957804629862od_b_a @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_196_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_197_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_198_subrelI,axiom,
    ! [R2: set_Pr6903500605879609269_c_nat,S2: set_Pr6903500605879609269_c_nat] :
      ( ! [X4: c,Y4: nat] :
          ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ R2 )
         => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le1411700432677832725_c_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_199_subrelI,axiom,
    ! [R2: set_Product_prod_c_a,S2: set_Product_prod_c_a] :
      ( ! [X4: c,Y4: a] :
          ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ R2 )
         => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le8698776994054418981od_c_a @ R2 @ S2 ) ) ).

% subrelI
thf(fact_200_subrelI,axiom,
    ! [R2: set_Pr8806432033423503795_c_nat,S2: set_Pr8806432033423503795_c_nat] :
      ( ! [X4: b,Y4: option7520157102916957007_c_nat] :
          ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ R2 )
         => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le7634486248364433939_c_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_201_subrelI,axiom,
    ! [R2: set_Product_prod_b_c,S2: set_Product_prod_b_c] :
      ( ! [X4: b,Y4: c] :
          ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) @ R2 )
         => ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le253122037897321832od_b_c @ R2 @ S2 ) ) ).

% subrelI
thf(fact_202_subrelI,axiom,
    ! [R2: set_Product_prod_b_a,S2: set_Product_prod_b_a] :
      ( ! [X4: b,Y4: a] :
          ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X4 @ Y4 ) @ R2 )
         => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le111053957804629862od_b_a @ R2 @ S2 ) ) ).

% subrelI
thf(fact_203_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_204_emptyE,axiom,
    ! [A: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ A @ bot_bo5327735625951526323at_nat ) ).

% emptyE
thf(fact_205_emptyE,axiom,
    ! [A: c] :
      ~ ( member_c @ A @ bot_bot_set_c ) ).

% emptyE
thf(fact_206_emptyE,axiom,
    ! [A: b] :
      ~ ( member_b @ A @ bot_bot_set_b ) ).

% emptyE
thf(fact_207_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_208_equals0D,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( A4 = bot_bo5327735625951526323at_nat )
     => ~ ( member8206827879206165904at_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_209_equals0D,axiom,
    ! [A4: set_c,A: c] :
      ( ( A4 = bot_bot_set_c )
     => ~ ( member_c @ A @ A4 ) ) ).

% equals0D
thf(fact_210_equals0D,axiom,
    ! [A4: set_b,A: b] :
      ( ( A4 = bot_bot_set_b )
     => ~ ( member_b @ A @ A4 ) ) ).

% equals0D
thf(fact_211_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_212_equals0I,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ! [Y4: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ Y4 @ A4 )
     => ( A4 = bot_bo5327735625951526323at_nat ) ) ).

% equals0I
thf(fact_213_equals0I,axiom,
    ! [A4: set_c] :
      ( ! [Y4: c] :
          ~ ( member_c @ Y4 @ A4 )
     => ( A4 = bot_bot_set_c ) ) ).

% equals0I
thf(fact_214_equals0I,axiom,
    ! [A4: set_b] :
      ( ! [Y4: b] :
          ~ ( member_b @ Y4 @ A4 )
     => ( A4 = bot_bot_set_b ) ) ).

% equals0I
thf(fact_215_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_216_ex__in__conv,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( ? [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A4 ) )
      = ( A4 != bot_bo5327735625951526323at_nat ) ) ).

% ex_in_conv
thf(fact_217_ex__in__conv,axiom,
    ! [A4: set_c] :
      ( ( ? [X: c] : ( member_c @ X @ A4 ) )
      = ( A4 != bot_bot_set_c ) ) ).

% ex_in_conv
thf(fact_218_ex__in__conv,axiom,
    ! [A4: set_b] :
      ( ( ? [X: b] : ( member_b @ X @ A4 ) )
      = ( A4 != bot_bot_set_b ) ) ).

% ex_in_conv
thf(fact_219_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_220_Set_Oempty__def,axiom,
    ( bot_bot_set_b
    = ( collect_b
      @ ^ [X: b] : $false ) ) ).

% Set.empty_def
thf(fact_221_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_222_not__finite__existsD,axiom,
    ! [P: b > $o] :
      ( ~ ( finite_finite_b @ ( collect_b @ P ) )
     => ? [X_1: b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_223_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_224_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B6: set_b,R: produc859450856879609959at_nat > b > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A4 )
     => ( ( finite_finite_b @ B6 )
       => ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A4 )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: b] :
              ( ( member_b @ X4 @ B6 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A6: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_225_pigeonhole__infinite__rel,axiom,
    ! [A4: set_c,B6: set_b,R: c > b > $o] :
      ( ~ ( finite_finite_c @ A4 )
     => ( ( finite_finite_b @ B6 )
       => ( ! [X4: c] :
              ( ( member_c @ X4 @ A4 )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: b] :
              ( ( member_b @ X4 @ B6 )
              & ~ ( finite_finite_c
                  @ ( collect_c
                    @ ^ [A6: c] :
                        ( ( member_c @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_226_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B6: set_nat,R: produc859450856879609959at_nat > nat > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B6 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A6: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_227_pigeonhole__infinite__rel,axiom,
    ! [A4: set_c,B6: set_nat,R: c > nat > $o] :
      ( ~ ( finite_finite_c @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X4: c] :
              ( ( member_c @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B6 )
              & ~ ( finite_finite_c
                  @ ( collect_c
                    @ ^ [A6: c] :
                        ( ( member_c @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_228_pigeonhole__infinite__rel,axiom,
    ! [A4: set_b,B6: set_b,R: b > b > $o] :
      ( ~ ( finite_finite_b @ A4 )
     => ( ( finite_finite_b @ B6 )
       => ( ! [X4: b] :
              ( ( member_b @ X4 @ A4 )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: b] :
              ( ( member_b @ X4 @ B6 )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A6: b] :
                        ( ( member_b @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_229_pigeonhole__infinite__rel,axiom,
    ! [A4: set_b,B6: set_nat,R: b > nat > $o] :
      ( ~ ( finite_finite_b @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X4: b] :
              ( ( member_b @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B6 )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A6: b] :
                        ( ( member_b @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_230_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B6: set_b,R: nat > b > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_b @ B6 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: b] :
              ( ( member_b @ X4 @ B6 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A6: nat] :
                        ( ( member_nat @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_231_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B6: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B6 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B6 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A6: nat] :
                        ( ( member_nat @ A6 @ A4 )
                        & ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_232_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( ord_less_eq_nat @ A @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_233_finite__has__maximal2,axiom,
    ! [A4: set_c,A: c] :
      ( ( finite_finite_c @ A4 )
     => ( ( member_c @ A @ A4 )
       => ? [X4: c] :
            ( ( member_c @ X4 @ A4 )
            & ( ord_less_eq_c @ A @ X4 )
            & ! [Xa: c] :
                ( ( member_c @ Xa @ A4 )
               => ( ( ord_less_eq_c @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_234_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( ord_less_eq_set_nat @ A @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_235_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( ord_less_eq_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_236_finite__has__minimal2,axiom,
    ! [A4: set_c,A: c] :
      ( ( finite_finite_c @ A4 )
     => ( ( member_c @ A @ A4 )
       => ? [X4: c] :
            ( ( member_c @ X4 @ A4 )
            & ( ord_less_eq_c @ X4 @ A )
            & ! [Xa: c] :
                ( ( member_c @ Xa @ A4 )
               => ( ( ord_less_eq_c @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_237_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( ord_less_eq_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_238_finite_OemptyI,axiom,
    finite_finite_b @ bot_bot_set_b ).

% finite.emptyI
thf(fact_239_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_240_infinite__imp__nonempty,axiom,
    ! [S: set_b] :
      ( ~ ( finite_finite_b @ S )
     => ( S != bot_bot_set_b ) ) ).

% infinite_imp_nonempty
thf(fact_241_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_242_finite__image__set2,axiom,
    ! [P: b > $o,Q7: b > $o,F: b > b > b] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_b @ ( collect_b @ Q7 ) )
       => ( finite_finite_b
          @ ( collect_b
            @ ^ [Uu: b] :
              ? [X: b,Y: b] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_243_finite__image__set2,axiom,
    ! [P: b > $o,Q7: b > $o,F: b > b > nat] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_b @ ( collect_b @ Q7 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: b,Y: b] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_244_finite__image__set2,axiom,
    ! [P: b > $o,Q7: nat > $o,F: b > nat > b] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q7 ) )
       => ( finite_finite_b
          @ ( collect_b
            @ ^ [Uu: b] :
              ? [X: b,Y: nat] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_245_finite__image__set2,axiom,
    ! [P: b > $o,Q7: nat > $o,F: b > nat > nat] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q7 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: b,Y: nat] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_246_finite__image__set2,axiom,
    ! [P: nat > $o,Q7: b > $o,F: nat > b > b] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_b @ ( collect_b @ Q7 ) )
       => ( finite_finite_b
          @ ( collect_b
            @ ^ [Uu: b] :
              ? [X: nat,Y: b] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_247_finite__image__set2,axiom,
    ! [P: nat > $o,Q7: b > $o,F: nat > b > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_b @ ( collect_b @ Q7 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: b] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_248_finite__image__set2,axiom,
    ! [P: nat > $o,Q7: nat > $o,F: nat > nat > b] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q7 ) )
       => ( finite_finite_b
          @ ( collect_b
            @ ^ [Uu: b] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_249_finite__image__set2,axiom,
    ! [P: nat > $o,Q7: nat > $o,F: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q7 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q7 @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_250_finite__image__set,axiom,
    ! [P: b > $o,F: b > b] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( finite_finite_b
        @ ( collect_b
          @ ^ [Uu: b] :
            ? [X: b] :
              ( ( Uu
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_251_finite__image__set,axiom,
    ! [P: b > $o,F: b > nat] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: b] :
              ( ( Uu
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_252_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > b] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_b
        @ ( collect_b
          @ ^ [Uu: b] :
            ? [X: nat] :
              ( ( Uu
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_253_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_254_valid__before_I2_J,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ J @ ( size_s2614380629626057239od_c_a @ rho ) )
       => ( ord_less_eq_c @ ( ts_at_c_a @ rho @ I ) @ ( ts_at_c_a @ rho @ J ) ) ) ) ).

% valid_before(2)
thf(fact_255_ts__at__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ J @ ( size_s2614380629626057239od_c_a @ rho2 ) )
       => ( ord_less_eq_c @ ( ts_at_c_a @ rho2 @ I ) @ ( ts_at_c_a @ rho2 @ J ) ) ) ) ).

% ts_at_mono
thf(fact_256_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_257_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_258_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_259_order__refl,axiom,
    ! [X3: c] : ( ord_less_eq_c @ X3 @ X3 ) ).

% order_refl
thf(fact_260_order__refl,axiom,
    ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_261_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_262_dual__order_Orefl,axiom,
    ! [A: c] : ( ord_less_eq_c @ A @ A ) ).

% dual_order.refl
thf(fact_263_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_264_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M6: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M6 @ N3 )
            & ( member_nat @ N3 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_265_subset__antisym,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( ord_less_eq_set_nat @ B6 @ A4 )
       => ( A4 = B6 ) ) ) ).

% subset_antisym
thf(fact_266_subsetI,axiom,
    ! [A4: set_b,B6: set_b] :
      ( ! [X4: b] :
          ( ( member_b @ X4 @ A4 )
         => ( member_b @ X4 @ B6 ) )
     => ( ord_less_eq_set_b @ A4 @ B6 ) ) ).

% subsetI
thf(fact_267_subsetI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
      ( ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A4 )
         => ( member8206827879206165904at_nat @ X4 @ B6 ) )
     => ( ord_le3000389064537975527at_nat @ A4 @ B6 ) ) ).

% subsetI
thf(fact_268_subsetI,axiom,
    ! [A4: set_c,B6: set_c] :
      ( ! [X4: c] :
          ( ( member_c @ X4 @ A4 )
         => ( member_c @ X4 @ B6 ) )
     => ( ord_less_eq_set_c @ A4 @ B6 ) ) ).

% subsetI
thf(fact_269_subsetI,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( member_nat @ X4 @ B6 ) )
     => ( ord_less_eq_set_nat @ A4 @ B6 ) ) ).

% subsetI
thf(fact_270_psubsetI,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( A4 != B6 )
       => ( ord_less_set_nat @ A4 @ B6 ) ) ) ).

% psubsetI
thf(fact_271_psubsetD,axiom,
    ! [A4: set_b,B6: set_b,C: b] :
      ( ( ord_less_set_b @ A4 @ B6 )
     => ( ( member_b @ C @ A4 )
       => ( member_b @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_272_psubsetD,axiom,
    ! [A4: set_nat,B6: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_273_psubsetD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A4 @ B6 )
     => ( ( member8206827879206165904at_nat @ C @ A4 )
       => ( member8206827879206165904at_nat @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_274_psubsetD,axiom,
    ! [A4: set_c,B6: set_c,C: c] :
      ( ( ord_less_set_c @ A4 @ B6 )
     => ( ( member_c @ C @ A4 )
       => ( member_c @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_275_less__set__def,axiom,
    ( ord_less_set_b
    = ( ^ [A7: set_b,B4: set_b] :
          ( ord_less_b_o
          @ ^ [X: b] : ( member_b @ X @ A7 )
          @ ^ [X: b] : ( member_b @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_276_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A7 )
          @ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_277_less__set__def,axiom,
    ( ord_le6428140832669894131at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
          ( ord_le7432393201274343594_nat_o
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A7 )
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_278_less__set__def,axiom,
    ( ord_less_set_c
    = ( ^ [A7: set_c,B4: set_c] :
          ( ord_less_c_o
          @ ^ [X: c] : ( member_c @ X @ A7 )
          @ ^ [X: c] : ( member_c @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_279_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ( ord_less_set_nat @ A7 @ B4 )
          | ( A7 = B4 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_280_subset__psubset__trans,axiom,
    ! [A4: set_nat,B6: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( ord_less_set_nat @ B6 @ C2 )
       => ( ord_less_set_nat @ A4 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_281_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B4 )
          & ~ ( ord_less_eq_set_nat @ B4 @ A7 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_282_psubset__subset__trans,axiom,
    ! [A4: set_nat,B6: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ( ( ord_less_eq_set_nat @ B6 @ C2 )
       => ( ord_less_set_nat @ A4 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_283_pred__subset__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( ord_less_eq_b_o
        @ ^ [X: b] : ( member_b @ X @ R )
        @ ^ [X: b] : ( member_b @ X @ S ) )
      = ( ord_less_eq_set_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_284_pred__subset__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ R )
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ S ) )
      = ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_285_pred__subset__eq,axiom,
    ! [R: set_c,S: set_c] :
      ( ( ord_less_eq_c_o
        @ ^ [X: c] : ( member_c @ X @ R )
        @ ^ [X: c] : ( member_c @ X @ S ) )
      = ( ord_less_eq_set_c @ R @ S ) ) ).

% pred_subset_eq
thf(fact_286_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_287_psubset__imp__subset,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ( ord_less_eq_set_nat @ A4 @ B6 ) ) ).

% psubset_imp_subset
thf(fact_288_Collect__mono__iff,axiom,
    ! [P: b > $o,Q7: b > $o] :
      ( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q7 ) )
      = ( ! [X: b] :
            ( ( P @ X )
           => ( Q7 @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_289_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q7: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q7 ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q7 @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_290_less__eq__set__def,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A7: set_b,B4: set_b] :
          ( ord_less_eq_b_o
          @ ^ [X: b] : ( member_b @ X @ A7 )
          @ ^ [X: b] : ( member_b @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_291_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A7 )
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_292_less__eq__set__def,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A7: set_c,B4: set_c] :
          ( ord_less_eq_c_o
          @ ^ [X: c] : ( member_c @ X @ A7 )
          @ ^ [X: c] : ( member_c @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_293_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A7 )
          @ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_294_Collect__subset,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_295_Collect__subset,axiom,
    ! [A4: set_c,P: c > $o] :
      ( ord_less_eq_set_c
      @ ( collect_c
        @ ^ [X: c] :
            ( ( member_c @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_296_Collect__subset,axiom,
    ! [A4: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_297_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_298_set__eq__subset,axiom,
    ( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_299_subset__trans,axiom,
    ! [A4: set_nat,B6: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( ord_less_eq_set_nat @ B6 @ C2 )
       => ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).

% subset_trans
thf(fact_300_Collect__mono,axiom,
    ! [P: b > $o,Q7: b > $o] :
      ( ! [X4: b] :
          ( ( P @ X4 )
         => ( Q7 @ X4 ) )
     => ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q7 ) ) ) ).

% Collect_mono
thf(fact_301_Collect__mono,axiom,
    ! [P: nat > $o,Q7: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( Q7 @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q7 ) ) ) ).

% Collect_mono
thf(fact_302_subset__refl,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).

% subset_refl
thf(fact_303_subset__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A7: set_b,B4: set_b] :
        ! [T3: b] :
          ( ( member_b @ T3 @ A7 )
         => ( member_b @ T3 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_304_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
        ! [T3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T3 @ A7 )
         => ( member8206827879206165904at_nat @ T3 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_305_subset__iff,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A7: set_c,B4: set_c] :
        ! [T3: c] :
          ( ( member_c @ T3 @ A7 )
         => ( member_c @ T3 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_306_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
        ! [T3: nat] :
          ( ( member_nat @ T3 @ A7 )
         => ( member_nat @ T3 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_307_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B4 )
          & ( A7 != B4 ) ) ) ) ).

% psubset_eq
thf(fact_308_equalityD2,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( A4 = B6 )
     => ( ord_less_eq_set_nat @ B6 @ A4 ) ) ).

% equalityD2
thf(fact_309_equalityD1,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( A4 = B6 )
     => ( ord_less_eq_set_nat @ A4 @ B6 ) ) ).

% equalityD1
thf(fact_310_subset__eq,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A7: set_b,B4: set_b] :
        ! [X: b] :
          ( ( member_b @ X @ A7 )
         => ( member_b @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_311_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
        ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ A7 )
         => ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_312_subset__eq,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A7: set_c,B4: set_c] :
        ! [X: c] :
          ( ( member_c @ X @ A7 )
         => ( member_c @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_313_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A7 )
         => ( member_nat @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_314_equalityE,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( A4 = B6 )
     => ~ ( ( ord_less_eq_set_nat @ A4 @ B6 )
         => ~ ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ).

% equalityE
thf(fact_315_psubsetE,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ~ ( ( ord_less_eq_set_nat @ A4 @ B6 )
         => ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ).

% psubsetE
thf(fact_316_subsetD,axiom,
    ! [A4: set_b,B6: set_b,C: b] :
      ( ( ord_less_eq_set_b @ A4 @ B6 )
     => ( ( member_b @ C @ A4 )
       => ( member_b @ C @ B6 ) ) ) ).

% subsetD
thf(fact_317_subsetD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B6 )
     => ( ( member8206827879206165904at_nat @ C @ A4 )
       => ( member8206827879206165904at_nat @ C @ B6 ) ) ) ).

% subsetD
thf(fact_318_subsetD,axiom,
    ! [A4: set_c,B6: set_c,C: c] :
      ( ( ord_less_eq_set_c @ A4 @ B6 )
     => ( ( member_c @ C @ A4 )
       => ( member_c @ C @ B6 ) ) ) ).

% subsetD
thf(fact_319_subsetD,axiom,
    ! [A4: set_nat,B6: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B6 ) ) ) ).

% subsetD
thf(fact_320_in__mono,axiom,
    ! [A4: set_b,B6: set_b,X3: b] :
      ( ( ord_less_eq_set_b @ A4 @ B6 )
     => ( ( member_b @ X3 @ A4 )
       => ( member_b @ X3 @ B6 ) ) ) ).

% in_mono
thf(fact_321_in__mono,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B6 )
     => ( ( member8206827879206165904at_nat @ X3 @ A4 )
       => ( member8206827879206165904at_nat @ X3 @ B6 ) ) ) ).

% in_mono
thf(fact_322_in__mono,axiom,
    ! [A4: set_c,B6: set_c,X3: c] :
      ( ( ord_less_eq_set_c @ A4 @ B6 )
     => ( ( member_c @ X3 @ A4 )
       => ( member_c @ X3 @ B6 ) ) ) ).

% in_mono
thf(fact_323_in__mono,axiom,
    ! [A4: set_nat,B6: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( member_nat @ X3 @ A4 )
       => ( member_nat @ X3 @ B6 ) ) ) ).

% in_mono
thf(fact_324_size__neq__size__imp__neq,axiom,
    ! [X3: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( size_s2614380629626057239od_c_a @ X3 )
       != ( size_s2614380629626057239od_c_a @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_325_size__neq__size__imp__neq,axiom,
    ! [X3: char,Y3: char] :
      ( ( ( size_size_char @ X3 )
       != ( size_size_char @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_326_bot__empty__eq,axiom,
    ( bot_bo7573314457883560170_nat_o
    = ( ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq
thf(fact_327_bot__empty__eq,axiom,
    ( bot_bot_c_o
    = ( ^ [X: c] : ( member_c @ X @ bot_bot_set_c ) ) ) ).

% bot_empty_eq
thf(fact_328_bot__empty__eq,axiom,
    ( bot_bot_b_o
    = ( ^ [X: b] : ( member_b @ X @ bot_bot_set_b ) ) ) ).

% bot_empty_eq
thf(fact_329_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_330_bot__set__def,axiom,
    ( bot_bot_set_b
    = ( collect_b @ bot_bot_b_o ) ) ).

% bot_set_def
thf(fact_331_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_332_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_333_bot__empty__eq2,axiom,
    ( bot_bot_c_nat_o
    = ( ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ bot_bo1794802125927853641_c_nat ) ) ) ).

% bot_empty_eq2
thf(fact_334_bot__empty__eq2,axiom,
    ( bot_bot_c_a_o
    = ( ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ bot_bo2086078286244720881od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_335_bot__empty__eq2,axiom,
    ( bot_bo7601773435270025082_nat_o
    = ( ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ bot_bo7474904860610844231_c_nat ) ) ) ).

% bot_empty_eq2
thf(fact_336_bot__empty__eq2,axiom,
    ( bot_bot_b_c_o
    = ( ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ bot_bo2863795366942399540od_b_c ) ) ) ).

% bot_empty_eq2
thf(fact_337_bot__empty__eq2,axiom,
    ( bot_bot_b_a_o
    = ( ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ bot_bo2721727286849707570od_b_a ) ) ) ).

% bot_empty_eq2
thf(fact_338_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_339_order__antisym__conv,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_340_order__antisym__conv,axiom,
    ! [Y3: c,X3: c] :
      ( ( ord_less_eq_c @ Y3 @ X3 )
     => ( ( ord_less_eq_c @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_341_order__antisym__conv,axiom,
    ! [Y3: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_set_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_342_linorder__le__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_343_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_344_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > c,C: c] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_345_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_346_ord__le__eq__subst,axiom,
    ! [A: c,B: c,F: c > nat,C: nat] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_347_ord__le__eq__subst,axiom,
    ! [A: c,B: c,F: c > c,C: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_348_ord__le__eq__subst,axiom,
    ! [A: c,B: c,F: c > set_nat,C: set_nat] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_349_ord__le__eq__subst,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_350_ord__le__eq__subst,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > c,C: c] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_351_ord__le__eq__subst,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_352_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_353_ord__eq__le__subst,axiom,
    ! [A: c,F: nat > c,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_354_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_355_ord__eq__le__subst,axiom,
    ! [A: nat,F: c > nat,B: c,C: c] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_356_ord__eq__le__subst,axiom,
    ! [A: c,F: c > c,B: c,C: c] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_357_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: c > set_nat,B: c,C: c] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_358_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_359_ord__eq__le__subst,axiom,
    ! [A: c,F: set_nat > c,B: set_nat,C: set_nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_360_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_361_linorder__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_362_order__eq__refl,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_363_order__eq__refl,axiom,
    ! [X3: c,Y3: c] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_c @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_364_order__eq__refl,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_365_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_366_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > c,C: c] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_c @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_367_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_368_order__subst2,axiom,
    ! [A: c,B: c,F: c > nat,C: nat] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_369_order__subst2,axiom,
    ! [A: c,B: c,F: c > c,C: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_eq_c @ ( F @ B ) @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_370_order__subst2,axiom,
    ! [A: c,B: c,F: c > set_nat,C: set_nat] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_371_order__subst2,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_372_order__subst2,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > c,C: c] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_c @ ( F @ B ) @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_373_order__subst2,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_374_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_375_order__subst1,axiom,
    ! [A: nat,F: c > nat,B: c,C: c] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_376_order__subst1,axiom,
    ! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_377_order__subst1,axiom,
    ! [A: c,F: nat > c,B: nat,C: nat] :
      ( ( ord_less_eq_c @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_378_order__subst1,axiom,
    ! [A: c,F: c > c,B: c,C: c] :
      ( ( ord_less_eq_c @ A @ ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_379_order__subst1,axiom,
    ! [A: c,F: set_nat > c,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_c @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_c @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_380_order__subst1,axiom,
    ! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_381_order__subst1,axiom,
    ! [A: set_nat,F: c > set_nat,B: c,C: c] :
      ( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_382_order__subst1,axiom,
    ! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_383_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_384_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: c,Z2: c] : ( Y6 = Z2 ) )
    = ( ^ [A6: c,B7: c] :
          ( ( ord_less_eq_c @ A6 @ B7 )
          & ( ord_less_eq_c @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_385_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_386_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_387_antisym,axiom,
    ! [A: c,B: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_eq_c @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_388_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_389_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_390_dual__order_Otrans,axiom,
    ! [B: c,A: c,C: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( ( ord_less_eq_c @ C @ B )
       => ( ord_less_eq_c @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_391_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_392_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_393_dual__order_Oantisym,axiom,
    ! [B: c,A: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( ( ord_less_eq_c @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_394_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_395_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_396_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: c,Z2: c] : ( Y6 = Z2 ) )
    = ( ^ [A6: c,B7: c] :
          ( ( ord_less_eq_c @ B7 @ A6 )
          & ( ord_less_eq_c @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_397_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ( ord_less_eq_set_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_398_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_399_order__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z3 )
       => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_400_order__trans,axiom,
    ! [X3: c,Y3: c,Z3: c] :
      ( ( ord_less_eq_c @ X3 @ Y3 )
     => ( ( ord_less_eq_c @ Y3 @ Z3 )
       => ( ord_less_eq_c @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_401_order__trans,axiom,
    ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ Z3 )
       => ( ord_less_eq_set_nat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_402_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_403_order_Otrans,axiom,
    ! [A: c,B: c,C: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ord_less_eq_c @ A @ C ) ) ) ).

% order.trans
thf(fact_404_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_405_order__antisym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_406_order__antisym,axiom,
    ! [X3: c,Y3: c] :
      ( ( ord_less_eq_c @ X3 @ Y3 )
     => ( ( ord_less_eq_c @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_407_order__antisym,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_408_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_409_ord__le__eq__trans,axiom,
    ! [A: c,B: c,C: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_c @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_410_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_411_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_412_ord__eq__le__trans,axiom,
    ! [A: c,B: c,C: c] :
      ( ( A = B )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ord_less_eq_c @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_413_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_414_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_415_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: c,Z2: c] : ( Y6 = Z2 ) )
    = ( ^ [X: c,Y: c] :
          ( ( ord_less_eq_c @ X @ Y )
          & ( ord_less_eq_c @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_416_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_417_le__cases3,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_418_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_419_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_420_less__imp__neq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_421_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_422_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_423_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_424_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X4 )
             => ( P @ Y5 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_425_antisym__conv3,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_426_linorder__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_427_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_428_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_429_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N3 )
             => ~ ( P4 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_430_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_431_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_432_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_433_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_434_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_435_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_436_linorder__neqE,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_437_order__less__asym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_438_linorder__neq__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_nat @ X3 @ Y3 )
        | ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_439_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_440_order__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_441_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_442_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_443_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_444_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_445_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_446_order__less__not__sym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_447_order__less__imp__triv,axiom,
    ! [X3: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_448_linorder__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_449_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_450_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_451_order__less__imp__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_452_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M7: nat] :
      ( ( P @ X3 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M7 ) )
       => ~ ! [M: nat] :
              ( ( P @ M )
             => ~ ! [X7: nat] :
                    ( ( P @ X7 )
                   => ( ord_less_eq_nat @ X7 @ M ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_453_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X3
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_454_leD,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_455_leD,axiom,
    ! [Y3: c,X3: c] :
      ( ( ord_less_eq_c @ Y3 @ X3 )
     => ~ ( ord_less_c @ X3 @ Y3 ) ) ).

% leD
thf(fact_456_leD,axiom,
    ! [Y3: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X3 )
     => ~ ( ord_less_set_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_457_leI,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% leI
thf(fact_458_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_459_nless__le,axiom,
    ! [A: c,B: c] :
      ( ( ~ ( ord_less_c @ A @ B ) )
      = ( ~ ( ord_less_eq_c @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_460_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_461_antisym__conv1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_462_antisym__conv1,axiom,
    ! [X3: c,Y3: c] :
      ( ~ ( ord_less_c @ X3 @ Y3 )
     => ( ( ord_less_eq_c @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_463_antisym__conv1,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ~ ( ord_less_set_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_set_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_464_antisym__conv2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_465_antisym__conv2,axiom,
    ! [X3: c,Y3: c] :
      ( ( ord_less_eq_c @ X3 @ Y3 )
     => ( ( ~ ( ord_less_c @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_466_antisym__conv2,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_set_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_467_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_468_less__le__not__le,axiom,
    ( ord_less_c
    = ( ^ [X: c,Y: c] :
          ( ( ord_less_eq_c @ X @ Y )
          & ~ ( ord_less_eq_c @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_469_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_470_not__le__imp__less,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ord_less_nat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_471_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_nat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_472_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_c
    = ( ^ [A6: c,B7: c] :
          ( ( ord_less_c @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_473_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_474_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_475_order_Ostrict__iff__order,axiom,
    ( ord_less_c
    = ( ^ [A6: c,B7: c] :
          ( ( ord_less_eq_c @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_476_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_477_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_478_order_Ostrict__trans1,axiom,
    ! [A: c,B: c,C: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_c @ B @ C )
       => ( ord_less_c @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_479_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_480_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_481_order_Ostrict__trans2,axiom,
    ! [A: c,B: c,C: c] :
      ( ( ord_less_c @ A @ B )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ord_less_c @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_482_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_483_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ~ ( ord_less_eq_nat @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_484_order_Ostrict__iff__not,axiom,
    ( ord_less_c
    = ( ^ [A6: c,B7: c] :
          ( ( ord_less_eq_c @ A6 @ B7 )
          & ~ ( ord_less_eq_c @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_485_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_486_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_less_nat @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_487_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_c
    = ( ^ [B7: c,A6: c] :
          ( ( ord_less_c @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_488_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( ord_less_set_nat @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_489_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_490_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_c
    = ( ^ [B7: c,A6: c] :
          ( ( ord_less_eq_c @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_491_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_492_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_493_dual__order_Ostrict__trans1,axiom,
    ! [B: c,A: c,C: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( ( ord_less_c @ C @ B )
       => ( ord_less_c @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_494_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_495_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_496_dual__order_Ostrict__trans2,axiom,
    ! [B: c,A: c,C: c] :
      ( ( ord_less_c @ B @ A )
     => ( ( ord_less_eq_c @ C @ B )
       => ( ord_less_c @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_497_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_498_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_499_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_c
    = ( ^ [B7: c,A6: c] :
          ( ( ord_less_eq_c @ B7 @ A6 )
          & ~ ( ord_less_eq_c @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_500_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ~ ( ord_less_eq_set_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_501_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_502_order_Ostrict__implies__order,axiom,
    ! [A: c,B: c] :
      ( ( ord_less_c @ A @ B )
     => ( ord_less_eq_c @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_503_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_504_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_505_dual__order_Ostrict__implies__order,axiom,
    ! [B: c,A: c] :
      ( ( ord_less_c @ B @ A )
     => ( ord_less_eq_c @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_506_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_507_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_508_order__le__less,axiom,
    ( ord_less_eq_c
    = ( ^ [X: c,Y: c] :
          ( ( ord_less_c @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_509_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_set_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_510_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_511_order__less__le,axiom,
    ( ord_less_c
    = ( ^ [X: c,Y: c] :
          ( ( ord_less_eq_c @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_512_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_513_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_514_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_515_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_516_order__less__imp__le,axiom,
    ! [X3: c,Y3: c] :
      ( ( ord_less_c @ X3 @ Y3 )
     => ( ord_less_eq_c @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_517_order__less__imp__le,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ord_less_set_nat @ X3 @ Y3 )
     => ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_518_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_519_order__le__neq__trans,axiom,
    ! [A: c,B: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( A != B )
       => ( ord_less_c @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_520_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_521_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_522_order__neq__le__trans,axiom,
    ! [A: c,B: c] :
      ( ( A != B )
     => ( ( ord_less_eq_c @ A @ B )
       => ( ord_less_c @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_523_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_524_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_525_order__le__less__trans,axiom,
    ! [X3: c,Y3: c,Z3: c] :
      ( ( ord_less_eq_c @ X3 @ Y3 )
     => ( ( ord_less_c @ Y3 @ Z3 )
       => ( ord_less_c @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_526_order__le__less__trans,axiom,
    ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y3 )
     => ( ( ord_less_set_nat @ Y3 @ Z3 )
       => ( ord_less_set_nat @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_527_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_528_order__less__le__trans,axiom,
    ! [X3: c,Y3: c,Z3: c] :
      ( ( ord_less_c @ X3 @ Y3 )
     => ( ( ord_less_eq_c @ Y3 @ Z3 )
       => ( ord_less_c @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_529_order__less__le__trans,axiom,
    ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
      ( ( ord_less_set_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_set_nat @ Y3 @ Z3 )
       => ( ord_less_set_nat @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_530_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_531_order__le__less__subst1,axiom,
    ! [A: c,F: nat > c,B: nat,C: nat] :
      ( ( ord_less_eq_c @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_532_order__le__less__subst1,axiom,
    ! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_533_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_534_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > c,C: c] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_c @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_535_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_536_order__le__less__subst2,axiom,
    ! [A: c,B: c,F: c > nat,C: nat] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_537_order__le__less__subst2,axiom,
    ! [A: c,B: c,F: c > c,C: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_c @ ( F @ B ) @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_538_order__le__less__subst2,axiom,
    ! [A: c,B: c,F: c > set_nat,C: set_nat] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( ord_less_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_539_order__le__less__subst2,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_540_order__le__less__subst2,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > c,C: c] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_c @ ( F @ B ) @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_541_order__le__less__subst2,axiom,
    ! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_542_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_543_order__less__le__subst1,axiom,
    ! [A: c,F: nat > c,B: nat,C: nat] :
      ( ( ord_less_c @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_544_order__less__le__subst1,axiom,
    ! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
      ( ( ord_less_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_545_order__less__le__subst1,axiom,
    ! [A: nat,F: c > nat,B: c,C: c] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_546_order__less__le__subst1,axiom,
    ! [A: c,F: c > c,B: c,C: c] :
      ( ( ord_less_c @ A @ ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_547_order__less__le__subst1,axiom,
    ! [A: set_nat,F: c > set_nat,B: c,C: c] :
      ( ( ord_less_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_c @ B @ C )
       => ( ! [X4: c,Y4: c] :
              ( ( ord_less_eq_c @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_548_order__less__le__subst1,axiom,
    ! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_549_order__less__le__subst1,axiom,
    ! [A: c,F: set_nat > c,B: set_nat,C: set_nat] :
      ( ( ord_less_c @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_550_order__less__le__subst1,axiom,
    ! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X4: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_551_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_552_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > c,C: c] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_c @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_c @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_c @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_553_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( ord_less_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_554_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_555_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_556_order__le__imp__less__or__eq,axiom,
    ! [X3: c,Y3: c] :
      ( ( ord_less_eq_c @ X3 @ Y3 )
     => ( ( ord_less_c @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_557_order__le__imp__less__or__eq,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y3 )
     => ( ( ord_less_set_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_558_bot_Oextremum__uniqueI,axiom,
    ! [A: set_b] :
      ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
     => ( A = bot_bot_set_b ) ) ).

% bot.extremum_uniqueI
thf(fact_559_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_560_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_561_bot_Oextremum__unique,axiom,
    ! [A: set_b] :
      ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
      = ( A = bot_bot_set_b ) ) ).

% bot.extremum_unique
thf(fact_562_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_563_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_564_bot_Oextremum,axiom,
    ! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).

% bot.extremum
thf(fact_565_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_566_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_567_bot_Oextremum__strict,axiom,
    ! [A: set_b] :
      ~ ( ord_less_set_b @ A @ bot_bot_set_b ) ).

% bot.extremum_strict
thf(fact_568_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_569_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_570_bot_Onot__eq__extremum,axiom,
    ! [A: set_b] :
      ( ( A != bot_bot_set_b )
      = ( ord_less_set_b @ bot_bot_set_b @ A ) ) ).

% bot.not_eq_extremum
thf(fact_571_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_572_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_573_finite__transitivity__chain,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ A4 )
     => ( ! [X4: produc859450856879609959at_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A4 )
               => ? [Y5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Y5 @ A4 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A4 = bot_bo5327735625951526323at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_574_finite__transitivity__chain,axiom,
    ! [A4: set_c,R: c > c > $o] :
      ( ( finite_finite_c @ A4 )
     => ( ! [X4: c] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: c,Y4: c,Z: c] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: c] :
                ( ( member_c @ X4 @ A4 )
               => ? [Y5: c] :
                    ( ( member_c @ Y5 @ A4 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A4 = bot_bot_set_c ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_575_finite__transitivity__chain,axiom,
    ! [A4: set_b,R: b > b > $o] :
      ( ( finite_finite_b @ A4 )
     => ( ! [X4: b] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: b,Y4: b,Z: b] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: b] :
                ( ( member_b @ X4 @ A4 )
               => ? [Y5: b] :
                    ( ( member_b @ Y5 @ A4 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A4 = bot_bot_set_b ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_576_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X4: nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y4: nat,Z: nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A4 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_577_unbounded__k__infinite,axiom,
    ! [K2: nat,S: set_nat] :
      ( ! [M: nat] :
          ( ( ord_less_nat @ K2 @ M )
         => ? [N5: nat] :
              ( ( ord_less_nat @ M @ N5 )
              & ( member_nat @ N5 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_578_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_nat @ X4 @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_579_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M6: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M6 @ N3 )
            & ( member_nat @ N3 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_580_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N7: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N7 )
         => ( ord_less_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_581_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N7: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N7 )
         => ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_582_in__finite__psubset,axiom,
    ! [A4: set_b,B6: set_b] :
      ( ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ A4 @ B6 ) @ finite_psubset_b )
      = ( ( ord_less_set_b @ A4 @ B6 )
        & ( finite_finite_b @ B6 ) ) ) ).

% in_finite_psubset
thf(fact_583_in__finite__psubset,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B6 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A4 @ B6 )
        & ( finite_finite_nat @ B6 ) ) ) ).

% in_finite_psubset
thf(fact_584_arg__min__if__finite_I2_J,axiom,
    ! [S: set_b,F: b > nat] :
      ( ( finite_finite_b @ S )
     => ( ( S != bot_bot_set_b )
       => ~ ? [X7: b] :
              ( ( member_b @ X7 @ S )
              & ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic7575731748627795062_b_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_585_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X7: nat] :
              ( ( member_nat @ X7 @ S )
              & ( ord_less_nat @ ( F @ X7 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_586_arg__min__least,axiom,
    ! [S: set_Pr8693737435421807431at_nat,Y3: produc859450856879609959at_nat,F: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( S != bot_bo5327735625951526323at_nat )
       => ( ( member8206827879206165904at_nat @ Y3 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic390166758595302878at_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_587_arg__min__least,axiom,
    ! [S: set_c,Y3: c,F: c > nat] :
      ( ( finite_finite_c @ S )
     => ( ( S != bot_bot_set_c )
       => ( ( member_c @ Y3 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic8811176077584189559_c_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_588_arg__min__least,axiom,
    ! [S: set_b,Y3: b,F: b > nat] :
      ( ( finite_finite_b @ S )
     => ( ( S != bot_bot_set_b )
       => ( ( member_b @ Y3 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7575731748627795062_b_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_589_arg__min__least,axiom,
    ! [S: set_nat,Y3: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y3 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_590_finite__maxlen,axiom,
    ! [M7: set_li4905775889721270027od_c_a] :
      ( ( finite2296398606116454484od_c_a @ M7 )
     => ? [N2: nat] :
        ! [X7: list_P125642481956313003od_c_a] :
          ( ( member5552703068553123156od_c_a @ X7 @ M7 )
         => ( ord_less_nat @ ( size_s2614380629626057239od_c_a @ X7 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_591_length__induct,axiom,
    ! [P: list_P125642481956313003od_c_a > $o,Xs: list_P125642481956313003od_c_a] :
      ( ! [Xs2: list_P125642481956313003od_c_a] :
          ( ! [Ys: list_P125642481956313003od_c_a] :
              ( ( ord_less_nat @ ( size_s2614380629626057239od_c_a @ Ys ) @ ( size_s2614380629626057239od_c_a @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_592_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_593_subset__emptyI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ! [X4: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ X4 @ A4 )
     => ( ord_le3000389064537975527at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ).

% subset_emptyI
thf(fact_594_subset__emptyI,axiom,
    ! [A4: set_c] :
      ( ! [X4: c] :
          ~ ( member_c @ X4 @ A4 )
     => ( ord_less_eq_set_c @ A4 @ bot_bot_set_c ) ) ).

% subset_emptyI
thf(fact_595_subset__emptyI,axiom,
    ! [A4: set_b] :
      ( ! [X4: b] :
          ~ ( member_b @ X4 @ A4 )
     => ( ord_less_eq_set_b @ A4 @ bot_bot_set_b ) ) ).

% subset_emptyI
thf(fact_596_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X4: nat] :
          ~ ( member_nat @ X4 @ A4 )
     => ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_597_sup__leadsto__SomeE,axiom,
    ! [I: nat,J: nat,Init: b,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b,Ts: c] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( sup_leadsto_b_a_c @ Init @ Step @ Rho @ I @ J @ Q )
          = ( some_c @ Ts ) )
       => ? [L3: nat] :
            ( ( ord_less_nat @ L3 @ I )
            & ( ( steps_b_a_c @ Step @ Rho @ Init @ ( product_Pair_nat_nat @ L3 @ J ) )
              = Q )
            & ( ( ts_at_c_a @ Rho @ L3 )
              = Ts ) ) ) ) ).

% sup_leadsto_SomeE
thf(fact_598_Mapping__keys__dest,axiom,
    ! [X3: nat,F: list_P3509250719794200884_c_nat] :
      ( ( member_nat @ X3 @ ( mmap_k6232465067979418574_c_nat @ F ) )
     => ? [Y4: produc4862256710654508797_c_nat] :
          ( ( mmap_l3750577438354026580_c_nat @ F @ X3 )
          = ( some_P2720002978653898840_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_599_Mapping__keys__dest,axiom,
    ! [X3: produc859450856879609959at_nat,F: list_P6274885450196660515_c_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( mmap_k8225290564293298213_c_nat @ F ) )
     => ? [Y4: produc4862256710654508797_c_nat] :
          ( ( mmap_l3052497582771463455_c_nat @ F @ X3 )
          = ( some_P2720002978653898840_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_600_Mapping__keys__dest,axiom,
    ! [X3: c,F: list_P4207096184057010108_c_nat] :
      ( ( member_c @ X3 @ ( mmap_k6678550602049508158_c_nat @ F ) )
     => ? [Y4: produc4862256710654508797_c_nat] :
          ( ( mmap_l8624657524578542648_c_nat @ F @ X3 )
          = ( some_P2720002978653898840_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_601_Mapping__keys__dest,axiom,
    ! [X3: b,F: list_P5567253521434164541_c_nat] :
      ( ( member_b @ X3 @ ( mmap_k3066642428818769471_c_nat @ F ) )
     => ? [Y4: product_prod_c_nat] :
          ( ( mmap_l4793845786776461369_c_nat @ F @ X3 )
          = ( some_P8722241760384591706_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_602_Mapping__keys__dest,axiom,
    ! [X3: nat,F: list_P8567156416134829366_c_nat] :
      ( ( member_nat @ X3 @ ( mmap_k4838861487035128272_c_nat @ F ) )
     => ? [Y4: product_prod_c_nat] :
          ( ( mmap_l851485680277214550_c_nat @ F @ X3 )
          = ( some_P8722241760384591706_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_603_Mapping__keys__dest,axiom,
    ! [X3: produc859450856879609959at_nat,F: list_P2747636819685220005_c_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( mmap_k151633161116578727_c_nat @ F ) )
     => ? [Y4: product_prod_c_nat] :
          ( ( mmap_l440427023202933153_c_nat @ F @ X3 )
          = ( some_P8722241760384591706_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_604_Mapping__keys__dest,axiom,
    ! [X3: c,F: list_P283830071204224958_c_nat] :
      ( ( member_c @ X3 @ ( mmap_k950214370828294464_c_nat @ F ) )
     => ? [Y4: product_prod_c_nat] :
          ( ( mmap_l2677417728785986362_c_nat @ F @ X3 )
          = ( some_P8722241760384591706_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_605_Mapping__keys__dest,axiom,
    ! [X3: b,F: list_P903359562653991662od_b_c] :
      ( ( member_b @ X3 @ ( mmap_keys_b_c @ F ) )
     => ? [Y4: c] :
          ( ( mmap_lookup_b_c @ F @ X3 )
          = ( some_c @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_606_Mapping__keys__dest,axiom,
    ! [X3: b,F: list_P7417839048565863355_c_nat] :
      ( ( member_b @ X3 @ ( mmap_k3080892797436903101_c_nat @ F ) )
     => ? [Y4: produc4862256710654508797_c_nat] :
          ( ( mmap_l5026999719965937591_c_nat @ F @ X3 )
          = ( some_P2720002978653898840_c_nat @ Y4 ) ) ) ).

% Mapping_keys_dest
thf(fact_607_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S2: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_608_ssubst__Pair__rhs,axiom,
    ! [R2: c,S2: nat,R: set_Pr6903500605879609269_c_nat,S3: nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_609_ssubst__Pair__rhs,axiom,
    ! [R2: c,S2: a,R: set_Product_prod_c_a,S3: a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_610_ssubst__Pair__rhs,axiom,
    ! [R2: b,S2: option7520157102916957007_c_nat,R: set_Pr8806432033423503795_c_nat,S3: option7520157102916957007_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_611_ssubst__Pair__rhs,axiom,
    ! [R2: b,S2: c,R: set_Product_prod_b_c,S3: c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_612_ssubst__Pair__rhs,axiom,
    ! [R2: b,S2: a,R: set_Product_prod_b_a,S3: a] :
      ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_613_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_614_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_P125642481956313003od_c_a] :
      ( ( size_s2614380629626057239od_c_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_615_neq__if__length__neq,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
       != ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_616_prop__restrict,axiom,
    ! [X3: produc859450856879609959at_nat,Z4: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ Z4 )
     => ( ( ord_le3000389064537975527at_nat @ Z4
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_617_prop__restrict,axiom,
    ! [X3: c,Z4: set_c,X5: set_c,P: c > $o] :
      ( ( member_c @ X3 @ Z4 )
     => ( ( ord_less_eq_set_c @ Z4
          @ ( collect_c
            @ ^ [X: c] :
                ( ( member_c @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_618_prop__restrict,axiom,
    ! [X3: b,Z4: set_b,X5: set_b,P: b > $o] :
      ( ( member_b @ X3 @ Z4 )
     => ( ( ord_less_eq_set_b @ Z4
          @ ( collect_b
            @ ^ [X: b] :
                ( ( member_b @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_619_prop__restrict,axiom,
    ! [X3: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_620_Collect__restrict,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_621_Collect__restrict,axiom,
    ! [X5: set_c,P: c > $o] :
      ( ord_less_eq_set_c
      @ ( collect_c
        @ ^ [X: c] :
            ( ( member_c @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_622_Collect__restrict,axiom,
    ! [X5: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_623_Collect__restrict,axiom,
    ! [X5: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_624_sup__leadsto__SomeI,axiom,
    ! [L2: nat,I: nat,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Init: b,J: nat,Q: b] :
      ( ( ord_less_nat @ L2 @ I )
     => ( ( ( steps_b_a_c @ Step @ Rho @ Init @ ( product_Pair_nat_nat @ L2 @ J ) )
          = Q )
       => ? [L4: nat] :
            ( ( ( sup_leadsto_b_a_c @ Init @ Step @ Rho @ I @ J @ Q )
              = ( some_c @ ( ts_at_c_a @ Rho @ L4 ) ) )
            & ( ord_less_eq_nat @ L2 @ L4 )
            & ( ord_less_nat @ L4 @ I ) ) ) ) ).

% sup_leadsto_SomeI
thf(fact_625_option_Oinject,axiom,
    ! [X2: produc4862256710654508797_c_nat,Y2: produc4862256710654508797_c_nat] :
      ( ( ( some_P2720002978653898840_c_nat @ X2 )
        = ( some_P2720002978653898840_c_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_626_option_Oinject,axiom,
    ! [X2: product_prod_c_nat,Y2: product_prod_c_nat] :
      ( ( ( some_P8722241760384591706_c_nat @ X2 )
        = ( some_P8722241760384591706_c_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_627_Collect__empty__eq__bot,axiom,
    ! [P: b > $o] :
      ( ( ( collect_b @ P )
        = bot_bot_set_b )
      = ( P = bot_bot_b_o ) ) ).

% Collect_empty_eq_bot
thf(fact_628_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_629_acc__app,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b,Q3: b,Accept: b > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) )
          = Q3 )
       => ( ( Accept @ Q3 )
         => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ ( suc @ J ) )
            = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ ( ts_at_c_a @ Rho @ J ) @ J ) ) ) ) ) ) ).

% acc_app
thf(fact_630_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X7: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X7 )
                    & ( ord_less_nat @ X7 @ C3 ) )
                 => ( P @ X7 ) )
              & ! [D: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X4 )
                        & ( ord_less_nat @ X4 @ D ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_631_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_632_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ~ ( ord_less_eq_nat @ X7 @ T ) ) ).

% pinf(6)
thf(fact_633_sup__acc__Some__ts,axiom,
    ! [Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,I: nat,J: nat,Ts: c,Tp: nat] :
      ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
        = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ Ts @ Tp ) ) )
     => ( Ts
        = ( ts_at_c_a @ Rho @ Tp ) ) ) ).

% sup_acc_Some_ts
thf(fact_634_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_635_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_636_verit__comp__simplify1_I2_J,axiom,
    ! [A: c] : ( ord_less_eq_c @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_637_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_638_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ~ ( ord_less_nat @ T @ X7 ) ) ).

% minf(7)
thf(fact_639_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ( ord_less_nat @ X7 @ T ) ) ).

% minf(5)
thf(fact_640_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ( X7 != T ) ) ).

% minf(4)
thf(fact_641_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ( X7 != T ) ) ).

% minf(3)
thf(fact_642_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q7: nat > $o,Q8: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q7 @ X4 )
              = ( Q8 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ X7 @ Z )
           => ( ( ( P @ X7 )
                | ( Q7 @ X7 ) )
              = ( ( P5 @ X7 )
                | ( Q8 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_643_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q7: nat > $o,Q8: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q7 @ X4 )
              = ( Q8 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ X7 @ Z )
           => ( ( ( P @ X7 )
                & ( Q7 @ X7 ) )
              = ( ( P5 @ X7 )
                & ( Q8 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_644_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ( ord_less_nat @ T @ X7 ) ) ).

% pinf(7)
thf(fact_645_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ~ ( ord_less_nat @ X7 @ T ) ) ).

% pinf(5)
thf(fact_646_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ( X7 != T ) ) ).

% pinf(4)
thf(fact_647_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ( X7 != T ) ) ).

% pinf(3)
thf(fact_648_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q7: nat > $o,Q8: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q7 @ X4 )
              = ( Q8 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ Z @ X7 )
           => ( ( ( P @ X7 )
                | ( Q7 @ X7 ) )
              = ( ( P5 @ X7 )
                | ( Q8 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_649_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q7: nat > $o,Q8: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z5 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( Q7 @ X4 )
              = ( Q8 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ Z @ X7 )
           => ( ( ( P @ X7 )
                & ( Q7 @ X7 ) )
              = ( ( P5 @ X7 )
                & ( Q8 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_650_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_651_acc__app__idle,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Rho: list_P125642481956313003od_c_a,Q: b,Q3: b,Accept: b > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) )
          = Q3 )
       => ( ~ ( Accept @ Q3 )
         => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ ( suc @ J ) )
            = ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J ) ) ) ) ) ).

% acc_app_idle
thf(fact_652_sup__acc__comp__Some__ge,axiom,
    ! [I: nat,L2: nat,J: nat,Tp: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,Ts: c] :
      ( ( ord_less_eq_nat @ I @ L2 )
     => ( ( ord_less_eq_nat @ L2 @ J )
       => ( ( ord_less_eq_nat @ L2 @ Tp )
         => ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ L2 ) ) @ L2 @ J )
              = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ Ts @ Tp ) ) )
           => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
              = ( sup_acc_b_a_c @ Step @ Accept @ Rho @ ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ L2 ) ) @ L2 @ J ) ) ) ) ) ) ).

% sup_acc_comp_Some_ge
thf(fact_653_sup__acc__l,axiom,
    ! [I: nat,J: nat,L2: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,Ts: c] :
      ( ( ord_less_nat @ I @ J )
     => ( ( I != L2 )
       => ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
            = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ Ts @ L2 ) ) )
         => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
            = ( sup_acc_b_a_c @ Step @ Accept @ Rho @ ( Step @ Q @ ( bs_at_c_a @ Rho @ I ) ) @ ( suc @ I ) @ J ) ) ) ) ) ).

% sup_acc_l
thf(fact_654_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ~ ( ord_less_eq_nat @ T @ X7 ) ) ).

% minf(8)
thf(fact_655_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ( ord_less_eq_nat @ X7 @ T ) ) ).

% minf(6)
thf(fact_656_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ( ord_less_eq_nat @ T @ X7 ) ) ).

% pinf(8)
thf(fact_657_sup__acc__ext,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( acc_b_a_c @ Step @ Accept @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) )
       => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ ( suc @ J ) )
          = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ ( ts_at_c_a @ Rho @ J ) @ J ) ) ) ) ) ).

% sup_acc_ext
thf(fact_658_lookup__s_H,axiom,
    ! [Q: b,Q3: b,Tstp: option7520157102916957007_c_nat] :
      ( ( ( mmap_l5026999719965937591_c_nat @ s @ Q )
        = ( some_P2720002978653898840_c_nat @ ( produc5716802255202478839_c_nat @ Q3 @ Tstp ) ) )
     => ( ( ( steps_b_a_c @ step @ rho2 @ Q @ ( product_Pair_nat_nat @ i @ ( suc @ j ) ) )
          = Q3 )
        & ( Tstp
          = ( sup_acc_b_a_c @ step @ accept @ rho2 @ Q @ i @ ( suc @ j ) ) ) ) ) ).

% lookup_s'
thf(fact_659_lookup__s,axiom,
    ! [Q: b,Q3: b,Tstp: option7520157102916957007_c_nat] :
      ( ( ( mmap_l5026999719965937591_c_nat @ s2 @ Q )
        = ( some_P2720002978653898840_c_nat @ ( produc5716802255202478839_c_nat @ Q3 @ Tstp ) ) )
     => ( ( ( steps_b_a_c @ step @ rho2 @ Q @ ( product_Pair_nat_nat @ i @ j ) )
          = Q3 )
        & ( Tstp
          = ( sup_acc_b_a_c @ step @ accept @ rho2 @ Q @ i @ j ) ) ) ) ).

% lookup_s
thf(fact_660_sup__acc__ext__idle,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ~ ( acc_b_a_c @ Step @ Accept @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) )
       => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ ( suc @ J ) )
          = ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J ) ) ) ) ).

% sup_acc_ext_idle
thf(fact_661_sup__acc__i,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,Ts: c] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
          = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ Ts @ I ) ) )
       => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ ( Step @ Q @ ( bs_at_c_a @ Rho @ I ) ) @ ( suc @ I ) @ J )
          = none_P6573378090780909534_c_nat ) ) ) ).

% sup_acc_i
thf(fact_662_map__values__lookup,axiom,
    ! [F: b > c > product_prod_c_nat,M2: list_P903359562653991662od_b_c,Z3: b,V: product_prod_c_nat] :
      ( ( ( mmap_l4793845786776461369_c_nat @ ( mmap_m2520143655228893010_c_nat @ F @ M2 ) @ Z3 )
        = ( some_P8722241760384591706_c_nat @ V ) )
     => ? [V2: c] :
          ( ( ( mmap_lookup_b_c @ M2 @ Z3 )
            = ( some_c @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_663_map__values__lookup,axiom,
    ! [F: b > produc4862256710654508797_c_nat > product_prod_c_nat,M2: list_P7417839048565863355_c_nat,Z3: b,V: product_prod_c_nat] :
      ( ( ( mmap_l4793845786776461369_c_nat @ ( mmap_m3116818598217024619_c_nat @ F @ M2 ) @ Z3 )
        = ( some_P8722241760384591706_c_nat @ V ) )
     => ? [V2: produc4862256710654508797_c_nat] :
          ( ( ( mmap_l5026999719965937591_c_nat @ M2 @ Z3 )
            = ( some_P2720002978653898840_c_nat @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_664_map__values__lookup,axiom,
    ! [F: b > product_prod_c_nat > c,M2: list_P5567253521434164541_c_nat,Z3: b,V: c] :
      ( ( ( mmap_lookup_b_c @ ( mmap_m8582951018221325842_nat_c @ F @ M2 ) @ Z3 )
        = ( some_c @ V ) )
     => ? [V2: product_prod_c_nat] :
          ( ( ( mmap_l4793845786776461369_c_nat @ M2 @ Z3 )
            = ( some_P8722241760384591706_c_nat @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_665_map__values__lookup,axiom,
    ! [F: b > c > c,M2: list_P903359562653991662od_b_c,Z3: b,V: c] :
      ( ( ( mmap_lookup_b_c @ ( mmap_map_b_c_c @ F @ M2 ) @ Z3 )
        = ( some_c @ V ) )
     => ? [V2: c] :
          ( ( ( mmap_lookup_b_c @ M2 @ Z3 )
            = ( some_c @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_666_map__values__lookup,axiom,
    ! [F: b > produc4862256710654508797_c_nat > c,M2: list_P7417839048565863355_c_nat,Z3: b,V: c] :
      ( ( ( mmap_lookup_b_c @ ( mmap_m3523663605641305108_nat_c @ F @ M2 ) @ Z3 )
        = ( some_c @ V ) )
     => ? [V2: produc4862256710654508797_c_nat] :
          ( ( ( mmap_l5026999719965937591_c_nat @ M2 @ Z3 )
            = ( some_P2720002978653898840_c_nat @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_667_map__values__lookup,axiom,
    ! [F: b > product_prod_c_nat > produc4862256710654508797_c_nat,M2: list_P5567253521434164541_c_nat,Z3: b,V: produc4862256710654508797_c_nat] :
      ( ( ( mmap_l5026999719965937591_c_nat @ ( mmap_m79571092140022379_c_nat @ F @ M2 ) @ Z3 )
        = ( some_P2720002978653898840_c_nat @ V ) )
     => ? [V2: product_prod_c_nat] :
          ( ( ( mmap_l4793845786776461369_c_nat @ M2 @ Z3 )
            = ( some_P8722241760384591706_c_nat @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_668_map__values__lookup,axiom,
    ! [F: b > c > produc4862256710654508797_c_nat,M2: list_P903359562653991662od_b_c,Z3: b,V: produc4862256710654508797_c_nat] :
      ( ( ( mmap_l5026999719965937591_c_nat @ ( mmap_m3921707120754670672_c_nat @ F @ M2 ) @ Z3 )
        = ( some_P2720002978653898840_c_nat @ V ) )
     => ? [V2: c] :
          ( ( ( mmap_lookup_b_c @ M2 @ Z3 )
            = ( some_c @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_669_map__values__lookup,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,M2: list_P7417839048565863355_c_nat,Z3: b,V: produc4862256710654508797_c_nat] :
      ( ( ( mmap_l5026999719965937591_c_nat @ ( mmap_m881505119971688169_c_nat @ F @ M2 ) @ Z3 )
        = ( some_P2720002978653898840_c_nat @ V ) )
     => ? [V2: produc4862256710654508797_c_nat] :
          ( ( ( mmap_l5026999719965937591_c_nat @ M2 @ Z3 )
            = ( some_P2720002978653898840_c_nat @ V2 ) )
          & ( V
            = ( F @ Z3 @ V2 ) ) ) ) ).

% map_values_lookup
thf(fact_670_keys__s_H,axiom,
    ( ( mmap_k3080892797436903101_c_nat @ s )
    = ( mmap_k3080892797436903101_c_nat @ s2 ) ) ).

% keys_s'
thf(fact_671_not__Some__eq,axiom,
    ! [X3: option2970150418924381261_c_nat] :
      ( ( ! [Y: produc4862256710654508797_c_nat] :
            ( X3
           != ( some_P2720002978653898840_c_nat @ Y ) ) )
      = ( X3 = none_P8487441334512977628_c_nat ) ) ).

% not_Some_eq
thf(fact_672_not__Some__eq,axiom,
    ! [X3: option7520157102916957007_c_nat] :
      ( ( ! [Y: product_prod_c_nat] :
            ( X3
           != ( some_P8722241760384591706_c_nat @ Y ) ) )
      = ( X3 = none_P6573378090780909534_c_nat ) ) ).

% not_Some_eq
thf(fact_673_not__None__eq,axiom,
    ! [X3: option2970150418924381261_c_nat] :
      ( ( X3 != none_P8487441334512977628_c_nat )
      = ( ? [Y: produc4862256710654508797_c_nat] :
            ( X3
            = ( some_P2720002978653898840_c_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_674_not__None__eq,axiom,
    ! [X3: option7520157102916957007_c_nat] :
      ( ( X3 != none_P6573378090780909534_c_nat )
      = ( ? [Y: product_prod_c_nat] :
            ( X3
            = ( some_P8722241760384591706_c_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_675_combine__options__cases,axiom,
    ! [X3: option2970150418924381261_c_nat,P: option2970150418924381261_c_nat > option2970150418924381261_c_nat > $o,Y3: option2970150418924381261_c_nat] :
      ( ( ( X3 = none_P8487441334512977628_c_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P8487441334512977628_c_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: produc4862256710654508797_c_nat,B3: produc4862256710654508797_c_nat] :
              ( ( X3
                = ( some_P2720002978653898840_c_nat @ A3 ) )
             => ( ( Y3
                  = ( some_P2720002978653898840_c_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_676_combine__options__cases,axiom,
    ! [X3: option2970150418924381261_c_nat,P: option2970150418924381261_c_nat > option7520157102916957007_c_nat > $o,Y3: option7520157102916957007_c_nat] :
      ( ( ( X3 = none_P8487441334512977628_c_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P6573378090780909534_c_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: produc4862256710654508797_c_nat,B3: product_prod_c_nat] :
              ( ( X3
                = ( some_P2720002978653898840_c_nat @ A3 ) )
             => ( ( Y3
                  = ( some_P8722241760384591706_c_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_677_combine__options__cases,axiom,
    ! [X3: option7520157102916957007_c_nat,P: option7520157102916957007_c_nat > option2970150418924381261_c_nat > $o,Y3: option2970150418924381261_c_nat] :
      ( ( ( X3 = none_P6573378090780909534_c_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P8487441334512977628_c_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: product_prod_c_nat,B3: produc4862256710654508797_c_nat] :
              ( ( X3
                = ( some_P8722241760384591706_c_nat @ A3 ) )
             => ( ( Y3
                  = ( some_P2720002978653898840_c_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_678_combine__options__cases,axiom,
    ! [X3: option7520157102916957007_c_nat,P: option7520157102916957007_c_nat > option7520157102916957007_c_nat > $o,Y3: option7520157102916957007_c_nat] :
      ( ( ( X3 = none_P6573378090780909534_c_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P6573378090780909534_c_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: product_prod_c_nat,B3: product_prod_c_nat] :
              ( ( X3
                = ( some_P8722241760384591706_c_nat @ A3 ) )
             => ( ( Y3
                  = ( some_P8722241760384591706_c_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_679_split__option__all,axiom,
    ( ( ^ [P3: option2970150418924381261_c_nat > $o] :
        ! [X6: option2970150418924381261_c_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option2970150418924381261_c_nat > $o] :
          ( ( P4 @ none_P8487441334512977628_c_nat )
          & ! [X: produc4862256710654508797_c_nat] : ( P4 @ ( some_P2720002978653898840_c_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_680_split__option__all,axiom,
    ( ( ^ [P3: option7520157102916957007_c_nat > $o] :
        ! [X6: option7520157102916957007_c_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option7520157102916957007_c_nat > $o] :
          ( ( P4 @ none_P6573378090780909534_c_nat )
          & ! [X: product_prod_c_nat] : ( P4 @ ( some_P8722241760384591706_c_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_681_split__option__ex,axiom,
    ( ( ^ [P3: option2970150418924381261_c_nat > $o] :
        ? [X6: option2970150418924381261_c_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option2970150418924381261_c_nat > $o] :
          ( ( P4 @ none_P8487441334512977628_c_nat )
          | ? [X: produc4862256710654508797_c_nat] : ( P4 @ ( some_P2720002978653898840_c_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_682_split__option__ex,axiom,
    ( ( ^ [P3: option7520157102916957007_c_nat > $o] :
        ? [X6: option7520157102916957007_c_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option7520157102916957007_c_nat > $o] :
          ( ( P4 @ none_P6573378090780909534_c_nat )
          | ? [X: product_prod_c_nat] : ( P4 @ ( some_P8722241760384591706_c_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_683_option_Oexhaust,axiom,
    ! [Y3: option2970150418924381261_c_nat] :
      ( ( Y3 != none_P8487441334512977628_c_nat )
     => ~ ! [X22: produc4862256710654508797_c_nat] :
            ( Y3
           != ( some_P2720002978653898840_c_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_684_option_Oexhaust,axiom,
    ! [Y3: option7520157102916957007_c_nat] :
      ( ( Y3 != none_P6573378090780909534_c_nat )
     => ~ ! [X22: product_prod_c_nat] :
            ( Y3
           != ( some_P8722241760384591706_c_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_685_option_OdiscI,axiom,
    ! [Option: option2970150418924381261_c_nat,X2: produc4862256710654508797_c_nat] :
      ( ( Option
        = ( some_P2720002978653898840_c_nat @ X2 ) )
     => ( Option != none_P8487441334512977628_c_nat ) ) ).

% option.discI
thf(fact_686_option_OdiscI,axiom,
    ! [Option: option7520157102916957007_c_nat,X2: product_prod_c_nat] :
      ( ( Option
        = ( some_P8722241760384591706_c_nat @ X2 ) )
     => ( Option != none_P6573378090780909534_c_nat ) ) ).

% option.discI
thf(fact_687_option_Odistinct_I1_J,axiom,
    ! [X2: produc4862256710654508797_c_nat] :
      ( none_P8487441334512977628_c_nat
     != ( some_P2720002978653898840_c_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_688_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_c_nat] :
      ( none_P6573378090780909534_c_nat
     != ( some_P8722241760384591706_c_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_689_sup__acc__same,axiom,
    ! [Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,I: nat] :
      ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ I )
      = none_P6573378090780909534_c_nat ) ).

% sup_acc_same
thf(fact_690_Window_Oacc__def,axiom,
    ( acc_b_a_c
    = ( ^ [Step2: b > a > b,Accept2: b > $o,Rho2: list_P125642481956313003od_c_a,Q2: b,Ij: product_prod_nat_nat] : ( Accept2 @ ( steps_b_a_c @ Step2 @ Rho2 @ Q2 @ Ij ) ) ) ) ).

% Window.acc_def
thf(fact_691_Mapping__keys__intro,axiom,
    ! [F: list_P903359562653991662od_b_c,X3: b] :
      ( ( ( mmap_lookup_b_c @ F @ X3 )
       != none_c )
     => ( member_b @ X3 @ ( mmap_keys_b_c @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_692_Mapping__keys__intro,axiom,
    ! [F: list_P7417839048565863355_c_nat,X3: b] :
      ( ( ( mmap_l5026999719965937591_c_nat @ F @ X3 )
       != none_P8487441334512977628_c_nat )
     => ( member_b @ X3 @ ( mmap_k3080892797436903101_c_nat @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_693_Mapping__not__keys__intro,axiom,
    ! [F: list_P903359562653991662od_b_c,X3: b] :
      ( ( ( mmap_lookup_b_c @ F @ X3 )
        = none_c )
     => ~ ( member_b @ X3 @ ( mmap_keys_b_c @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_694_Mapping__not__keys__intro,axiom,
    ! [F: list_P7417839048565863355_c_nat,X3: b] :
      ( ( ( mmap_l5026999719965937591_c_nat @ F @ X3 )
        = none_P8487441334512977628_c_nat )
     => ~ ( member_b @ X3 @ ( mmap_k3080892797436903101_c_nat @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_695_Mapping__lookup__None__intro,axiom,
    ! [X3: b,F: list_P903359562653991662od_b_c] :
      ( ~ ( member_b @ X3 @ ( mmap_keys_b_c @ F ) )
     => ( ( mmap_lookup_b_c @ F @ X3 )
        = none_c ) ) ).

% Mapping_lookup_None_intro
thf(fact_696_Mapping__lookup__None__intro,axiom,
    ! [X3: b,F: list_P7417839048565863355_c_nat] :
      ( ~ ( member_b @ X3 @ ( mmap_k3080892797436903101_c_nat @ F ) )
     => ( ( mmap_l5026999719965937591_c_nat @ F @ X3 )
        = none_P8487441334512977628_c_nat ) ) ).

% Mapping_lookup_None_intro
thf(fact_697_mmap__map__keys,axiom,
    ! [F: b > c > c,M2: list_P903359562653991662od_b_c] :
      ( ( mmap_keys_b_c @ ( mmap_map_b_c_c @ F @ M2 ) )
      = ( mmap_keys_b_c @ M2 ) ) ).

% mmap_map_keys
thf(fact_698_mmap__map__keys,axiom,
    ! [F: b > produc4862256710654508797_c_nat > c,M2: list_P7417839048565863355_c_nat] :
      ( ( mmap_keys_b_c @ ( mmap_m3523663605641305108_nat_c @ F @ M2 ) )
      = ( mmap_k3080892797436903101_c_nat @ M2 ) ) ).

% mmap_map_keys
thf(fact_699_mmap__map__keys,axiom,
    ! [F: b > c > produc4862256710654508797_c_nat,M2: list_P903359562653991662od_b_c] :
      ( ( mmap_k3080892797436903101_c_nat @ ( mmap_m3921707120754670672_c_nat @ F @ M2 ) )
      = ( mmap_keys_b_c @ M2 ) ) ).

% mmap_map_keys
thf(fact_700_mmap__map__keys,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,M2: list_P7417839048565863355_c_nat] :
      ( ( mmap_k3080892797436903101_c_nat @ ( mmap_m881505119971688169_c_nat @ F @ M2 ) )
      = ( mmap_k3080892797436903101_c_nat @ M2 ) ) ).

% mmap_map_keys
thf(fact_701_sup__acc__comp__None,axiom,
    ! [I: nat,L2: nat,J: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b] :
      ( ( ord_less_eq_nat @ I @ L2 )
     => ( ( ord_less_eq_nat @ L2 @ J )
       => ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ ( steps_b_a_c @ Step @ Rho @ Q @ ( product_Pair_nat_nat @ I @ L2 ) ) @ L2 @ J )
            = none_P6573378090780909534_c_nat )
         => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
            = ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ L2 ) ) ) ) ) ).

% sup_acc_comp_None
thf(fact_702_sup__acc__None__restrict,axiom,
    ! [I: nat,J: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
          = none_P6573378090780909534_c_nat )
       => ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ ( Step @ Q @ ( bs_at_c_a @ Rho @ I ) ) @ ( suc @ I ) @ J )
          = none_P6573378090780909534_c_nat ) ) ) ).

% sup_acc_None_restrict
thf(fact_703_valid__before_I7_J,axiom,
    valid_s_b_a_c @ init @ step @ st3 @ accept @ rho @ i @ i @ j @ s2 ).

% valid_before(7)
thf(fact_704_s_H__mmap__map_I1_J,axiom,
    ( s
    = ( mmap_m881505119971688169_c_nat
      @ ^ [Q2: b] :
          ( produc3722540595886809633_c_nat
          @ ^ [Q9: b,Tstp2: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ ( step @ Q9 @ bs ) @ ( if_opt8655011569862983689_c_nat @ ( accept @ ( step @ Q9 @ bs ) ) @ ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ t @ j ) ) @ Tstp2 ) ) )
      @ s2 ) ) ).

% s'_mmap_map(1)
thf(fact_705__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_H_Ast_H_Aac_H_O_Ammap__fold__s_Astep_Ast_Aaccept_Aac_Abs_At_Aj_As_A_061_A_Is_H_M_Ast_H_M_Aac_H_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [S4: list_P7417839048565863355_c_nat,St: mappin8597647756751374250_b_a_b,Ac: mapping_b_o] :
        ( ( mmap_fold_s_b_a_c @ step @ st3 @ accept @ ac2 @ bs @ t @ j @ s2 )
       != ( produc6596061091177265825ng_b_o @ S4 @ ( produc2576847906786980013ng_b_o @ St @ Ac ) ) ) ).

% \<open>\<And>thesis. (\<And>s' st' ac'. mmap_fold_s step st accept ac bs t j s = (s', st', ac') \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_706_sup__acc__SomeI,axiom,
    ! [Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,I: nat,L2: nat,J: nat] :
      ( ( acc_b_a_c @ Step @ Accept @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ L2 ) ) )
     => ( ( member_nat @ L2 @ ( set_or4665077453230672383an_nat @ I @ J ) )
       => ? [Tp2: nat] :
            ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
              = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ ( ts_at_c_a @ Rho @ Tp2 ) @ Tp2 ) ) )
            & ( ord_less_eq_nat @ L2 @ Tp2 )
            & ( ord_less_nat @ Tp2 @ J ) ) ) ) ).

% sup_acc_SomeI
thf(fact_707_s_H__def,axiom,
    ( ( mmap_fold_s_b_a_c @ step @ st3 @ accept @ ac2 @ bs @ t @ j @ s2 )
    = ( produc6596061091177265825ng_b_o @ s @ ( produc2576847906786980013ng_b_o @ st2 @ ac ) ) ) ).

% s'_def
thf(fact_708_finite__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).

% finite_atLeastLessThan
thf(fact_709_case__prod__conv,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc3722540595886809633_c_nat @ F @ ( produc5716802255202478839_c_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_710_case__prod__conv,axiom,
    ! [F: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,A: product_prod_b_c,B: mappin8597647756751374250_b_a_b] :
      ( ( produc5539993262610490793_b_a_b @ F @ ( produc5370114720211116987_b_a_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_711_case__prod__conv,axiom,
    ! [F: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,A: b,B: c] :
      ( ( produc8710285878991682935_b_a_b @ F @ ( product_Pair_b_c @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_712_case__prod__conv,axiom,
    ! [F: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,A: b,B: mappin8597647756751374250_b_a_b] :
      ( ( produc7520394990416881216_b_a_b @ F @ ( produc5420919026241514322_b_a_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_713_atLeastLessThan__iff,axiom,
    ! [I: c,L2: c,U: c] :
      ( ( member_c @ I @ ( set_or5139330845457685137Than_c @ L2 @ U ) )
      = ( ( ord_less_eq_c @ L2 @ I )
        & ( ord_less_c @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_714_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L2: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L2 @ U ) )
      = ( ( ord_less_eq_set_nat @ L2 @ I )
        & ( ord_less_set_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_715_atLeastLessThan__iff,axiom,
    ! [I: nat,L2: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
      = ( ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_716_atLeastLessThan__empty,axiom,
    ! [B: c,A: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( ( set_or5139330845457685137Than_c @ A @ B )
        = bot_bot_set_c ) ) ).

% atLeastLessThan_empty
thf(fact_717_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( set_or3540276404033026485et_nat @ A @ B )
        = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_718_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_719_ivl__subset,axiom,
    ! [I: nat,J: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ J @ I )
        | ( ( ord_less_eq_nat @ M2 @ I )
          & ( ord_less_eq_nat @ J @ N ) ) ) ) ).

% ivl_subset
thf(fact_720_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_721_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_722_inv__st_H,axiom,
    inv @ st2 ).

% inv_st'
thf(fact_723_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_724_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_725_Ico__eq__Ico,axiom,
    ! [L2: nat,H: nat,L5: nat,H2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ L2 @ H )
        = ( set_or4665077453230672383an_nat @ L5 @ H2 ) )
      = ( ( ( L2 = L5 )
          & ( H = H2 ) )
        | ( ~ ( ord_less_nat @ L2 @ H )
          & ~ ( ord_less_nat @ L5 @ H2 ) ) ) ) ).

% Ico_eq_Ico
thf(fact_726_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_727_prod_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Prod: produc4862256710654508797_c_nat] :
      ( ( H @ ( produc3722540595886809633_c_nat @ F @ Prod ) )
      = ( produc3722540595886809633_c_nat
        @ ^ [X12: b,X23: option7520157102916957007_c_nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_728_prod_Ocase__distrib,axiom,
    ! [H: produc6741251563483866561_b_a_b > produc6741251563483866561_b_a_b,F: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,Prod: produc6741251563483866561_b_a_b] :
      ( ( H @ ( produc5539993262610490793_b_a_b @ F @ Prod ) )
      = ( produc5539993262610490793_b_a_b
        @ ^ [X12: product_prod_b_c,X23: mappin8597647756751374250_b_a_b] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_729_prod_Ocase__distrib,axiom,
    ! [H: ( mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b ) > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,F: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,Prod: product_prod_b_c] :
      ( ( H @ ( produc8710285878991682935_b_a_b @ F @ Prod ) )
      = ( produc8710285878991682935_b_a_b
        @ ^ [X12: b,X23: c] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_730_prod_Ocase__distrib,axiom,
    ! [H: produc6741251563483866561_b_a_b > produc6741251563483866561_b_a_b,F: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,Prod: produc69853407467193432_b_a_b] :
      ( ( H @ ( produc7520394990416881216_b_a_b @ F @ Prod ) )
      = ( produc7520394990416881216_b_a_b
        @ ^ [X12: b,X23: mappin8597647756751374250_b_a_b] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_731_old_Oprod_Ocase,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,X1: b,X2: option7520157102916957007_c_nat] :
      ( ( produc3722540595886809633_c_nat @ F @ ( produc5716802255202478839_c_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_732_old_Oprod_Ocase,axiom,
    ! [F: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,X1: product_prod_b_c,X2: mappin8597647756751374250_b_a_b] :
      ( ( produc5539993262610490793_b_a_b @ F @ ( produc5370114720211116987_b_a_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_733_old_Oprod_Ocase,axiom,
    ! [F: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,X1: b,X2: c] :
      ( ( produc8710285878991682935_b_a_b @ F @ ( product_Pair_b_c @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_734_old_Oprod_Ocase,axiom,
    ! [F: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,X1: b,X2: mappin8597647756751374250_b_a_b] :
      ( ( produc7520394990416881216_b_a_b @ F @ ( produc5420919026241514322_b_a_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_735_split__cong,axiom,
    ! [Q: produc4862256710654508797_c_nat,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,G: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,P2: produc4862256710654508797_c_nat] :
      ( ! [X4: b,Y4: option7520157102916957007_c_nat] :
          ( ( ( produc5716802255202478839_c_nat @ X4 @ Y4 )
            = Q )
         => ( ( F @ X4 @ Y4 )
            = ( G @ X4 @ Y4 ) ) )
     => ( ( P2 = Q )
       => ( ( produc3722540595886809633_c_nat @ F @ P2 )
          = ( produc3722540595886809633_c_nat @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_736_split__cong,axiom,
    ! [Q: produc6741251563483866561_b_a_b,F: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,G: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,P2: produc6741251563483866561_b_a_b] :
      ( ! [X4: product_prod_b_c,Y4: mappin8597647756751374250_b_a_b] :
          ( ( ( produc5370114720211116987_b_a_b @ X4 @ Y4 )
            = Q )
         => ( ( F @ X4 @ Y4 )
            = ( G @ X4 @ Y4 ) ) )
     => ( ( P2 = Q )
       => ( ( produc5539993262610490793_b_a_b @ F @ P2 )
          = ( produc5539993262610490793_b_a_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_737_split__cong,axiom,
    ! [Q: product_prod_b_c,F: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,G: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,P2: product_prod_b_c] :
      ( ! [X4: b,Y4: c] :
          ( ( ( product_Pair_b_c @ X4 @ Y4 )
            = Q )
         => ( ( F @ X4 @ Y4 )
            = ( G @ X4 @ Y4 ) ) )
     => ( ( P2 = Q )
       => ( ( produc8710285878991682935_b_a_b @ F @ P2 )
          = ( produc8710285878991682935_b_a_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_738_split__cong,axiom,
    ! [Q: produc69853407467193432_b_a_b,F: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,G: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,P2: produc69853407467193432_b_a_b] :
      ( ! [X4: b,Y4: mappin8597647756751374250_b_a_b] :
          ( ( ( produc5420919026241514322_b_a_b @ X4 @ Y4 )
            = Q )
         => ( ( F @ X4 @ Y4 )
            = ( G @ X4 @ Y4 ) ) )
     => ( ( P2 = Q )
       => ( ( produc7520394990416881216_b_a_b @ F @ P2 )
          = ( produc7520394990416881216_b_a_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_739_case__prodE2,axiom,
    ! [Q7: produc4862256710654508797_c_nat > $o,P: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Z3: produc4862256710654508797_c_nat] :
      ( ( Q7 @ ( produc3722540595886809633_c_nat @ P @ Z3 ) )
     => ~ ! [X4: b,Y4: option7520157102916957007_c_nat] :
            ( ( Z3
              = ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
           => ~ ( Q7 @ ( P @ X4 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_740_case__prodE2,axiom,
    ! [Q7: produc6741251563483866561_b_a_b > $o,P: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,Z3: produc6741251563483866561_b_a_b] :
      ( ( Q7 @ ( produc5539993262610490793_b_a_b @ P @ Z3 ) )
     => ~ ! [X4: product_prod_b_c,Y4: mappin8597647756751374250_b_a_b] :
            ( ( Z3
              = ( produc5370114720211116987_b_a_b @ X4 @ Y4 ) )
           => ~ ( Q7 @ ( P @ X4 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_741_case__prodE2,axiom,
    ! [Q7: ( mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b ) > $o,P: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,Z3: product_prod_b_c] :
      ( ( Q7 @ ( produc8710285878991682935_b_a_b @ P @ Z3 ) )
     => ~ ! [X4: b,Y4: c] :
            ( ( Z3
              = ( product_Pair_b_c @ X4 @ Y4 ) )
           => ~ ( Q7 @ ( P @ X4 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_742_case__prodE2,axiom,
    ! [Q7: produc6741251563483866561_b_a_b > $o,P: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,Z3: produc69853407467193432_b_a_b] :
      ( ( Q7 @ ( produc7520394990416881216_b_a_b @ P @ Z3 ) )
     => ~ ! [X4: b,Y4: mappin8597647756751374250_b_a_b] :
            ( ( Z3
              = ( produc5420919026241514322_b_a_b @ X4 @ Y4 ) )
           => ~ ( Q7 @ ( P @ X4 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_743_case__prod__eta,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( produc3722540595886809633_c_nat
        @ ^ [X: b,Y: option7520157102916957007_c_nat] : ( F @ ( produc5716802255202478839_c_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_744_case__prod__eta,axiom,
    ! [F: produc6741251563483866561_b_a_b > produc6741251563483866561_b_a_b] :
      ( ( produc5539993262610490793_b_a_b
        @ ^ [X: product_prod_b_c,Y: mappin8597647756751374250_b_a_b] : ( F @ ( produc5370114720211116987_b_a_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_745_case__prod__eta,axiom,
    ! [F: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b] :
      ( ( produc8710285878991682935_b_a_b
        @ ^ [X: b,Y: c] : ( F @ ( product_Pair_b_c @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_746_case__prod__eta,axiom,
    ! [F: produc69853407467193432_b_a_b > produc6741251563483866561_b_a_b] :
      ( ( produc7520394990416881216_b_a_b
        @ ^ [X: b,Y: mappin8597647756751374250_b_a_b] : ( F @ ( produc5420919026241514322_b_a_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_747_cond__case__prod__eta,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,G: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ! [X4: b,Y4: option7520157102916957007_c_nat] :
          ( ( F @ X4 @ Y4 )
          = ( G @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) ) )
     => ( ( produc3722540595886809633_c_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_748_cond__case__prod__eta,axiom,
    ! [F: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,G: produc6741251563483866561_b_a_b > produc6741251563483866561_b_a_b] :
      ( ! [X4: product_prod_b_c,Y4: mappin8597647756751374250_b_a_b] :
          ( ( F @ X4 @ Y4 )
          = ( G @ ( produc5370114720211116987_b_a_b @ X4 @ Y4 ) ) )
     => ( ( produc5539993262610490793_b_a_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_749_cond__case__prod__eta,axiom,
    ! [F: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,G: product_prod_b_c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b] :
      ( ! [X4: b,Y4: c] :
          ( ( F @ X4 @ Y4 )
          = ( G @ ( product_Pair_b_c @ X4 @ Y4 ) ) )
     => ( ( produc8710285878991682935_b_a_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_750_cond__case__prod__eta,axiom,
    ! [F: b > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,G: produc69853407467193432_b_a_b > produc6741251563483866561_b_a_b] :
      ( ! [X4: b,Y4: mappin8597647756751374250_b_a_b] :
          ( ( F @ X4 @ Y4 )
          = ( G @ ( produc5420919026241514322_b_a_b @ X4 @ Y4 ) ) )
     => ( ( produc7520394990416881216_b_a_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_751_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_752_sup__acc__NoneE,axiom,
    ! [L2: nat,I: nat,J: nat,Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b] :
      ( ( member_nat @ L2 @ ( set_or4665077453230672383an_nat @ I @ J ) )
     => ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
          = none_P6573378090780909534_c_nat )
       => ~ ( acc_b_a_c @ Step @ Accept @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ L2 ) ) ) ) ) ).

% sup_acc_NoneE
thf(fact_753_sup__acc__SomeE,axiom,
    ! [Step: b > a > b,Accept: b > $o,Rho: list_P125642481956313003od_c_a,Q: b,I: nat,J: nat,Ts: c,Tp: nat] :
      ( ( ( sup_acc_b_a_c @ Step @ Accept @ Rho @ Q @ I @ J )
        = ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ Ts @ Tp ) ) )
     => ( ( member_nat @ Tp @ ( set_or4665077453230672383an_nat @ I @ J ) )
        & ( acc_b_a_c @ Step @ Accept @ Rho @ Q @ ( product_Pair_nat_nat @ I @ ( suc @ Tp ) ) ) ) ) ).

% sup_acc_SomeE
thf(fact_754_sup__acc__def,axiom,
    ( sup_acc_b_a_c
    = ( ^ [Step2: b > a > b,Accept2: b > $o,Rho2: list_P125642481956313003od_c_a,Q2: b,I2: nat,J3: nat] :
          ( if_opt8655011569862983689_c_nat
          @ ( ( collect_nat
              @ ^ [L: nat] :
                  ( ( member_nat @ L @ ( set_or4665077453230672383an_nat @ I2 @ J3 ) )
                  & ( acc_b_a_c @ Step2 @ Accept2 @ Rho2 @ Q2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ L ) ) ) ) )
            = bot_bot_set_nat )
          @ none_P6573378090780909534_c_nat
          @ ( some_P8722241760384591706_c_nat
            @ ( product_Pair_c_nat
              @ ( ts_at_c_a @ Rho2
                @ ( lattic8265883725875713057ax_nat
                  @ ( collect_nat
                    @ ^ [L: nat] :
                        ( ( member_nat @ L @ ( set_or4665077453230672383an_nat @ I2 @ J3 ) )
                        & ( acc_b_a_c @ Step2 @ Accept2 @ Rho2 @ Q2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ L ) ) ) ) ) ) )
              @ ( lattic8265883725875713057ax_nat
                @ ( collect_nat
                  @ ^ [L: nat] :
                      ( ( member_nat @ L @ ( set_or4665077453230672383an_nat @ I2 @ J3 ) )
                      & ( acc_b_a_c @ Step2 @ Accept2 @ Rho2 @ Q2 @ ( product_Pair_nat_nat @ I2 @ ( suc @ L ) ) ) ) ) ) ) ) ) ) ) ).

% sup_acc_def
thf(fact_755_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_c_nat] :
      ( ( produc1049061115736377381_c_nat @ product_Pair_c_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_756_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_c_a] :
      ( ( produc2713253650426281595od_c_a @ product_Pair_c_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_757_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_b_c] :
      ( ( produc281880053716946747od_b_c @ product_Pair_b_c @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_758_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_b_a] :
      ( ( produc1560760645774121403od_b_a @ product_Pair_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_759_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_760_case__prod__Pair__iden,axiom,
    ! [P2: produc4862256710654508797_c_nat] :
      ( ( produc3722540595886809633_c_nat @ produc5716802255202478839_c_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_761_case__prod__Pair__iden,axiom,
    ! [P2: produc6741251563483866561_b_a_b] :
      ( ( produc5539993262610490793_b_a_b @ produc5370114720211116987_b_a_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_762_sup__leadsto__def,axiom,
    ( sup_leadsto_b_a_c
    = ( ^ [Init2: b,Step2: b > a > b,Rho2: list_P125642481956313003od_c_a,I2: nat,J3: nat,Q2: b] :
          ( if_option_c
          @ ( ( collect_nat
              @ ^ [L: nat] :
                  ( ( ord_less_nat @ L @ I2 )
                  & ( ( steps_b_a_c @ Step2 @ Rho2 @ Init2 @ ( product_Pair_nat_nat @ L @ J3 ) )
                    = Q2 ) ) )
            = bot_bot_set_nat )
          @ none_c
          @ ( some_c
            @ ( ts_at_c_a @ Rho2
              @ ( lattic8265883725875713057ax_nat
                @ ( collect_nat
                  @ ^ [L: nat] :
                      ( ( ord_less_nat @ L @ I2 )
                      & ( ( steps_b_a_c @ Step2 @ Rho2 @ Init2 @ ( product_Pair_nat_nat @ L @ J3 ) )
                        = Q2 ) ) ) ) ) ) ) ) ) ).

% sup_leadsto_def
thf(fact_763_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( ( finite6177210948735845034at_nat @ Y7 )
       => ( ( Y7 != bot_bo2099793752762293965at_nat )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X5 )
               => ? [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ Y7 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Xa ) @ R ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y7 ) @ ( max_ex8135407076693332796at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_764_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,Y7: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( finite4392333629123659920at_nat @ X5 )
     => ( ( finite4392333629123659920at_nat @ Y7 )
       => ( ( Y7 != bot_bo5327735625951526323at_nat )
         => ( ! [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ X5 )
               => ? [Xa: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Xa @ Y7 )
                    & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Xa ) @ R ) ) )
           => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ X5 @ Y7 ) @ ( max_ex4511810952740877858at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_765_max__ext_Omax__extI,axiom,
    ! [X5: set_c,Y7: set_c,R: set_Product_prod_c_c] :
      ( ( finite_finite_c @ X5 )
     => ( ( finite_finite_c @ Y7 )
       => ( ( Y7 != bot_bot_set_c )
         => ( ! [X4: c] :
                ( ( member_c @ X4 @ X5 )
               => ? [Xa: c] :
                    ( ( member_c @ Xa @ Y7 )
                    & ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X4 @ Xa ) @ R ) ) )
           => ( member1877963456866042576_set_c @ ( produc2840744799846408087_set_c @ X5 @ Y7 ) @ ( max_ext_c @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_766_max__ext_Omax__extI,axiom,
    ! [X5: set_b,Y7: set_b,R: set_Product_prod_b_b] :
      ( ( finite_finite_b @ X5 )
     => ( ( finite_finite_b @ Y7 )
       => ( ( Y7 != bot_bot_set_b )
         => ( ! [X4: b] :
                ( ( member_b @ X4 @ X5 )
               => ? [Xa: b] :
                    ( ( member_b @ Xa @ Y7 )
                    & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Xa ) @ R ) ) )
           => ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ X5 @ Y7 ) @ ( max_ext_b @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_767_max__ext_Omax__extI,axiom,
    ! [X5: set_nat,Y7: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( finite_finite_nat @ Y7 )
       => ( ( Y7 != bot_bot_set_nat )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ X5 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ Y7 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Xa ) @ R ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y7 ) @ ( max_ext_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_768_case__prodI2,axiom,
    ! [P2: product_prod_c_nat,C: c > nat > $o] :
      ( ! [A3: c,B3: nat] :
          ( ( P2
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc7133942929724870258_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_769_case__prodI2,axiom,
    ! [P2: product_prod_c_a,C: c > a > $o] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc8907516716866730350_c_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_770_case__prodI2,axiom,
    ! [P2: produc4862256710654508797_c_nat,C: b > option7520157102916957007_c_nat > $o] :
      ( ! [A3: b,B3: option7520157102916957007_c_nat] :
          ( ( P2
            = ( produc5716802255202478839_c_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc2544960137906461044_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_771_case__prodI2,axiom,
    ! [P2: product_prod_b_c,C: b > c > $o] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc1873627588981547243_b_c_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_772_case__prodI2,axiom,
    ! [P2: product_prod_b_a,C: b > a > $o] :
      ( ! [A3: b,B3: a] :
          ( ( P2
            = ( product_Pair_b_a @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc7672072387910335853_b_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_773_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_774_case__prodI,axiom,
    ! [F: c > nat > $o,A: c,B: nat] :
      ( ( F @ A @ B )
     => ( produc7133942929724870258_nat_o @ F @ ( product_Pair_c_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_775_case__prodI,axiom,
    ! [F: c > a > $o,A: c,B: a] :
      ( ( F @ A @ B )
     => ( produc8907516716866730350_c_a_o @ F @ ( product_Pair_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_776_case__prodI,axiom,
    ! [F: b > option7520157102916957007_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( F @ A @ B )
     => ( produc2544960137906461044_nat_o @ F @ ( produc5716802255202478839_c_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_777_case__prodI,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( F @ A @ B )
     => ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) ) ) ).

% case_prodI
thf(fact_778_case__prodI,axiom,
    ! [F: b > a > $o,A: b,B: a] :
      ( ( F @ A @ B )
     => ( produc7672072387910335853_b_a_o @ F @ ( product_Pair_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_779_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_780_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_nat,Z3: b,C: c > nat > set_b] :
      ( ! [A3: c,B3: nat] :
          ( ( P2
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( member_b @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z3 @ ( produc454361093202844089_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_781_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_nat,Z3: nat,C: c > nat > set_nat] :
      ( ! [A3: c,B3: nat] :
          ( ( P2
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc4234106380533341996et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_782_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_nat,Z3: c,C: c > nat > set_c] :
      ( ! [A3: c,B3: nat] :
          ( ( P2
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( member_c @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_c @ Z3 @ ( produc454361097506072890_set_c @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_783_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_a,Z3: b,C: c > a > set_b] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( member_b @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z3 @ ( produc2545871484943226549_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_784_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_a,Z3: nat,C: c > a > set_nat] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc5989027250228583216et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_785_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_a,Z3: c,C: c > a > set_c] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( member_c @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_c @ Z3 @ ( produc2545871489246455350_set_c @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_786_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_c,Z3: b,C: b > c > set_b] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_b @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z3 @ ( produc3958908711191305138_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_787_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_c,Z3: nat,C: b > c > set_nat] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc8716184571927494899et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_788_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_c,Z3: c,C: b > c > set_c] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_c @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_c @ Z3 @ ( produc3958908715494533939_set_c @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_789_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_a,Z3: b,C: b > a > set_b] :
      ( ! [A3: b,B3: a] :
          ( ( P2
            = ( product_Pair_b_a @ A3 @ B3 ) )
         => ( member_b @ Z3 @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z3 @ ( produc1269978637572329268_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_790_mem__case__prodI,axiom,
    ! [Z3: b,C: c > nat > set_b,A: c,B: nat] :
      ( ( member_b @ Z3 @ ( C @ A @ B ) )
     => ( member_b @ Z3 @ ( produc454361093202844089_set_b @ C @ ( product_Pair_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_791_mem__case__prodI,axiom,
    ! [Z3: nat,C: c > nat > set_nat,A: c,B: nat] :
      ( ( member_nat @ Z3 @ ( C @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc4234106380533341996et_nat @ C @ ( product_Pair_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_792_mem__case__prodI,axiom,
    ! [Z3: c,C: c > nat > set_c,A: c,B: nat] :
      ( ( member_c @ Z3 @ ( C @ A @ B ) )
     => ( member_c @ Z3 @ ( produc454361097506072890_set_c @ C @ ( product_Pair_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_793_mem__case__prodI,axiom,
    ! [Z3: b,C: c > a > set_b,A: c,B: a] :
      ( ( member_b @ Z3 @ ( C @ A @ B ) )
     => ( member_b @ Z3 @ ( produc2545871484943226549_set_b @ C @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_794_mem__case__prodI,axiom,
    ! [Z3: nat,C: c > a > set_nat,A: c,B: a] :
      ( ( member_nat @ Z3 @ ( C @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc5989027250228583216et_nat @ C @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_795_mem__case__prodI,axiom,
    ! [Z3: c,C: c > a > set_c,A: c,B: a] :
      ( ( member_c @ Z3 @ ( C @ A @ B ) )
     => ( member_c @ Z3 @ ( produc2545871489246455350_set_c @ C @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_796_mem__case__prodI,axiom,
    ! [Z3: b,C: b > c > set_b,A: b,B: c] :
      ( ( member_b @ Z3 @ ( C @ A @ B ) )
     => ( member_b @ Z3 @ ( produc3958908711191305138_set_b @ C @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_797_mem__case__prodI,axiom,
    ! [Z3: nat,C: b > c > set_nat,A: b,B: c] :
      ( ( member_nat @ Z3 @ ( C @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc8716184571927494899et_nat @ C @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_798_mem__case__prodI,axiom,
    ! [Z3: c,C: b > c > set_c,A: b,B: c] :
      ( ( member_c @ Z3 @ ( C @ A @ B ) )
     => ( member_c @ Z3 @ ( produc3958908715494533939_set_c @ C @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_799_mem__case__prodI,axiom,
    ! [Z3: b,C: b > a > set_b,A: b,B: a] :
      ( ( member_b @ Z3 @ ( C @ A @ B ) )
     => ( member_b @ Z3 @ ( produc1269978637572329268_set_b @ C @ ( product_Pair_b_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_800_Max_Obounded__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_801_Max__less__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_802__092_060open_062_092_060And_062p_H_Ap_Ae_H_Ae_O_A_092_060lbrakk_062inv_Ae_059_A_Icase_A_Ip_M_Ae_J_Aof_A_Ix_M_Axa_J_A_092_060Rightarrow_062_A_Icase_Ax_Aof_A_Ix_M_Ay_J_A_092_060Rightarrow_062_A_092_060lambda_062st_O_Alet_A_Iq_H_M_Ast_H_J_A_061_Acstep_Astep_Ast_Ax_Abs_Ain_A_I_Iq_H_M_Ay_J_M_Ast_H_J_J_Axa_J_A_061_A_Ip_H_M_Ae_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ap_H_A_061_A_Icase_Ap_Aof_A_Ix_M_Ay_J_A_092_060Rightarrow_062_A_Istep_Ax_Abs_M_Ay_J_J_A_092_060and_062_Ainv_Ae_H_092_060close_062,axiom,
    ! [E: mappin8597647756751374250_b_a_b,P2: product_prod_b_a,P6: product_prod_b_a,E2: mappin8597647756751374250_b_a_b] :
      ( ( inv @ E )
     => ( ( ( produc4109059922087624101_b_a_b
            @ ( produc4226495961226819699_b_a_b
              @ ^ [X: b,Y: a,St2: mappin8597647756751374250_b_a_b] :
                  ( produc8669413817617329470_b_a_b
                  @ ^ [Q9: b] : ( produc6519133547411565241_b_a_b @ ( product_Pair_b_a @ Q9 @ Y ) )
                  @ ( cstep_b_a @ step @ St2 @ X @ bs ) ) )
            @ ( produc6519133547411565241_b_a_b @ P2 @ E ) )
          = ( produc6519133547411565241_b_a_b @ P6 @ E2 ) )
       => ( ( P6
            = ( produc1560760645774121403od_b_a
              @ ^ [X: b] : ( product_Pair_b_a @ ( step @ X @ bs ) )
              @ P2 ) )
          & ( inv @ E2 ) ) ) ) ).

% \<open>\<And>p' p e' e. \<lbrakk>inv e; (case (p, e) of (x, xa) \<Rightarrow> (case x of (x, y) \<Rightarrow> \<lambda>st. let (q', st') = cstep step st x bs in ((q', y), st')) xa) = (p', e')\<rbrakk> \<Longrightarrow> p' = (case p of (x, y) \<Rightarrow> (step x bs, y)) \<and> inv e'\<close>
thf(fact_803__092_060open_062_092_060And_062p_H_Ap_Ae_H_Ae_O_A_092_060lbrakk_062inv_Ae_059_A_Icase_A_Ip_M_Ae_J_Aof_A_Ix_M_Axa_J_A_092_060Rightarrow_062_A_Icase_Ax_Aof_A_Ix_M_Ay_J_A_092_060Rightarrow_062_A_092_060lambda_062st_O_Alet_A_Iq_H_M_Ast_H_J_A_061_Acstep_Astep_Ast_Ax_Abs_Ain_A_I_Iq_H_M_Ay_J_M_Ast_H_J_J_Axa_J_A_061_A_Ip_H_M_Ae_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ap_H_A_061_A_Icase_Ap_Aof_A_Ix_M_Ay_J_A_092_060Rightarrow_062_A_Istep_Ax_Abs_M_Ay_J_J_A_092_060and_062_Ainv_Ae_H_092_060close_062,axiom,
    ! [E: mappin8597647756751374250_b_a_b,P2: produc4862256710654508797_c_nat,P6: produc4862256710654508797_c_nat,E2: mappin8597647756751374250_b_a_b] :
      ( ( inv @ E )
     => ( ( ( produc1020174056063876467_b_a_b
            @ ( produc1347186926687317233_b_a_b
              @ ^ [X: b,Y: option7520157102916957007_c_nat,St2: mappin8597647756751374250_b_a_b] :
                  ( produc796255709968299465_b_a_b
                  @ ^ [Q9: b] : ( produc73199367669405504_b_a_b @ ( produc5716802255202478839_c_nat @ Q9 @ Y ) )
                  @ ( cstep_b_a @ step @ St2 @ X @ bs ) ) )
            @ ( produc73199367669405504_b_a_b @ P2 @ E ) )
          = ( produc73199367669405504_b_a_b @ P6 @ E2 ) )
       => ( ( P6
            = ( produc3722540595886809633_c_nat
              @ ^ [X: b] : ( produc5716802255202478839_c_nat @ ( step @ X @ bs ) )
              @ P2 ) )
          & ( inv @ E2 ) ) ) ) ).

% \<open>\<And>p' p e' e. \<lbrakk>inv e; (case (p, e) of (x, xa) \<Rightarrow> (case x of (x, y) \<Rightarrow> \<lambda>st. let (q', st') = cstep step st x bs in ((q', y), st')) xa) = (p', e')\<rbrakk> \<Longrightarrow> p' = (case p of (x, y) \<Rightarrow> (step x bs, y)) \<and> inv e'\<close>
thf(fact_804__092_060open_062_092_060And_062p_H_Ap_Ae_H_Ae_O_A_092_060lbrakk_062inv_Ae_059_A_Icase_A_Ip_M_Ae_J_Aof_A_Ix_M_Axa_J_A_092_060Rightarrow_062_A_Icase_Ax_Aof_A_Ix_M_Ay_J_A_092_060Rightarrow_062_A_092_060lambda_062st_O_Alet_A_Iq_H_M_Ast_H_J_A_061_Acstep_Astep_Ast_Ax_Abs_Ain_A_I_Iq_H_M_Ay_J_M_Ast_H_J_J_Axa_J_A_061_A_Ip_H_M_Ae_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ap_H_A_061_A_Icase_Ap_Aof_A_Ix_M_Ay_J_A_092_060Rightarrow_062_A_Istep_Ax_Abs_M_Ay_J_J_A_092_060and_062_Ainv_Ae_H_092_060close_062,axiom,
    ! [E: mappin8597647756751374250_b_a_b,P2: product_prod_b_c,P6: product_prod_b_c,E2: mappin8597647756751374250_b_a_b] :
      ( ( inv @ E )
     => ( ( ( produc5539993262610490793_b_a_b
            @ ( produc8710285878991682935_b_a_b
              @ ^ [X: b,Y: c,St2: mappin8597647756751374250_b_a_b] :
                  ( produc7520394990416881216_b_a_b
                  @ ^ [Q9: b] : ( produc5370114720211116987_b_a_b @ ( product_Pair_b_c @ Q9 @ Y ) )
                  @ ( cstep_b_a @ step @ St2 @ X @ bs ) ) )
            @ ( produc5370114720211116987_b_a_b @ P2 @ E ) )
          = ( produc5370114720211116987_b_a_b @ P6 @ E2 ) )
       => ( ( P6
            = ( produc281880053716946747od_b_c
              @ ^ [X: b] : ( product_Pair_b_c @ ( step @ X @ bs ) )
              @ P2 ) )
          & ( inv @ E2 ) ) ) ) ).

% \<open>\<And>p' p e' e. \<lbrakk>inv e; (case (p, e) of (x, xa) \<Rightarrow> (case x of (x, y) \<Rightarrow> \<lambda>st. let (q', st') = cstep step st x bs in ((q', y), st')) xa) = (p', e')\<rbrakk> \<Longrightarrow> p' = (case p of (x, y) \<Rightarrow> (step x bs, y)) \<and> inv e'\<close>
thf(fact_805_mem__case__prodE,axiom,
    ! [Z3: b,C: c > nat > set_b,P2: product_prod_c_nat] :
      ( ( member_b @ Z3 @ ( produc454361093202844089_set_b @ C @ P2 ) )
     => ~ ! [X4: c,Y4: nat] :
            ( ( P2
              = ( product_Pair_c_nat @ X4 @ Y4 ) )
           => ~ ( member_b @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_806_mem__case__prodE,axiom,
    ! [Z3: nat,C: c > nat > set_nat,P2: product_prod_c_nat] :
      ( ( member_nat @ Z3 @ ( produc4234106380533341996et_nat @ C @ P2 ) )
     => ~ ! [X4: c,Y4: nat] :
            ( ( P2
              = ( product_Pair_c_nat @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_807_mem__case__prodE,axiom,
    ! [Z3: c,C: c > nat > set_c,P2: product_prod_c_nat] :
      ( ( member_c @ Z3 @ ( produc454361097506072890_set_c @ C @ P2 ) )
     => ~ ! [X4: c,Y4: nat] :
            ( ( P2
              = ( product_Pair_c_nat @ X4 @ Y4 ) )
           => ~ ( member_c @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_808_mem__case__prodE,axiom,
    ! [Z3: b,C: c > a > set_b,P2: product_prod_c_a] :
      ( ( member_b @ Z3 @ ( produc2545871484943226549_set_b @ C @ P2 ) )
     => ~ ! [X4: c,Y4: a] :
            ( ( P2
              = ( product_Pair_c_a @ X4 @ Y4 ) )
           => ~ ( member_b @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_809_mem__case__prodE,axiom,
    ! [Z3: nat,C: c > a > set_nat,P2: product_prod_c_a] :
      ( ( member_nat @ Z3 @ ( produc5989027250228583216et_nat @ C @ P2 ) )
     => ~ ! [X4: c,Y4: a] :
            ( ( P2
              = ( product_Pair_c_a @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_810_mem__case__prodE,axiom,
    ! [Z3: c,C: c > a > set_c,P2: product_prod_c_a] :
      ( ( member_c @ Z3 @ ( produc2545871489246455350_set_c @ C @ P2 ) )
     => ~ ! [X4: c,Y4: a] :
            ( ( P2
              = ( product_Pair_c_a @ X4 @ Y4 ) )
           => ~ ( member_c @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_811_mem__case__prodE,axiom,
    ! [Z3: b,C: b > c > set_b,P2: product_prod_b_c] :
      ( ( member_b @ Z3 @ ( produc3958908711191305138_set_b @ C @ P2 ) )
     => ~ ! [X4: b,Y4: c] :
            ( ( P2
              = ( product_Pair_b_c @ X4 @ Y4 ) )
           => ~ ( member_b @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_812_mem__case__prodE,axiom,
    ! [Z3: nat,C: b > c > set_nat,P2: product_prod_b_c] :
      ( ( member_nat @ Z3 @ ( produc8716184571927494899et_nat @ C @ P2 ) )
     => ~ ! [X4: b,Y4: c] :
            ( ( P2
              = ( product_Pair_b_c @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_813_mem__case__prodE,axiom,
    ! [Z3: c,C: b > c > set_c,P2: product_prod_b_c] :
      ( ( member_c @ Z3 @ ( produc3958908715494533939_set_c @ C @ P2 ) )
     => ~ ! [X4: b,Y4: c] :
            ( ( P2
              = ( product_Pair_b_c @ X4 @ Y4 ) )
           => ~ ( member_c @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_814_mem__case__prodE,axiom,
    ! [Z3: b,C: b > a > set_b,P2: product_prod_b_a] :
      ( ( member_b @ Z3 @ ( produc1269978637572329268_set_b @ C @ P2 ) )
     => ~ ! [X4: b,Y4: a] :
            ( ( P2
              = ( product_Pair_b_a @ X4 @ Y4 ) )
           => ~ ( member_b @ Z3 @ ( C @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_815_case__prodE,axiom,
    ! [C: c > nat > $o,P2: product_prod_c_nat] :
      ( ( produc7133942929724870258_nat_o @ C @ P2 )
     => ~ ! [X4: c,Y4: nat] :
            ( ( P2
              = ( product_Pair_c_nat @ X4 @ Y4 ) )
           => ~ ( C @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_816_case__prodE,axiom,
    ! [C: c > a > $o,P2: product_prod_c_a] :
      ( ( produc8907516716866730350_c_a_o @ C @ P2 )
     => ~ ! [X4: c,Y4: a] :
            ( ( P2
              = ( product_Pair_c_a @ X4 @ Y4 ) )
           => ~ ( C @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_817_case__prodE,axiom,
    ! [C: b > option7520157102916957007_c_nat > $o,P2: produc4862256710654508797_c_nat] :
      ( ( produc2544960137906461044_nat_o @ C @ P2 )
     => ~ ! [X4: b,Y4: option7520157102916957007_c_nat] :
            ( ( P2
              = ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
           => ~ ( C @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_818_case__prodE,axiom,
    ! [C: b > c > $o,P2: product_prod_b_c] :
      ( ( produc1873627588981547243_b_c_o @ C @ P2 )
     => ~ ! [X4: b,Y4: c] :
            ( ( P2
              = ( product_Pair_b_c @ X4 @ Y4 ) )
           => ~ ( C @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_819_case__prodE,axiom,
    ! [C: b > a > $o,P2: product_prod_b_a] :
      ( ( produc7672072387910335853_b_a_o @ C @ P2 )
     => ~ ! [X4: b,Y4: a] :
            ( ( P2
              = ( product_Pair_b_a @ X4 @ Y4 ) )
           => ~ ( C @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_820_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y4 ) )
           => ~ ( C @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_821_case__prodD,axiom,
    ! [F: c > nat > $o,A: c,B: nat] :
      ( ( produc7133942929724870258_nat_o @ F @ ( product_Pair_c_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_822_case__prodD,axiom,
    ! [F: c > a > $o,A: c,B: a] :
      ( ( produc8907516716866730350_c_a_o @ F @ ( product_Pair_c_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_823_case__prodD,axiom,
    ! [F: b > option7520157102916957007_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc2544960137906461044_nat_o @ F @ ( produc5716802255202478839_c_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_824_case__prodD,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_825_case__prodD,axiom,
    ! [F: b > a > $o,A: b,B: a] :
      ( ( produc7672072387910335853_b_a_o @ F @ ( product_Pair_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_826_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_827_Max_OcoboundedI,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).

% Max.coboundedI
thf(fact_828_Max__eq__if,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B6 )
                  & ( ord_less_eq_nat @ X4 @ Xa ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ B6 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ A4 )
                    & ( ord_less_eq_nat @ X4 @ Xa ) ) )
           => ( ( lattic8265883725875713057ax_nat @ A4 )
              = ( lattic8265883725875713057ax_nat @ B6 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_829_Max__eqI,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ A4 )
           => ( ord_less_eq_nat @ Y4 @ X3 ) )
       => ( ( member_nat @ X3 @ A4 )
         => ( ( lattic8265883725875713057ax_nat @ A4 )
            = X3 ) ) ) ) ).

% Max_eqI
thf(fact_830_Max__ge,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).

% Max_ge
thf(fact_831_Max__in,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( member_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ A4 ) ) ) ).

% Max_in
thf(fact_832_Max__eq__iff,axiom,
    ! [A4: set_nat,M2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ( lattic8265883725875713057ax_nat @ A4 )
            = M2 )
          = ( ( member_nat @ M2 @ A4 )
            & ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_833_Max__ge__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A4 )
                & ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_834_eq__Max__iff,axiom,
    ! [A4: set_nat,M2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( M2
            = ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( ( member_nat @ M2 @ A4 )
            & ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_835_Max_OboundedE,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X3 )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A4 )
             => ( ord_less_eq_nat @ A8 @ X3 ) ) ) ) ) ).

% Max.boundedE
thf(fact_836_Max_OboundedI,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( ord_less_eq_nat @ A3 @ X3 ) )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X3 ) ) ) ) ).

% Max.boundedI
thf(fact_837_Max__gr__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A4 ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A4 )
                & ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_838_finite__psubset__def,axiom,
    ( finite_psubset_b
    = ( collec595060174259921746_set_b
      @ ( produc1429663378616588076et_b_o
        @ ^ [A7: set_b,B4: set_b] :
            ( ( ord_less_set_b @ A7 @ B4 )
            & ( finite_finite_b @ B4 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_839_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A7: set_nat,B4: set_nat] :
            ( ( ord_less_set_nat @ A7 @ B4 )
            & ( finite_finite_nat @ B4 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_840_Max__mono,axiom,
    ! [M7: set_nat,N6: set_nat] :
      ( ( ord_less_eq_set_nat @ M7 @ N6 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N6 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M7 ) @ ( lattic8265883725875713057ax_nat @ N6 ) ) ) ) ) ).

% Max_mono
thf(fact_841_Max_Osubset__imp,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B6 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ B6 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_842_case__prod__app,axiom,
    ( produc8710285878991682935_b_a_b
    = ( ^ [F3: b > c > mappin8597647756751374250_b_a_b > produc6741251563483866561_b_a_b,X: product_prod_b_c,Y: mappin8597647756751374250_b_a_b] :
          ( produc1653379926913576494_b_a_b
          @ ^ [L: b,R3: c] : ( F3 @ L @ R3 @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_843_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X7: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X7 @ A1 )
                   => ? [Xa2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa2 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_844_max__ext_Ocases,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R ) )
     => ~ ( ( finite4392333629123659920at_nat @ A1 )
         => ( ( finite4392333629123659920at_nat @ A22 )
           => ( ( A22 != bot_bo5327735625951526323at_nat )
             => ~ ! [X7: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ X7 @ A1 )
                   => ? [Xa2: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ Xa2 @ A22 )
                        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_845_max__ext_Ocases,axiom,
    ! [A1: set_c,A22: set_c,R: set_Product_prod_c_c] :
      ( ( member1877963456866042576_set_c @ ( produc2840744799846408087_set_c @ A1 @ A22 ) @ ( max_ext_c @ R ) )
     => ~ ( ( finite_finite_c @ A1 )
         => ( ( finite_finite_c @ A22 )
           => ( ( A22 != bot_bot_set_c )
             => ~ ! [X7: c] :
                    ( ( member_c @ X7 @ A1 )
                   => ? [Xa2: c] :
                        ( ( member_c @ Xa2 @ A22 )
                        & ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_846_max__ext_Ocases,axiom,
    ! [A1: set_b,A22: set_b,R: set_Product_prod_b_b] :
      ( ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ A1 @ A22 ) @ ( max_ext_b @ R ) )
     => ~ ( ( finite_finite_b @ A1 )
         => ( ( finite_finite_b @ A22 )
           => ( ( A22 != bot_bot_set_b )
             => ~ ! [X7: b] :
                    ( ( member_b @ X7 @ A1 )
                   => ? [Xa2: b] :
                        ( ( member_b @ Xa2 @ A22 )
                        & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_847_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A22 )
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X7: nat] :
                    ( ( member_nat @ X7 @ A1 )
                   => ? [Xa2: nat] :
                        ( ( member_nat @ Xa2 @ A22 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_848_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A1 )
           => ? [Y: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_849_max__ext_Osimps,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R ) )
      = ( ( finite4392333629123659920at_nat @ A1 )
        & ( finite4392333629123659920at_nat @ A22 )
        & ( A22 != bot_bo5327735625951526323at_nat )
        & ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A1 )
           => ? [Y: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y @ A22 )
                & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_850_max__ext_Osimps,axiom,
    ! [A1: set_c,A22: set_c,R: set_Product_prod_c_c] :
      ( ( member1877963456866042576_set_c @ ( produc2840744799846408087_set_c @ A1 @ A22 ) @ ( max_ext_c @ R ) )
      = ( ( finite_finite_c @ A1 )
        & ( finite_finite_c @ A22 )
        & ( A22 != bot_bot_set_c )
        & ! [X: c] :
            ( ( member_c @ X @ A1 )
           => ? [Y: c] :
                ( ( member_c @ Y @ A22 )
                & ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X @ Y ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_851_max__ext_Osimps,axiom,
    ! [A1: set_b,A22: set_b,R: set_Product_prod_b_b] :
      ( ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ A1 @ A22 ) @ ( max_ext_b @ R ) )
      = ( ( finite_finite_b @ A1 )
        & ( finite_finite_b @ A22 )
        & ( A22 != bot_bot_set_b )
        & ! [X: b] :
            ( ( member_b @ X @ A1 )
           => ? [Y: b] :
                ( ( member_b @ Y @ A22 )
                & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_852_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A22 )
        & ( A22 != bot_bot_set_nat )
        & ! [X: nat] :
            ( ( member_nat @ X @ A1 )
           => ? [Y: nat] :
                ( ( member_nat @ Y @ A22 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_853_dual__Min,axiom,
    ( ( lattices_Min_nat
      @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X ) )
    = lattic8265883725875713057ax_nat ) ).

% dual_Min
thf(fact_854_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( max_ex4864111882549613972at_nat
        @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R ) )
      = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( max_ex8135407076693332796at_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_855_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
      = ( ^ [X: set_nat,Y: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( max_ext_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_856_finite__lists__length__le,axiom,
    ! [A4: set_c,N: nat] :
      ( ( finite_finite_c @ A4 )
     => ( finite_finite_list_c
        @ ( collect_list_c
          @ ^ [Xs3: list_c] :
              ( ( ord_less_eq_set_c @ ( set_c2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_c @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_857_finite__lists__length__le,axiom,
    ! [A4: set_b,N: nat] :
      ( ( finite_finite_b @ A4 )
     => ( finite_finite_list_b
        @ ( collect_list_b
          @ ^ [Xs3: list_b] :
              ( ( ord_less_eq_set_b @ ( set_b2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_b @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_858_finite__lists__length__le,axiom,
    ! [A4: set_Product_prod_c_a,N: nat] :
      ( ( finite969547431062635214od_c_a @ A4 )
     => ( finite2296398606116454484od_c_a
        @ ( collec1503280309375431318od_c_a
          @ ^ [Xs3: list_P125642481956313003od_c_a] :
              ( ( ord_le8698776994054418981od_c_a @ ( set_Product_prod_c_a2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s2614380629626057239od_c_a @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_859_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_860_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_861_List_Ofinite__set,axiom,
    ! [Xs: list_c] : ( finite_finite_c @ ( set_c2 @ Xs ) ) ).

% List.finite_set
thf(fact_862_List_Ofinite__set,axiom,
    ! [Xs: list_b] : ( finite_finite_b @ ( set_b2 @ Xs ) ) ).

% List.finite_set
thf(fact_863_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_864_pair__less__iff1,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ X3 @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z3 ) ) ).

% pair_less_iff1
thf(fact_865_subset__code_I1_J,axiom,
    ! [Xs: list_b,B6: set_b] :
      ( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B6 )
      = ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ Xs ) )
           => ( member_b @ X @ B6 ) ) ) ) ).

% subset_code(1)
thf(fact_866_subset__code_I1_J,axiom,
    ! [Xs: list_P8469869581646625389at_nat,B6: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ B6 )
      = ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ).

% subset_code(1)
thf(fact_867_subset__code_I1_J,axiom,
    ! [Xs: list_c,B6: set_c] :
      ( ( ord_less_eq_set_c @ ( set_c2 @ Xs ) @ B6 )
      = ( ! [X: c] :
            ( ( member_c @ X @ ( set_c2 @ Xs ) )
           => ( member_c @ X @ B6 ) ) ) ) ).

% subset_code(1)
thf(fact_868_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B6 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B6 ) ) ) ) ).

% subset_code(1)
thf(fact_869_finite__list,axiom,
    ! [A4: set_c] :
      ( ( finite_finite_c @ A4 )
     => ? [Xs2: list_c] :
          ( ( set_c2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_870_finite__list,axiom,
    ! [A4: set_b] :
      ( ( finite_finite_b @ A4 )
     => ? [Xs2: list_b] :
          ( ( set_b2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_871_finite__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs2: list_nat] :
          ( ( set_nat2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_872_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_873_finite__lists__length__eq,axiom,
    ! [A4: set_c,N: nat] :
      ( ( finite_finite_c @ A4 )
     => ( finite_finite_list_c
        @ ( collect_list_c
          @ ^ [Xs3: list_c] :
              ( ( ord_less_eq_set_c @ ( set_c2 @ Xs3 ) @ A4 )
              & ( ( size_size_list_c @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_874_finite__lists__length__eq,axiom,
    ! [A4: set_b,N: nat] :
      ( ( finite_finite_b @ A4 )
     => ( finite_finite_list_b
        @ ( collect_list_b
          @ ^ [Xs3: list_b] :
              ( ( ord_less_eq_set_b @ ( set_b2 @ Xs3 ) @ A4 )
              & ( ( size_size_list_b @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_875_finite__lists__length__eq,axiom,
    ! [A4: set_Product_prod_c_a,N: nat] :
      ( ( finite969547431062635214od_c_a @ A4 )
     => ( finite2296398606116454484od_c_a
        @ ( collec1503280309375431318od_c_a
          @ ^ [Xs3: list_P125642481956313003od_c_a] :
              ( ( ord_le8698776994054418981od_c_a @ ( set_Product_prod_c_a2 @ Xs3 ) @ A4 )
              & ( ( size_s2614380629626057239od_c_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_876_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
              & ( ( size_size_list_nat @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_877_max__ext__def,axiom,
    ( max_ex8135407076693332796at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ( max_ex4864111882549613972at_nat
              @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R4 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_878_max__ext__def,axiom,
    ( max_ext_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ( max_extp_nat
              @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_879_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_880_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_881_plus__prod_Ocases,axiom,
    ! [X3: produc5740701590023533791_c_nat] :
      ~ ! [A3: c,B3: nat,C3: c,D3: nat] :
          ( X3
         != ( produc3560461569812927383_c_nat @ ( product_Pair_c_nat @ A3 @ B3 ) @ ( product_Pair_c_nat @ C3 @ D3 ) ) ) ).

% plus_prod.cases
thf(fact_882_plus__prod_Ocases,axiom,
    ! [X3: produc859450856879609959at_nat] :
      ~ ! [A3: nat,B3: nat,C3: nat,D3: nat] :
          ( X3
         != ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ C3 @ D3 ) ) ) ).

% plus_prod.cases
thf(fact_883_timestamp__total,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
      | ( ord_less_eq_nat @ B @ A ) ) ).

% timestamp_total
thf(fact_884_mlex__eq,axiom,
    ( mlex_p6366001652026297872at_nat
    = ( ^ [F3: product_prod_nat_nat > nat,R4: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
                ( ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_885_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F3: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_886_e_H__def,axiom,
    ( ( mmap_f2656902001591654721_b_a_b @ e2 @ st2
      @ ( produc5539993262610490793_b_a_b
        @ ( produc8710285878991682935_b_a_b
          @ ^ [X: b,Y: c,St2: mappin8597647756751374250_b_a_b] :
              ( produc7520394990416881216_b_a_b
              @ ^ [Q9: b] : ( produc5370114720211116987_b_a_b @ ( product_Pair_b_c @ Q9 @ Y ) )
              @ ( cstep_b_a @ step @ St2 @ X @ bs ) ) ) )
      @ sup_sup_c
      @ nil_Product_prod_b_c )
    = ( produc2325102696694170177_b_a_b @ e @ st ) ) ).

% e'_def
thf(fact_887_set__n__lists,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( set_list_c2 @ ( n_lists_c @ N @ Xs ) )
      = ( collect_list_c
        @ ^ [Ys3: list_c] :
            ( ( ( size_size_list_c @ Ys3 )
              = N )
            & ( ord_less_eq_set_c @ ( set_c2 @ Ys3 ) @ ( set_c2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_888_set__n__lists,axiom,
    ! [N: nat,Xs: list_P125642481956313003od_c_a] :
      ( ( set_li3702951541821976000od_c_a @ ( n_list2437481335113170867od_c_a @ N @ Xs ) )
      = ( collec1503280309375431318od_c_a
        @ ^ [Ys3: list_P125642481956313003od_c_a] :
            ( ( ( size_s2614380629626057239od_c_a @ Ys3 )
              = N )
            & ( ord_le8698776994054418981od_c_a @ ( set_Product_prod_c_a2 @ Ys3 ) @ ( set_Product_prod_c_a2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_889_set__n__lists,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
      = ( collect_list_nat
        @ ^ [Ys3: list_nat] :
            ( ( ( size_size_list_nat @ Ys3 )
              = N )
            & ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_890_mmap__fold__s_Osimps_I2_J,axiom,
    ! [Step: b > a > b,St3: mappin8597647756751374250_b_a_b,Accept: b > $o,Ac2: mapping_b_o,Bs: a,T: nat,J: nat,Q: b,Q3: b,Tstp: option4927543243414619207at_nat,Qbss: list_P8320790736095886109at_nat] :
      ( ( mmap_fold_s_b_a_nat @ Step @ St3 @ Accept @ Ac2 @ Bs @ T @ J @ ( cons_P4202341208877677261at_nat @ ( produc6611121650401626247at_nat @ Q @ ( produc4505668108609368735at_nat @ Q3 @ Tstp ) ) @ Qbss ) )
      = ( produc307282345960993920ng_b_o
        @ ^ [Q10: b,St4: mappin8597647756751374250_b_a_b] :
            ( produc7366830474715801831ng_b_o
            @ ^ [Beta: $o,Ac3: mapping_b_o] :
                ( produc6578064564710026747ng_b_o
                @ ^ [Qbss2: list_P8320790736095886109at_nat] :
                    ( produc9063590996320064893ng_b_o
                    @ ^ [St5: mappin8597647756751374250_b_a_b,Ac4: mapping_b_o] : ( produc5882103236339388335ng_b_o @ ( cons_P4202341208877677261at_nat @ ( produc6611121650401626247at_nat @ Q @ ( produc4505668108609368735at_nat @ Q10 @ ( if_opt6109864365331422477at_nat @ Beta @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ T @ J ) ) @ Tstp ) ) ) @ Qbss2 ) @ ( produc2576847906786980013ng_b_o @ St5 @ Ac4 ) ) )
                @ ( mmap_fold_s_b_a_nat @ Step @ St4 @ Accept @ Ac3 @ Bs @ T @ J @ Qbss ) )
            @ ( cac_b @ Accept @ Ac2 @ Q10 ) )
        @ ( cstep_b_a @ Step @ St3 @ Q3 @ Bs ) ) ) ).

% mmap_fold_s.simps(2)
thf(fact_891_mmap__fold__s_Osimps_I2_J,axiom,
    ! [Step: b > a > b,St3: mappin8597647756751374250_b_a_b,Accept: b > $o,Ac2: mapping_b_o,Bs: a,T: c,J: nat,Q: b,Q3: b,Tstp: option7520157102916957007_c_nat,Qbss: list_P7417839048565863355_c_nat] :
      ( ( mmap_fold_s_b_a_c @ Step @ St3 @ Accept @ Ac2 @ Bs @ T @ J @ ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ Q @ ( produc5716802255202478839_c_nat @ Q3 @ Tstp ) ) @ Qbss ) )
      = ( produc3526667420415499886ng_b_o
        @ ^ [Q10: b,St4: mappin8597647756751374250_b_a_b] :
            ( produc3765276298755831239ng_b_o
            @ ^ [Beta: $o,Ac3: mapping_b_o] :
                ( produc7725563706580016317ng_b_o
                @ ^ [Qbss2: list_P7417839048565863355_c_nat] :
                    ( produc3349605360071246513ng_b_o
                    @ ^ [St5: mappin8597647756751374250_b_a_b,Ac4: mapping_b_o] : ( produc6596061091177265825ng_b_o @ ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ Q @ ( produc5716802255202478839_c_nat @ Q10 @ ( if_opt8655011569862983689_c_nat @ Beta @ ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ T @ J ) ) @ Tstp ) ) ) @ Qbss2 ) @ ( produc2576847906786980013ng_b_o @ St5 @ Ac4 ) ) )
                @ ( mmap_fold_s_b_a_c @ Step @ St4 @ Accept @ Ac3 @ Bs @ T @ J @ Qbss ) )
            @ ( cac_b @ Accept @ Ac2 @ Q10 ) )
        @ ( cstep_b_a @ Step @ St3 @ Q3 @ Bs ) ) ) ).

% mmap_fold_s.simps(2)
thf(fact_892_list_Oinject,axiom,
    ! [X21: product_prod_c_a,X222: list_P125642481956313003od_c_a,Y21: product_prod_c_a,Y22: list_P125642481956313003od_c_a] :
      ( ( ( cons_P1742027962761213787od_c_a @ X21 @ X222 )
        = ( cons_P1742027962761213787od_c_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_893_set__empty,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( ( set_Product_prod_b_c2 @ Xs )
        = bot_bo2863795366942399540od_b_c )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% set_empty
thf(fact_894_set__empty,axiom,
    ! [Xs: list_P125642481956313003od_c_a] :
      ( ( ( set_Product_prod_c_a2 @ Xs )
        = bot_bo2086078286244720881od_c_a )
      = ( Xs = nil_Product_prod_c_a ) ) ).

% set_empty
thf(fact_895_set__empty,axiom,
    ! [Xs: list_c] :
      ( ( ( set_c2 @ Xs )
        = bot_bot_set_c )
      = ( Xs = nil_c ) ) ).

% set_empty
thf(fact_896_set__empty,axiom,
    ! [Xs: list_b] :
      ( ( ( set_b2 @ Xs )
        = bot_bot_set_b )
      = ( Xs = nil_b ) ) ).

% set_empty
thf(fact_897_set__empty,axiom,
    ! [Xs: list_nat] :
      ( ( ( set_nat2 @ Xs )
        = bot_bot_set_nat )
      = ( Xs = nil_nat ) ) ).

% set_empty
thf(fact_898_set__empty2,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( bot_bo2863795366942399540od_b_c
        = ( set_Product_prod_b_c2 @ Xs ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% set_empty2
thf(fact_899_set__empty2,axiom,
    ! [Xs: list_P125642481956313003od_c_a] :
      ( ( bot_bo2086078286244720881od_c_a
        = ( set_Product_prod_c_a2 @ Xs ) )
      = ( Xs = nil_Product_prod_c_a ) ) ).

% set_empty2
thf(fact_900_set__empty2,axiom,
    ! [Xs: list_c] :
      ( ( bot_bot_set_c
        = ( set_c2 @ Xs ) )
      = ( Xs = nil_c ) ) ).

% set_empty2
thf(fact_901_set__empty2,axiom,
    ! [Xs: list_b] :
      ( ( bot_bot_set_b
        = ( set_b2 @ Xs ) )
      = ( Xs = nil_b ) ) ).

% set_empty2
thf(fact_902_set__empty2,axiom,
    ! [Xs: list_nat] :
      ( ( bot_bot_set_nat
        = ( set_nat2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% set_empty2
thf(fact_903__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062e_H_Ast_H_H_O_Ammap__fold_H_Ae_Ast_H_A_I_092_060lambda_062_I_Ix_M_Ay_J_M_Ast_J_O_Alet_A_Iq_H_M_Ast_H_J_A_061_Acstep_Astep_Ast_Ax_Abs_Ain_A_I_Iq_H_M_Ay_J_M_Ast_H_J_J_Asup_A_091_093_A_061_A_Ie_H_M_Ast_H_H_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [E3: list_P903359562653991662od_b_c,St6: mappin8597647756751374250_b_a_b] :
        ( ( mmap_f2656902001591654721_b_a_b @ e2 @ st2
          @ ( produc5539993262610490793_b_a_b
            @ ( produc8710285878991682935_b_a_b
              @ ^ [X: b,Y: c,St2: mappin8597647756751374250_b_a_b] :
                  ( produc7520394990416881216_b_a_b
                  @ ^ [Q9: b] : ( produc5370114720211116987_b_a_b @ ( product_Pair_b_c @ Q9 @ Y ) )
                  @ ( cstep_b_a @ step @ St2 @ X @ bs ) ) ) )
          @ sup_sup_c
          @ nil_Product_prod_b_c )
       != ( produc2325102696694170177_b_a_b @ E3 @ St6 ) ) ).

% \<open>\<And>thesis. (\<And>e' st''. mmap_fold' e st' (\<lambda>((x, y), st). let (q', st') = cstep step st x bs in ((q', y), st')) sup [] = (e', st'') \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_904_successively_Ocases,axiom,
    ! [X3: produc2416979464461413415od_b_c] :
      ( ! [P7: product_prod_b_c > product_prod_b_c > $o] :
          ( X3
         != ( produc6092892548594537111od_b_c @ P7 @ nil_Product_prod_b_c ) )
     => ( ! [P7: product_prod_b_c > product_prod_b_c > $o,X4: product_prod_b_c] :
            ( X3
           != ( produc6092892548594537111od_b_c @ P7 @ ( cons_P4529483553340347422od_b_c @ X4 @ nil_Product_prod_b_c ) ) )
       => ~ ! [P7: product_prod_b_c > product_prod_b_c > $o,X4: product_prod_b_c,Y4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( X3
             != ( produc6092892548594537111od_b_c @ P7 @ ( cons_P4529483553340347422od_b_c @ X4 @ ( cons_P4529483553340347422od_b_c @ Y4 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_905_successively_Ocases,axiom,
    ! [X3: produc8993033250616892836od_c_a] :
      ( ! [P7: product_prod_c_a > product_prod_c_a > $o] :
          ( X3
         != ( produc4198189786975670932od_c_a @ P7 @ nil_Product_prod_c_a ) )
     => ( ! [P7: product_prod_c_a > product_prod_c_a > $o,X4: product_prod_c_a] :
            ( X3
           != ( produc4198189786975670932od_c_a @ P7 @ ( cons_P1742027962761213787od_c_a @ X4 @ nil_Product_prod_c_a ) ) )
       => ~ ! [P7: product_prod_c_a > product_prod_c_a > $o,X4: product_prod_c_a,Y4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a] :
              ( X3
             != ( produc4198189786975670932od_c_a @ P7 @ ( cons_P1742027962761213787od_c_a @ X4 @ ( cons_P1742027962761213787od_c_a @ Y4 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_906_sorted__wrt_Ocases,axiom,
    ! [X3: produc2416979464461413415od_b_c] :
      ( ! [P7: product_prod_b_c > product_prod_b_c > $o] :
          ( X3
         != ( produc6092892548594537111od_b_c @ P7 @ nil_Product_prod_b_c ) )
     => ~ ! [P7: product_prod_b_c > product_prod_b_c > $o,X4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] :
            ( X3
           != ( produc6092892548594537111od_b_c @ P7 @ ( cons_P4529483553340347422od_b_c @ X4 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_907_sorted__wrt_Ocases,axiom,
    ! [X3: produc8993033250616892836od_c_a] :
      ( ! [P7: product_prod_c_a > product_prod_c_a > $o] :
          ( X3
         != ( produc4198189786975670932od_c_a @ P7 @ nil_Product_prod_c_a ) )
     => ~ ! [P7: product_prod_c_a > product_prod_c_a > $o,X4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] :
            ( X3
           != ( produc4198189786975670932od_c_a @ P7 @ ( cons_P1742027962761213787od_c_a @ X4 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_908_shuffles_Ocases,axiom,
    ! [X3: produc5878855887281848999od_b_c] :
      ( ! [Ys4: list_P903359562653991662od_b_c] :
          ( X3
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys4 ) )
     => ( ! [Xs2: list_P903359562653991662od_b_c] :
            ( X3
           != ( produc3076307724246486423od_b_c @ Xs2 @ nil_Product_prod_b_c ) )
       => ~ ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] :
              ( X3
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_909_shuffles_Ocases,axiom,
    ! [X3: produc8538575299855860967od_c_a] :
      ( ! [Ys4: list_P125642481956313003od_c_a] :
          ( X3
         != ( produc6488642625900928599od_c_a @ nil_Product_prod_c_a @ Ys4 ) )
     => ( ! [Xs2: list_P125642481956313003od_c_a] :
            ( X3
           != ( produc6488642625900928599od_c_a @ Xs2 @ nil_Product_prod_c_a ) )
       => ~ ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] :
              ( X3
             != ( produc6488642625900928599od_c_a @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_910_splice_Ocases,axiom,
    ! [X3: produc5878855887281848999od_b_c] :
      ( ! [Ys4: list_P903359562653991662od_b_c] :
          ( X3
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys4 ) )
     => ~ ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys4: list_P903359562653991662od_b_c] :
            ( X3
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ Ys4 ) ) ) ).

% splice.cases
thf(fact_911_splice_Ocases,axiom,
    ! [X3: produc8538575299855860967od_c_a] :
      ( ! [Ys4: list_P125642481956313003od_c_a] :
          ( X3
         != ( produc6488642625900928599od_c_a @ nil_Product_prod_c_a @ Ys4 ) )
     => ~ ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Ys4: list_P125642481956313003od_c_a] :
            ( X3
           != ( produc6488642625900928599od_c_a @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ Ys4 ) ) ) ).

% splice.cases
thf(fact_912_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ws: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s3392097710323735898od_b_c @ Ys4 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                      = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs2 )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_913_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ws: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs )
            = ( size_s2614380629626057239od_c_a @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c,W: product_prod_c_a,Ws2: list_P125642481956313003od_c_a] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s3392097710323735898od_b_c @ Ys4 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                      = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs2 )
                        = ( size_s2614380629626057239od_c_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) @ ( cons_P1742027962761213787od_c_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_914_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,Zs: list_P125642481956313003od_c_a,Ws: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( ( size_s2614380629626057239od_c_a @ Zs )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_b_c )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s3392097710323735898od_b_c @ Ys4 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                      = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                   => ( ( ( size_s2614380629626057239od_c_a @ Zs2 )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_915_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,Zs: list_P125642481956313003od_c_a,Ws: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( ( size_s2614380629626057239od_c_a @ Zs )
            = ( size_s2614380629626057239od_c_a @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_c_a )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a,W: product_prod_c_a,Ws2: list_P125642481956313003od_c_a] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s3392097710323735898od_b_c @ Ys4 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                      = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                   => ( ( ( size_s2614380629626057239od_c_a @ Zs2 )
                        = ( size_s2614380629626057239od_c_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) @ ( cons_P1742027962761213787od_c_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_916_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,Zs: list_P903359562653991662od_b_c,Ws: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s2614380629626057239od_c_a @ Ys4 ) )
                 => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                      = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs2 )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_917_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,Zs: list_P903359562653991662od_b_c,Ws: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs )
            = ( size_s2614380629626057239od_c_a @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c,W: product_prod_c_a,Ws2: list_P125642481956313003od_c_a] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s2614380629626057239od_c_a @ Ys4 ) )
                 => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                      = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs2 )
                        = ( size_s2614380629626057239od_c_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) @ ( cons_P1742027962761213787od_c_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_918_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a,Ws: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( ( size_s2614380629626057239od_c_a @ Zs )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_c_a @ nil_Product_prod_b_c )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s2614380629626057239od_c_a @ Ys4 ) )
                 => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                      = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                   => ( ( ( size_s2614380629626057239od_c_a @ Zs2 )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_919_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a,Ws: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( ( size_s2614380629626057239od_c_a @ Zs )
            = ( size_s2614380629626057239od_c_a @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_c_a @ nil_Product_prod_c_a )
           => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a,W: product_prod_c_a,Ws2: list_P125642481956313003od_c_a] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                    = ( size_s2614380629626057239od_c_a @ Ys4 ) )
                 => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                      = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                   => ( ( ( size_s2614380629626057239od_c_a @ Zs2 )
                        = ( size_s2614380629626057239od_c_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) @ ( cons_P1742027962761213787od_c_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_920_list__induct4,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ws: list_P903359562653991662od_b_c,P: list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
           => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                    = ( size_s3392097710323735898od_b_c @ Ys4 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                      = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs2 )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_921_list__induct4,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ws: list_P125642481956313003od_c_a,P: list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs )
            = ( size_s2614380629626057239od_c_a @ Ws ) )
         => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
           => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c,W: product_prod_c_a,Ws2: list_P125642481956313003od_c_a] :
                  ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                    = ( size_s3392097710323735898od_b_c @ Ys4 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                      = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs2 )
                        = ( size_s2614380629626057239od_c_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) @ ( cons_P1742027962761213787od_c_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_922_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
         => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                  = ( size_s3392097710323735898od_b_c @ Ys4 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                    = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_923_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,Zs: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
         => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                  = ( size_s3392097710323735898od_b_c @ Ys4 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                    = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_924_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,Zs: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_b_c )
         => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                  = ( size_s2614380629626057239od_c_a @ Ys4 ) )
               => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                    = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_925_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a @ nil_Product_prod_c_a )
         => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                  = ( size_s2614380629626057239od_c_a @ Ys4 ) )
               => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                    = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_926_list__induct3,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,P: list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
         => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
                ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                  = ( size_s3392097710323735898od_b_c @ Ys4 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                    = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_927_list__induct3,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P903359562653991662od_b_c,Zs: list_P125642481956313003od_c_a,P: list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
         => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a] :
                ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                  = ( size_s3392097710323735898od_b_c @ Ys4 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys4 )
                    = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_928_list__induct3,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,Zs: list_P903359562653991662od_b_c,P: list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s3392097710323735898od_b_c @ Zs ) )
       => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_c_a @ nil_Product_prod_b_c )
         => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
                ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                  = ( size_s2614380629626057239od_c_a @ Ys4 ) )
               => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                    = ( size_s3392097710323735898od_b_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_929_list__induct3,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a,P: list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( ( size_s2614380629626057239od_c_a @ Ys2 )
          = ( size_s2614380629626057239od_c_a @ Zs ) )
       => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_c_a @ nil_Product_prod_c_a )
         => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a,Z: product_prod_c_a,Zs2: list_P125642481956313003od_c_a] :
                ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                  = ( size_s2614380629626057239od_c_a @ Ys4 ) )
               => ( ( ( size_s2614380629626057239od_c_a @ Ys4 )
                    = ( size_s2614380629626057239od_c_a @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys4 @ Zs2 )
                   => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) @ ( cons_P1742027962761213787od_c_a @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_930_list__induct2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
       => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] :
              ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                = ( size_s3392097710323735898od_b_c @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_931_list__induct2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a,P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
       => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] :
              ( ( ( size_s3392097710323735898od_b_c @ Xs2 )
                = ( size_s2614380629626057239od_c_a @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_932_list__induct2,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P903359562653991662od_b_c,P: list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys2 ) )
     => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_b_c )
       => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] :
              ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                = ( size_s3392097710323735898od_b_c @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_933_list__induct2,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,P: list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > $o] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_c_a )
       => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] :
              ( ( ( size_s2614380629626057239od_c_a @ Xs2 )
                = ( size_s2614380629626057239od_c_a @ Ys4 ) )
             => ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_934_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
      ( nil_Product_prod_b_c
     != ( cons_P4529483553340347422od_b_c @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_935_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_c_a,X222: list_P125642481956313003od_c_a] :
      ( nil_Product_prod_c_a
     != ( cons_P1742027962761213787od_c_a @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_936_list_OdiscI,axiom,
    ! [List: list_P903359562653991662od_b_c,X21: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
      ( ( List
        = ( cons_P4529483553340347422od_b_c @ X21 @ X222 ) )
     => ( List != nil_Product_prod_b_c ) ) ).

% list.discI
thf(fact_937_list_OdiscI,axiom,
    ! [List: list_P125642481956313003od_c_a,X21: product_prod_c_a,X222: list_P125642481956313003od_c_a] :
      ( ( List
        = ( cons_P1742027962761213787od_c_a @ X21 @ X222 ) )
     => ( List != nil_Product_prod_c_a ) ) ).

% list.discI
thf(fact_938_list_Oexhaust,axiom,
    ! [Y3: list_P903359562653991662od_b_c] :
      ( ( Y3 != nil_Product_prod_b_c )
     => ~ ! [X212: product_prod_b_c,X223: list_P903359562653991662od_b_c] :
            ( Y3
           != ( cons_P4529483553340347422od_b_c @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_939_list_Oexhaust,axiom,
    ! [Y3: list_P125642481956313003od_c_a] :
      ( ( Y3 != nil_Product_prod_c_a )
     => ~ ! [X212: product_prod_c_a,X223: list_P125642481956313003od_c_a] :
            ( Y3
           != ( cons_P1742027962761213787od_c_a @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_940_transpose_Ocases,axiom,
    ! [X3: list_l8907847357763382004od_b_c] :
      ( ( X3 != nil_li8071162985807626740od_b_c )
     => ( ! [Xss: list_l8907847357763382004od_b_c] :
            ( X3
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
       => ~ ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
              ( X3
             != ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_941_transpose_Ocases,axiom,
    ! [X3: list_l7377514787985273137od_c_a] :
      ( ( X3 != nil_li7293445905109948081od_c_a )
     => ( ! [Xss: list_l7377514787985273137od_c_a] :
            ( X3
           != ( cons_l8788426223280736865od_c_a @ nil_Product_prod_c_a @ Xss ) )
       => ~ ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Xss: list_l7377514787985273137od_c_a] :
              ( X3
             != ( cons_l8788426223280736865od_c_a @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_942_remdups__adj_Ocases,axiom,
    ! [X3: list_P903359562653991662od_b_c] :
      ( ( X3 != nil_Product_prod_b_c )
     => ( ! [X4: product_prod_b_c] :
            ( X3
           != ( cons_P4529483553340347422od_b_c @ X4 @ nil_Product_prod_b_c ) )
       => ~ ! [X4: product_prod_b_c,Y4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( X3
             != ( cons_P4529483553340347422od_b_c @ X4 @ ( cons_P4529483553340347422od_b_c @ Y4 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_943_remdups__adj_Ocases,axiom,
    ! [X3: list_P125642481956313003od_c_a] :
      ( ( X3 != nil_Product_prod_c_a )
     => ( ! [X4: product_prod_c_a] :
            ( X3
           != ( cons_P1742027962761213787od_c_a @ X4 @ nil_Product_prod_c_a ) )
       => ~ ! [X4: product_prod_c_a,Y4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a] :
              ( X3
             != ( cons_P1742027962761213787od_c_a @ X4 @ ( cons_P1742027962761213787od_c_a @ Y4 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_944_neq__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( Xs != nil_Product_prod_b_c )
      = ( ? [Y: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
            ( Xs
            = ( cons_P4529483553340347422od_b_c @ Y @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_945_neq__Nil__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a] :
      ( ( Xs != nil_Product_prod_c_a )
      = ( ? [Y: product_prod_c_a,Ys3: list_P125642481956313003od_c_a] :
            ( Xs
            = ( cons_P1742027962761213787od_c_a @ Y @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_946_list__induct2_H,axiom,
    ! [P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
     => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ nil_Product_prod_b_c )
       => ( ! [Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] : ( P @ nil_Product_prod_b_c @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) )
         => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_947_list__induct2_H,axiom,
    ! [P: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a > $o,Xs: list_P903359562653991662od_b_c,Ys2: list_P125642481956313003od_c_a] :
      ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_c_a )
     => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ nil_Product_prod_c_a )
       => ( ! [Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] : ( P @ nil_Product_prod_b_c @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) )
         => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_948_list__induct2_H,axiom,
    ! [P: list_P125642481956313003od_c_a > list_P903359562653991662od_b_c > $o,Xs: list_P125642481956313003od_c_a,Ys2: list_P903359562653991662od_b_c] :
      ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_b_c )
     => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a] : ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ nil_Product_prod_b_c )
       => ( ! [Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] : ( P @ nil_Product_prod_c_a @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) )
         => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_b_c,Ys4: list_P903359562653991662od_b_c] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_949_list__induct2_H,axiom,
    ! [P: list_P125642481956313003od_c_a > list_P125642481956313003od_c_a > $o,Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( P @ nil_Product_prod_c_a @ nil_Product_prod_c_a )
     => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a] : ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ nil_Product_prod_c_a )
       => ( ! [Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] : ( P @ nil_Product_prod_c_a @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) )
         => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a,Y4: product_prod_c_a,Ys4: list_P125642481956313003od_c_a] :
                ( ( P @ Xs2 @ Ys4 )
               => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) @ ( cons_P1742027962761213787od_c_a @ Y4 @ Ys4 ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_950_not__Cons__self2,axiom,
    ! [X3: product_prod_c_a,Xs: list_P125642481956313003od_c_a] :
      ( ( cons_P1742027962761213787od_c_a @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_951_list__nonempty__induct,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ! [X4: product_prod_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ nil_Product_prod_b_c ) )
       => ( ! [X4: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( ( Xs2 != nil_Product_prod_b_c )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_952_list__nonempty__induct,axiom,
    ! [Xs: list_P125642481956313003od_c_a,P: list_P125642481956313003od_c_a > $o] :
      ( ( Xs != nil_Product_prod_c_a )
     => ( ! [X4: product_prod_c_a] : ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ nil_Product_prod_c_a ) )
       => ( ! [X4: product_prod_c_a,Xs2: list_P125642481956313003od_c_a] :
              ( ( Xs2 != nil_Product_prod_c_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_P1742027962761213787od_c_a @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_953_set__ConsD,axiom,
    ! [Y3: b,X3: b,Xs: list_b] :
      ( ( member_b @ Y3 @ ( set_b2 @ ( cons_b @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_b @ Y3 @ ( set_b2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_954_set__ConsD,axiom,
    ! [Y3: nat,X3: nat,Xs: list_nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_nat @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_955_set__ConsD,axiom,
    ! [Y3: produc859450856879609959at_nat,X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_956_set__ConsD,axiom,
    ! [Y3: c,X3: c,Xs: list_c] :
      ( ( member_c @ Y3 @ ( set_c2 @ ( cons_c @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_c @ Y3 @ ( set_c2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_957_set__ConsD,axiom,
    ! [Y3: product_prod_c_a,X3: product_prod_c_a,Xs: list_P125642481956313003od_c_a] :
      ( ( member5074992350434858958od_c_a @ Y3 @ ( set_Product_prod_c_a2 @ ( cons_P1742027962761213787od_c_a @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member5074992350434858958od_c_a @ Y3 @ ( set_Product_prod_c_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_958_list_Oset__cases,axiom,
    ! [E: b,A: list_b] :
      ( ( member_b @ E @ ( set_b2 @ A ) )
     => ( ! [Z22: list_b] :
            ( A
           != ( cons_b @ E @ Z22 ) )
       => ~ ! [Z1: b,Z22: list_b] :
              ( ( A
                = ( cons_b @ Z1 @ Z22 ) )
             => ~ ( member_b @ E @ ( set_b2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_959_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_960_list_Oset__cases,axiom,
    ! [E: produc859450856879609959at_nat,A: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( set_Pr5518436109238095868at_nat @ A ) )
     => ( ! [Z22: list_P8469869581646625389at_nat] :
            ( A
           != ( cons_P8732206157123786781at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc859450856879609959at_nat,Z22: list_P8469869581646625389at_nat] :
              ( ( A
                = ( cons_P8732206157123786781at_nat @ Z1 @ Z22 ) )
             => ~ ( member8206827879206165904at_nat @ E @ ( set_Pr5518436109238095868at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_961_list_Oset__cases,axiom,
    ! [E: c,A: list_c] :
      ( ( member_c @ E @ ( set_c2 @ A ) )
     => ( ! [Z22: list_c] :
            ( A
           != ( cons_c @ E @ Z22 ) )
       => ~ ! [Z1: c,Z22: list_c] :
              ( ( A
                = ( cons_c @ Z1 @ Z22 ) )
             => ~ ( member_c @ E @ ( set_c2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_962_list_Oset__cases,axiom,
    ! [E: product_prod_c_a,A: list_P125642481956313003od_c_a] :
      ( ( member5074992350434858958od_c_a @ E @ ( set_Product_prod_c_a2 @ A ) )
     => ( ! [Z22: list_P125642481956313003od_c_a] :
            ( A
           != ( cons_P1742027962761213787od_c_a @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_c_a,Z22: list_P125642481956313003od_c_a] :
              ( ( A
                = ( cons_P1742027962761213787od_c_a @ Z1 @ Z22 ) )
             => ~ ( member5074992350434858958od_c_a @ E @ ( set_Product_prod_c_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_963_list_Oset__intros_I1_J,axiom,
    ! [X21: b,X222: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_964_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_965_list_Oset__intros_I1_J,axiom,
    ! [X21: produc859450856879609959at_nat,X222: list_P8469869581646625389at_nat] : ( member8206827879206165904at_nat @ X21 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_966_list_Oset__intros_I1_J,axiom,
    ! [X21: c,X222: list_c] : ( member_c @ X21 @ ( set_c2 @ ( cons_c @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_967_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_c_a,X222: list_P125642481956313003od_c_a] : ( member5074992350434858958od_c_a @ X21 @ ( set_Product_prod_c_a2 @ ( cons_P1742027962761213787od_c_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_968_list_Oset__intros_I2_J,axiom,
    ! [Y3: b,X222: list_b,X21: b] :
      ( ( member_b @ Y3 @ ( set_b2 @ X222 ) )
     => ( member_b @ Y3 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_969_list_Oset__intros_I2_J,axiom,
    ! [Y3: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_970_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc859450856879609959at_nat,X222: list_P8469869581646625389at_nat,X21: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ X222 ) )
     => ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_971_list_Oset__intros_I2_J,axiom,
    ! [Y3: c,X222: list_c,X21: c] :
      ( ( member_c @ Y3 @ ( set_c2 @ X222 ) )
     => ( member_c @ Y3 @ ( set_c2 @ ( cons_c @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_972_list_Oset__intros_I2_J,axiom,
    ! [Y3: product_prod_c_a,X222: list_P125642481956313003od_c_a,X21: product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ Y3 @ ( set_Product_prod_c_a2 @ X222 ) )
     => ( member5074992350434858958od_c_a @ Y3 @ ( set_Product_prod_c_a2 @ ( cons_P1742027962761213787od_c_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_973_set__subset__Cons,axiom,
    ! [Xs: list_c,X3: c] : ( ord_less_eq_set_c @ ( set_c2 @ Xs ) @ ( set_c2 @ ( cons_c @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_974_set__subset__Cons,axiom,
    ! [Xs: list_P125642481956313003od_c_a,X3: product_prod_c_a] : ( ord_le8698776994054418981od_c_a @ ( set_Product_prod_c_a2 @ Xs ) @ ( set_Product_prod_c_a2 @ ( cons_P1742027962761213787od_c_a @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_975_set__subset__Cons,axiom,
    ! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_976_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P125642481956313003od_c_a] :
      ( ( ( suc @ N )
        = ( size_s2614380629626057239od_c_a @ Xs ) )
      = ( ? [Y: product_prod_c_a,Ys3: list_P125642481956313003od_c_a] :
            ( ( Xs
              = ( cons_P1742027962761213787od_c_a @ Y @ Ys3 ) )
            & ( ( size_s2614380629626057239od_c_a @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_977_length__Suc__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,N: nat] :
      ( ( ( size_s2614380629626057239od_c_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Y: product_prod_c_a,Ys3: list_P125642481956313003od_c_a] :
            ( ( Xs
              = ( cons_P1742027962761213787od_c_a @ Y @ Ys3 ) )
            & ( ( size_s2614380629626057239od_c_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_978_impossible__Cons,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,X3: product_prod_c_a] :
      ( ( ord_less_eq_nat @ ( size_s2614380629626057239od_c_a @ Xs ) @ ( size_s2614380629626057239od_c_a @ Ys2 ) )
     => ( Xs
       != ( cons_P1742027962761213787od_c_a @ X3 @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_979_empty__set,axiom,
    ( bot_bo2863795366942399540od_b_c
    = ( set_Product_prod_b_c2 @ nil_Product_prod_b_c ) ) ).

% empty_set
thf(fact_980_empty__set,axiom,
    ( bot_bo2086078286244720881od_c_a
    = ( set_Product_prod_c_a2 @ nil_Product_prod_c_a ) ) ).

% empty_set
thf(fact_981_empty__set,axiom,
    ( bot_bot_set_c
    = ( set_c2 @ nil_c ) ) ).

% empty_set
thf(fact_982_empty__set,axiom,
    ( bot_bot_set_b
    = ( set_b2 @ nil_b ) ) ).

% empty_set
thf(fact_983_empty__set,axiom,
    ( bot_bot_set_nat
    = ( set_nat2 @ nil_nat ) ) ).

% empty_set
thf(fact_984_length__n__lists__elem,axiom,
    ! [Ys2: list_P125642481956313003od_c_a,N: nat,Xs: list_P125642481956313003od_c_a] :
      ( ( member5552703068553123156od_c_a @ Ys2 @ ( set_li3702951541821976000od_c_a @ ( n_list2437481335113170867od_c_a @ N @ Xs ) ) )
     => ( ( size_s2614380629626057239od_c_a @ Ys2 )
        = N ) ) ).

% length_n_lists_elem
thf(fact_985_mmap__lookup__empty,axiom,
    ! [Z3: c] :
      ( ( mmap_lookup_c_a @ nil_Product_prod_c_a @ Z3 )
      = none_a ) ).

% mmap_lookup_empty
thf(fact_986_mmap__lookup__empty,axiom,
    ! [Z3: b] :
      ( ( mmap_lookup_b_c @ nil_Product_prod_b_c @ Z3 )
      = none_c ) ).

% mmap_lookup_empty
thf(fact_987_mmap__lookup__empty,axiom,
    ! [Z3: b] :
      ( ( mmap_l5026999719965937591_c_nat @ nil_Pr7000167559216756261_c_nat @ Z3 )
      = none_P8487441334512977628_c_nat ) ).

% mmap_lookup_empty
thf(fact_988_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P125642481956313003od_c_a] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s2614380629626057239od_c_a @ Xs ) )
      = ( ? [X: product_prod_c_a,Ys3: list_P125642481956313003od_c_a] :
            ( ( Xs
              = ( cons_P1742027962761213787od_c_a @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s2614380629626057239od_c_a @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_989_mlex__iff,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,F: product_prod_nat_nat > nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( mlex_p6366001652026297872at_nat @ F @ R ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X3 )
            = ( F @ Y3 ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_990_mlex__iff,axiom,
    ! [X3: nat,Y3: nat,F: nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( mlex_prod_nat @ F @ R ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X3 )
            = ( F @ Y3 ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_991_mlex__less,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( mlex_p6366001652026297872at_nat @ F @ R ) ) ) ).

% mlex_less
thf(fact_992_mlex__less,axiom,
    ! [F: nat > nat,X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( mlex_prod_nat @ F @ R ) ) ) ).

% mlex_less
thf(fact_993_mlex__leq,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( mlex_p6366001652026297872at_nat @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_994_mlex__leq,axiom,
    ! [F: nat > nat,X3: nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( mlex_prod_nat @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_995_mmap__fold__s_Oelims,axiom,
    ! [X3: b > a > b,Xa3: mappin8597647756751374250_b_a_b,Xb: b > $o,Xc: mapping_b_o,Xd: a,Xe: nat,Xf: nat,Xg: list_P8320790736095886109at_nat,Y3: produc9159043268023579839ng_b_o] :
      ( ( ( mmap_fold_s_b_a_nat @ X3 @ Xa3 @ Xb @ Xc @ Xd @ Xe @ Xf @ Xg )
        = Y3 )
     => ( ( ( Xg = nil_Pr5085655994973978397at_nat )
         => ( Y3
           != ( produc5882103236339388335ng_b_o @ nil_Pr5085655994973978397at_nat @ ( produc2576847906786980013ng_b_o @ Xa3 @ Xc ) ) ) )
       => ~ ! [Q11: b,Q12: b,Tstp3: option4927543243414619207at_nat,Qbss3: list_P8320790736095886109at_nat] :
              ( ( Xg
                = ( cons_P4202341208877677261at_nat @ ( produc6611121650401626247at_nat @ Q11 @ ( produc4505668108609368735at_nat @ Q12 @ Tstp3 ) ) @ Qbss3 ) )
             => ( Y3
               != ( produc307282345960993920ng_b_o
                  @ ^ [Q10: b,St4: mappin8597647756751374250_b_a_b] :
                      ( produc7366830474715801831ng_b_o
                      @ ^ [Beta: $o,Ac3: mapping_b_o] :
                          ( produc6578064564710026747ng_b_o
                          @ ^ [Qbss2: list_P8320790736095886109at_nat] :
                              ( produc9063590996320064893ng_b_o
                              @ ^ [St5: mappin8597647756751374250_b_a_b,Ac4: mapping_b_o] : ( produc5882103236339388335ng_b_o @ ( cons_P4202341208877677261at_nat @ ( produc6611121650401626247at_nat @ Q11 @ ( produc4505668108609368735at_nat @ Q10 @ ( if_opt6109864365331422477at_nat @ Beta @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xe @ Xf ) ) @ Tstp3 ) ) ) @ Qbss2 ) @ ( produc2576847906786980013ng_b_o @ St5 @ Ac4 ) ) )
                          @ ( mmap_fold_s_b_a_nat @ X3 @ St4 @ Xb @ Ac3 @ Xd @ Xe @ Xf @ Qbss3 ) )
                      @ ( cac_b @ Xb @ Xc @ Q10 ) )
                  @ ( cstep_b_a @ X3 @ Xa3 @ Q12 @ Xd ) ) ) ) ) ) ).

% mmap_fold_s.elims
thf(fact_996_mmap__fold__s_Oelims,axiom,
    ! [X3: b > a > b,Xa3: mappin8597647756751374250_b_a_b,Xb: b > $o,Xc: mapping_b_o,Xd: a,Xe: c,Xf: nat,Xg: list_P7417839048565863355_c_nat,Y3: produc217993615331160047ng_b_o] :
      ( ( ( mmap_fold_s_b_a_c @ X3 @ Xa3 @ Xb @ Xc @ Xd @ Xe @ Xf @ Xg )
        = Y3 )
     => ( ( ( Xg = nil_Pr7000167559216756261_c_nat )
         => ( Y3
           != ( produc6596061091177265825ng_b_o @ nil_Pr7000167559216756261_c_nat @ ( produc2576847906786980013ng_b_o @ Xa3 @ Xc ) ) ) )
       => ~ ! [Q11: b,Q12: b,Tstp3: option7520157102916957007_c_nat,Qbss3: list_P7417839048565863355_c_nat] :
              ( ( Xg
                = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ Q11 @ ( produc5716802255202478839_c_nat @ Q12 @ Tstp3 ) ) @ Qbss3 ) )
             => ( Y3
               != ( produc3526667420415499886ng_b_o
                  @ ^ [Q10: b,St4: mappin8597647756751374250_b_a_b] :
                      ( produc3765276298755831239ng_b_o
                      @ ^ [Beta: $o,Ac3: mapping_b_o] :
                          ( produc7725563706580016317ng_b_o
                          @ ^ [Qbss2: list_P7417839048565863355_c_nat] :
                              ( produc3349605360071246513ng_b_o
                              @ ^ [St5: mappin8597647756751374250_b_a_b,Ac4: mapping_b_o] : ( produc6596061091177265825ng_b_o @ ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ Q11 @ ( produc5716802255202478839_c_nat @ Q10 @ ( if_opt8655011569862983689_c_nat @ Beta @ ( some_P8722241760384591706_c_nat @ ( product_Pair_c_nat @ Xe @ Xf ) ) @ Tstp3 ) ) ) @ Qbss2 ) @ ( produc2576847906786980013ng_b_o @ St5 @ Ac4 ) ) )
                          @ ( mmap_fold_s_b_a_c @ X3 @ St4 @ Xb @ Ac3 @ Xd @ Xe @ Xf @ Qbss3 ) )
                      @ ( cac_b @ Xb @ Xc @ Q10 ) )
                  @ ( cstep_b_a @ X3 @ Xa3 @ Q12 @ Xd ) ) ) ) ) ) ).

% mmap_fold_s.elims
thf(fact_997_sup__bot__left,axiom,
    ! [X3: set_b] :
      ( ( sup_sup_set_b @ bot_bot_set_b @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_998_sup__bot__left,axiom,
    ! [X3: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_999_sup__bot__right,axiom,
    ! [X3: set_b] :
      ( ( sup_sup_set_b @ X3 @ bot_bot_set_b )
      = X3 ) ).

% sup_bot_right
thf(fact_1000_sup__bot__right,axiom,
    ! [X3: set_nat] :
      ( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
      = X3 ) ).

% sup_bot_right
thf(fact_1001_bot__eq__sup__iff,axiom,
    ! [X3: set_b,Y3: set_b] :
      ( ( bot_bot_set_b
        = ( sup_sup_set_b @ X3 @ Y3 ) )
      = ( ( X3 = bot_bot_set_b )
        & ( Y3 = bot_bot_set_b ) ) ) ).

% bot_eq_sup_iff
thf(fact_1002_bot__eq__sup__iff,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X3 @ Y3 ) )
      = ( ( X3 = bot_bot_set_nat )
        & ( Y3 = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_1003_Un__empty,axiom,
    ! [A4: set_b,B6: set_b] :
      ( ( ( sup_sup_set_b @ A4 @ B6 )
        = bot_bot_set_b )
      = ( ( A4 = bot_bot_set_b )
        & ( B6 = bot_bot_set_b ) ) ) ).

% Un_empty
thf(fact_1004_Un__empty,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ( sup_sup_set_nat @ A4 @ B6 )
        = bot_bot_set_nat )
      = ( ( A4 = bot_bot_set_nat )
        & ( B6 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_1005_finite__Un,axiom,
    ! [F4: set_b,G2: set_b] :
      ( ( finite_finite_b @ ( sup_sup_set_b @ F4 @ G2 ) )
      = ( ( finite_finite_b @ F4 )
        & ( finite_finite_b @ G2 ) ) ) ).

% finite_Un
thf(fact_1006_finite__Un,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G2 ) )
      = ( ( finite_finite_nat @ F4 )
        & ( finite_finite_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_1007_Un__subset__iff,axiom,
    ! [A4: set_nat,B6: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B6 ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A4 @ C2 )
        & ( ord_less_eq_set_nat @ B6 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1008_le__sup__iff,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ Z3 )
      = ( ( ord_less_eq_nat @ X3 @ Z3 )
        & ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1009_le__sup__iff,axiom,
    ! [X3: c,Y3: c,Z3: c] :
      ( ( ord_less_eq_c @ ( sup_sup_c @ X3 @ Y3 ) @ Z3 )
      = ( ( ord_less_eq_c @ X3 @ Z3 )
        & ( ord_less_eq_c @ Y3 @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1010_le__sup__iff,axiom,
    ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z3 )
      = ( ( ord_less_eq_set_nat @ X3 @ Z3 )
        & ( ord_less_eq_set_nat @ Y3 @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1011_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1012_sup_Obounded__iff,axiom,
    ! [B: c,C: c,A: c] :
      ( ( ord_less_eq_c @ ( sup_sup_c @ B @ C ) @ A )
      = ( ( ord_less_eq_c @ B @ A )
        & ( ord_less_eq_c @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1013_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1014_sup__bot_Oright__neutral,axiom,
    ! [A: set_b] :
      ( ( sup_sup_set_b @ A @ bot_bot_set_b )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1015_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1016_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_b,B: set_b] :
      ( ( bot_bot_set_b
        = ( sup_sup_set_b @ A @ B ) )
      = ( ( A = bot_bot_set_b )
        & ( B = bot_bot_set_b ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1017_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1018_sup__bot_Oleft__neutral,axiom,
    ! [A: set_b] :
      ( ( sup_sup_set_b @ bot_bot_set_b @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1019_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1020_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ( sup_sup_set_b @ A @ B )
        = bot_bot_set_b )
      = ( ( A = bot_bot_set_b )
        & ( B = bot_bot_set_b ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1021_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1022_sup__eq__bot__iff,axiom,
    ! [X3: set_b,Y3: set_b] :
      ( ( ( sup_sup_set_b @ X3 @ Y3 )
        = bot_bot_set_b )
      = ( ( X3 = bot_bot_set_b )
        & ( Y3 = bot_bot_set_b ) ) ) ).

% sup_eq_bot_iff
thf(fact_1023_sup__eq__bot__iff,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ( sup_sup_set_nat @ X3 @ Y3 )
        = bot_bot_set_nat )
      = ( ( X3 = bot_bot_set_nat )
        & ( Y3 = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_1024_sup__Un__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( sup_su362511073950362882_nat_o
        @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
        @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ S ) )
      = ( ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( sup_su718114333110466843at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1025_sup__Un__eq2,axiom,
    ! [R: set_Pr6903500605879609269_c_nat,S: set_Pr6903500605879609269_c_nat] :
      ( ( sup_sup_c_nat_o
        @ ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R )
        @ ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ S ) )
      = ( ^ [X: c,Y: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ ( sup_su2428977078917480673_c_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1026_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_c_a,S: set_Product_prod_c_a] :
      ( ( sup_sup_c_a_o
        @ ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ R )
        @ ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ S ) )
      = ( ^ [X: c,Y: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ ( sup_su1776960780389684313od_c_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1027_sup__Un__eq2,axiom,
    ! [R: set_Pr8806432033423503795_c_nat,S: set_Pr8806432033423503795_c_nat] :
      ( ( sup_su2667635397647816210_nat_o
        @ ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R )
        @ ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ S ) )
      = ( ^ [X: b,Y: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ ( sup_su1518086237216066783_c_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1028_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( sup_sup_b_c_o
        @ ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ R )
        @ ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ S ) )
      = ( ^ [X: b,Y: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( sup_su2554677861087362972od_b_c @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1029_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_b_a,S: set_Product_prod_b_a] :
      ( ( sup_sup_b_a_o
        @ ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ R )
        @ ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ S ) )
      = ( ^ [X: b,Y: a] : ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X @ Y ) @ ( sup_su2412609780994671002od_b_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1030_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1031_Un__empty__right,axiom,
    ! [A4: set_b] :
      ( ( sup_sup_set_b @ A4 @ bot_bot_set_b )
      = A4 ) ).

% Un_empty_right
thf(fact_1032_Un__empty__right,axiom,
    ! [A4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1033_Un__empty__left,axiom,
    ! [B6: set_b] :
      ( ( sup_sup_set_b @ bot_bot_set_b @ B6 )
      = B6 ) ).

% Un_empty_left
thf(fact_1034_Un__empty__left,axiom,
    ! [B6: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B6 )
      = B6 ) ).

% Un_empty_left
thf(fact_1035_Un__mono,axiom,
    ! [A4: set_nat,C2: set_nat,B6: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C2 )
     => ( ( ord_less_eq_set_nat @ B6 @ D4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B6 ) @ ( sup_sup_set_nat @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_1036_Un__least,axiom,
    ! [A4: set_nat,C2: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C2 )
     => ( ( ord_less_eq_set_nat @ B6 @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B6 ) @ C2 ) ) ) ).

% Un_least
thf(fact_1037_Un__upper1,axiom,
    ! [A4: set_nat,B6: set_nat] : ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B6 ) ) ).

% Un_upper1
thf(fact_1038_Un__upper2,axiom,
    ! [B6: set_nat,A4: set_nat] : ( ord_less_eq_set_nat @ B6 @ ( sup_sup_set_nat @ A4 @ B6 ) ) ).

% Un_upper2
thf(fact_1039_Un__absorb1,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( sup_sup_set_nat @ A4 @ B6 )
        = B6 ) ) ).

% Un_absorb1
thf(fact_1040_Un__absorb2,axiom,
    ! [B6: set_nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B6 @ A4 )
     => ( ( sup_sup_set_nat @ A4 @ B6 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_1041_subset__UnE,axiom,
    ! [C2: set_nat,A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B6 ) )
     => ~ ! [A9: set_nat] :
            ( ( ord_less_eq_set_nat @ A9 @ A4 )
           => ! [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ B8 @ B6 )
               => ( C2
                 != ( sup_sup_set_nat @ A9 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_1042_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( ( sup_sup_set_nat @ A7 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_1043_finite__UnI,axiom,
    ! [F4: set_b,G2: set_b] :
      ( ( finite_finite_b @ F4 )
     => ( ( finite_finite_b @ G2 )
       => ( finite_finite_b @ ( sup_sup_set_b @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_1044_finite__UnI,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ( finite_finite_nat @ G2 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_1045_Un__infinite,axiom,
    ! [S: set_b,T2: set_b] :
      ( ~ ( finite_finite_b @ S )
     => ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_1046_Un__infinite,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_1047_infinite__Un,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( ~ ( finite_finite_b @ ( sup_sup_set_b @ S @ T2 ) ) )
      = ( ~ ( finite_finite_b @ S )
        | ~ ( finite_finite_b @ T2 ) ) ) ).

% infinite_Un
thf(fact_1048_infinite__Un,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_1049_ivl__disj__un__two_I3_J,axiom,
    ! [L2: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L2 @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L2 @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
          = ( set_or4665077453230672383an_nat @ L2 @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_1050_mmap__fold__s_Osimps_I1_J,axiom,
    ! [Step: b > a > b,St3: mappin8597647756751374250_b_a_b,Accept: b > $o,Ac2: mapping_b_o,Bs: a,T: c,J: nat] :
      ( ( mmap_fold_s_b_a_c @ Step @ St3 @ Accept @ Ac2 @ Bs @ T @ J @ nil_Pr7000167559216756261_c_nat )
      = ( produc6596061091177265825ng_b_o @ nil_Pr7000167559216756261_c_nat @ ( produc2576847906786980013ng_b_o @ St3 @ Ac2 ) ) ) ).

% mmap_fold_s.simps(1)
thf(fact_1051_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1052_sup_OcoboundedI2,axiom,
    ! [C: c,B: c,A: c] :
      ( ( ord_less_eq_c @ C @ B )
     => ( ord_less_eq_c @ C @ ( sup_sup_c @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1053_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1054_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1055_sup_OcoboundedI1,axiom,
    ! [C: c,A: c,B: c] :
      ( ( ord_less_eq_c @ C @ A )
     => ( ord_less_eq_c @ C @ ( sup_sup_c @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1056_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1057_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( sup_sup_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_1058_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_c
    = ( ^ [A6: c,B7: c] :
          ( ( sup_sup_c @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_1059_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_1060_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( sup_sup_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_1061_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_c
    = ( ^ [B7: c,A6: c] :
          ( ( sup_sup_c @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_1062_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_1063_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1064_sup_Ocobounded2,axiom,
    ! [B: c,A: c] : ( ord_less_eq_c @ B @ ( sup_sup_c @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1065_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1066_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1067_sup_Ocobounded1,axiom,
    ! [A: c,B: c] : ( ord_less_eq_c @ A @ ( sup_sup_c @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1068_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1069_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( A6
          = ( sup_sup_nat @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_1070_sup_Oorder__iff,axiom,
    ( ord_less_eq_c
    = ( ^ [B7: c,A6: c] :
          ( A6
          = ( sup_sup_c @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_1071_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( A6
          = ( sup_sup_set_nat @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_1072_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1073_sup_OboundedI,axiom,
    ! [B: c,A: c,C: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( ( ord_less_eq_c @ C @ A )
       => ( ord_less_eq_c @ ( sup_sup_c @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1074_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1075_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1076_sup_OboundedE,axiom,
    ! [B: c,C: c,A: c] :
      ( ( ord_less_eq_c @ ( sup_sup_c @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_c @ B @ A )
         => ~ ( ord_less_eq_c @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1077_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1078_sup__absorb2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( sup_sup_nat @ X3 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_1079_sup__absorb2,axiom,
    ! [X3: c,Y3: c] :
      ( ( ord_less_eq_c @ X3 @ Y3 )
     => ( ( sup_sup_c @ X3 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_1080_sup__absorb2,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y3 )
     => ( ( sup_sup_set_nat @ X3 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_1081_sup__absorb1,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( sup_sup_nat @ X3 @ Y3 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_1082_sup__absorb1,axiom,
    ! [Y3: c,X3: c] :
      ( ( ord_less_eq_c @ Y3 @ X3 )
     => ( ( sup_sup_c @ X3 @ Y3 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_1083_sup__absorb1,axiom,
    ! [Y3: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X3 )
     => ( ( sup_sup_set_nat @ X3 @ Y3 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_1084_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1085_sup_Oabsorb2,axiom,
    ! [A: c,B: c] :
      ( ( ord_less_eq_c @ A @ B )
     => ( ( sup_sup_c @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1086_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1087_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1088_sup_Oabsorb1,axiom,
    ! [B: c,A: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( ( sup_sup_c @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1089_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1090_sup__unique,axiom,
    ! [F: nat > nat > nat,X3: nat,Y3: nat] :
      ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y4 ) )
     => ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X4 @ Y4 ) )
       => ( ! [X4: nat,Y4: nat,Z: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X4 )
             => ( ( ord_less_eq_nat @ Z @ X4 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z ) @ X4 ) ) )
         => ( ( sup_sup_nat @ X3 @ Y3 )
            = ( F @ X3 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_1091_sup__unique,axiom,
    ! [F: c > c > c,X3: c,Y3: c] :
      ( ! [X4: c,Y4: c] : ( ord_less_eq_c @ X4 @ ( F @ X4 @ Y4 ) )
     => ( ! [X4: c,Y4: c] : ( ord_less_eq_c @ Y4 @ ( F @ X4 @ Y4 ) )
       => ( ! [X4: c,Y4: c,Z: c] :
              ( ( ord_less_eq_c @ Y4 @ X4 )
             => ( ( ord_less_eq_c @ Z @ X4 )
               => ( ord_less_eq_c @ ( F @ Y4 @ Z ) @ X4 ) ) )
         => ( ( sup_sup_c @ X3 @ Y3 )
            = ( F @ X3 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_1092_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
      ( ! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y4 ) )
     => ( ! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X4 @ Y4 ) )
       => ( ! [X4: set_nat,Y4: set_nat,Z: set_nat] :
              ( ( ord_less_eq_set_nat @ Y4 @ X4 )
             => ( ( ord_less_eq_set_nat @ Z @ X4 )
               => ( ord_less_eq_set_nat @ ( F @ Y4 @ Z ) @ X4 ) ) )
         => ( ( sup_sup_set_nat @ X3 @ Y3 )
            = ( F @ X3 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_1093_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_1094_sup_OorderI,axiom,
    ! [A: c,B: c] :
      ( ( A
        = ( sup_sup_c @ A @ B ) )
     => ( ord_less_eq_c @ B @ A ) ) ).

% sup.orderI
thf(fact_1095_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_1096_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1097_sup_OorderE,axiom,
    ! [B: c,A: c] :
      ( ( ord_less_eq_c @ B @ A )
     => ( A
        = ( sup_sup_c @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1098_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( A
        = ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1099_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( sup_sup_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_1100_le__iff__sup,axiom,
    ( ord_less_eq_c
    = ( ^ [X: c,Y: c] :
          ( ( sup_sup_c @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_1101_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( sup_sup_set_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_1102_sup__least,axiom,
    ! [Y3: nat,X3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ Z3 @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z3 ) @ X3 ) ) ) ).

% sup_least
thf(fact_1103_sup__least,axiom,
    ! [Y3: c,X3: c,Z3: c] :
      ( ( ord_less_eq_c @ Y3 @ X3 )
     => ( ( ord_less_eq_c @ Z3 @ X3 )
       => ( ord_less_eq_c @ ( sup_sup_c @ Y3 @ Z3 ) @ X3 ) ) ) ).

% sup_least
thf(fact_1104_sup__least,axiom,
    ! [Y3: set_nat,X3: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_set_nat @ Z3 @ X3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z3 ) @ X3 ) ) ) ).

% sup_least
thf(fact_1105_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1106_sup__mono,axiom,
    ! [A: c,C: c,B: c,D2: c] :
      ( ( ord_less_eq_c @ A @ C )
     => ( ( ord_less_eq_c @ B @ D2 )
       => ( ord_less_eq_c @ ( sup_sup_c @ A @ B ) @ ( sup_sup_c @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1107_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1108_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1109_sup_Omono,axiom,
    ! [C: c,A: c,D2: c,B: c] :
      ( ( ord_less_eq_c @ C @ A )
     => ( ( ord_less_eq_c @ D2 @ B )
       => ( ord_less_eq_c @ ( sup_sup_c @ C @ D2 ) @ ( sup_sup_c @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1110_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ( ord_less_eq_set_nat @ D2 @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1111_le__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X3 @ B )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1112_le__supI2,axiom,
    ! [X3: c,B: c,A: c] :
      ( ( ord_less_eq_c @ X3 @ B )
     => ( ord_less_eq_c @ X3 @ ( sup_sup_c @ A @ B ) ) ) ).

% le_supI2
thf(fact_1113_le__supI2,axiom,
    ! [X3: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ B )
     => ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1114_le__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ A )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1115_le__supI1,axiom,
    ! [X3: c,A: c,B: c] :
      ( ( ord_less_eq_c @ X3 @ A )
     => ( ord_less_eq_c @ X3 @ ( sup_sup_c @ A @ B ) ) ) ).

% le_supI1
thf(fact_1116_le__supI1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ A )
     => ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1117_sup__ge2,axiom,
    ! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).

% sup_ge2
thf(fact_1118_sup__ge2,axiom,
    ! [Y3: c,X3: c] : ( ord_less_eq_c @ Y3 @ ( sup_sup_c @ X3 @ Y3 ) ) ).

% sup_ge2
thf(fact_1119_sup__ge2,axiom,
    ! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).

% sup_ge2
thf(fact_1120_sup__ge1,axiom,
    ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).

% sup_ge1
thf(fact_1121_sup__ge1,axiom,
    ! [X3: c,Y3: c] : ( ord_less_eq_c @ X3 @ ( sup_sup_c @ X3 @ Y3 ) ) ).

% sup_ge1
thf(fact_1122_sup__ge1,axiom,
    ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).

% sup_ge1
thf(fact_1123_le__supI,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X3 )
     => ( ( ord_less_eq_nat @ B @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1124_le__supI,axiom,
    ! [A: c,X3: c,B: c] :
      ( ( ord_less_eq_c @ A @ X3 )
     => ( ( ord_less_eq_c @ B @ X3 )
       => ( ord_less_eq_c @ ( sup_sup_c @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1125_le__supI,axiom,
    ! [A: set_nat,X3: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X3 )
     => ( ( ord_less_eq_set_nat @ B @ X3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1126_le__supE,axiom,
    ! [A: nat,B: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_nat @ A @ X3 )
         => ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1127_le__supE,axiom,
    ! [A: c,B: c,X3: c] :
      ( ( ord_less_eq_c @ ( sup_sup_c @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_c @ A @ X3 )
         => ~ ( ord_less_eq_c @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1128_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_set_nat @ A @ X3 )
         => ~ ( ord_less_eq_set_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1129_inf__sup__ord_I3_J,axiom,
    ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_1130_inf__sup__ord_I3_J,axiom,
    ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_1131_inf__sup__ord_I4_J,axiom,
    ! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_1132_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_1133_less__supI1,axiom,
    ! [X3: c,A: c,B: c] :
      ( ( ord_less_c @ X3 @ A )
     => ( ord_less_c @ X3 @ ( sup_sup_c @ A @ B ) ) ) ).

% less_supI1
thf(fact_1134_less__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X3 @ A )
     => ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_1135_less__supI2,axiom,
    ! [X3: c,B: c,A: c] :
      ( ( ord_less_c @ X3 @ B )
     => ( ord_less_c @ X3 @ ( sup_sup_c @ A @ B ) ) ) ).

% less_supI2
thf(fact_1136_less__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X3 @ B )
     => ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_1137_sup_Oabsorb3,axiom,
    ! [B: c,A: c] :
      ( ( ord_less_c @ B @ A )
     => ( ( sup_sup_c @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1138_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1139_sup_Oabsorb4,axiom,
    ! [A: c,B: c] :
      ( ( ord_less_c @ A @ B )
     => ( ( sup_sup_c @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1140_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1141_sup_Ostrict__boundedE,axiom,
    ! [B: c,C: c,A: c] :
      ( ( ord_less_c @ ( sup_sup_c @ B @ C ) @ A )
     => ~ ( ( ord_less_c @ B @ A )
         => ~ ( ord_less_c @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1142_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1143_sup_Ostrict__order__iff,axiom,
    ( ord_less_c
    = ( ^ [B7: c,A6: c] :
          ( ( A6
            = ( sup_sup_c @ A6 @ B7 ) )
          & ( A6 != B7 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1144_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( A6
            = ( sup_sup_nat @ A6 @ B7 ) )
          & ( A6 != B7 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1145_sup_Ostrict__coboundedI1,axiom,
    ! [C: c,A: c,B: c] :
      ( ( ord_less_c @ C @ A )
     => ( ord_less_c @ C @ ( sup_sup_c @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1146_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1147_sup_Ostrict__coboundedI2,axiom,
    ! [C: c,B: c,A: c] :
      ( ( ord_less_c @ C @ B )
     => ( ord_less_c @ C @ ( sup_sup_c @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1148_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1149_rho_H__def,axiom,
    ( rho2
    = ( append8983669691956257088od_c_a @ rho @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ t @ bs ) @ nil_Product_prod_c_a ) ) ) ).

% rho'_def
thf(fact_1150_fold__sup__def,axiom,
    ( fold_sup_c_nat
    = ( ^ [M6: list_P5561950507774946575_c_nat,F3: c > c] :
          ( mmap_fold_c_nat @ M6
          @ ( produc1049061115736377381_c_nat
            @ ^ [X: c] : ( product_Pair_c_nat @ ( F3 @ X ) ) )
          @ sup_sup_nat
          @ nil_Pr650041864559007481_c_nat ) ) ) ).

% fold_sup_def
thf(fact_1151_fold__sup__def,axiom,
    ( fold_sup_nat_nat
    = ( ^ [M6: list_P6011104703257516679at_nat,F3: nat > nat] :
          ( mmap_fold_nat_nat @ M6
          @ ( produc2626176000494625587at_nat
            @ ^ [X: nat] : ( product_Pair_nat_nat @ ( F3 @ X ) ) )
          @ sup_sup_nat
          @ nil_Pr5478986624290739719at_nat ) ) ) ).

% fold_sup_def
thf(fact_1152_fold__sup__def,axiom,
    ( fold_sup_b_c
    = ( ^ [M6: list_P903359562653991662od_b_c,F3: b > b] :
          ( mmap_fold_b_c @ M6
          @ ( produc281880053716946747od_b_c
            @ ^ [X: b] : ( product_Pair_b_c @ ( F3 @ X ) ) )
          @ sup_sup_c
          @ nil_Product_prod_b_c ) ) ) ).

% fold_sup_def
thf(fact_1153_UnCI,axiom,
    ! [C: b,B6: set_b,A4: set_b] :
      ( ( ~ ( member_b @ C @ B6 )
       => ( member_b @ C @ A4 ) )
     => ( member_b @ C @ ( sup_sup_set_b @ A4 @ B6 ) ) ) ).

% UnCI
thf(fact_1154_UnCI,axiom,
    ! [C: nat,B6: set_nat,A4: set_nat] :
      ( ( ~ ( member_nat @ C @ B6 )
       => ( member_nat @ C @ A4 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B6 ) ) ) ).

% UnCI
thf(fact_1155_UnCI,axiom,
    ! [C: produc859450856879609959at_nat,B6: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ C @ B6 )
       => ( member8206827879206165904at_nat @ C @ A4 ) )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B6 ) ) ) ).

% UnCI
thf(fact_1156_UnCI,axiom,
    ! [C: c,B6: set_c,A4: set_c] :
      ( ( ~ ( member_c @ C @ B6 )
       => ( member_c @ C @ A4 ) )
     => ( member_c @ C @ ( sup_sup_set_c @ A4 @ B6 ) ) ) ).

% UnCI
thf(fact_1157_Un__iff,axiom,
    ! [C: b,A4: set_b,B6: set_b] :
      ( ( member_b @ C @ ( sup_sup_set_b @ A4 @ B6 ) )
      = ( ( member_b @ C @ A4 )
        | ( member_b @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_1158_Un__iff,axiom,
    ! [C: nat,A4: set_nat,B6: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B6 ) )
      = ( ( member_nat @ C @ A4 )
        | ( member_nat @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_1159_Un__iff,axiom,
    ! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B6 ) )
      = ( ( member8206827879206165904at_nat @ C @ A4 )
        | ( member8206827879206165904at_nat @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_1160_Un__iff,axiom,
    ! [C: c,A4: set_c,B6: set_c] :
      ( ( member_c @ C @ ( sup_sup_set_c @ A4 @ B6 ) )
      = ( ( member_c @ C @ A4 )
        | ( member_c @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_1161_append_Oassoc,axiom,
    ! [A: list_P125642481956313003od_c_a,B: list_P125642481956313003od_c_a,C: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ ( append8983669691956257088od_c_a @ A @ B ) @ C )
      = ( append8983669691956257088od_c_a @ A @ ( append8983669691956257088od_c_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_1162_append__assoc,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ ( append8983669691956257088od_c_a @ Xs @ Ys2 ) @ Zs )
      = ( append8983669691956257088od_c_a @ Xs @ ( append8983669691956257088od_c_a @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_1163_append__same__eq,axiom,
    ! [Ys2: list_P125642481956313003od_c_a,Xs: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Ys2 @ Xs )
        = ( append8983669691956257088od_c_a @ Zs @ Xs ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_1164_same__append__eq,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys2 )
        = ( append8983669691956257088od_c_a @ Xs @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_1165_append_Oright__neutral,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ A @ nil_Product_prod_b_c )
      = A ) ).

% append.right_neutral
thf(fact_1166_append_Oright__neutral,axiom,
    ! [A: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ A @ nil_Product_prod_c_a )
      = A ) ).

% append.right_neutral
thf(fact_1167_append__Nil2,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ Xs @ nil_Product_prod_b_c )
      = Xs ) ).

% append_Nil2
thf(fact_1168_append__Nil2,axiom,
    ! [Xs: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ Xs @ nil_Product_prod_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_1169_append__self__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_Product_prod_b_c ) ) ).

% append_self_conv
thf(fact_1170_append__self__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_Product_prod_c_a ) ) ).

% append_self_conv
thf(fact_1171_self__append__conv,axiom,
    ! [Y3: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( Y3
        = ( append2547753245680614915od_b_c @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_Product_prod_b_c ) ) ).

% self_append_conv
thf(fact_1172_self__append__conv,axiom,
    ! [Y3: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( Y3
        = ( append8983669691956257088od_c_a @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_Product_prod_c_a ) ) ).

% self_append_conv
thf(fact_1173_append__self__conv2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% append_self_conv2
thf(fact_1174_append__self__conv2,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_Product_prod_c_a ) ) ).

% append_self_conv2
thf(fact_1175_self__append__conv2,axiom,
    ! [Y3: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( Y3
        = ( append2547753245680614915od_b_c @ Xs @ Y3 ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% self_append_conv2
thf(fact_1176_self__append__conv2,axiom,
    ! [Y3: list_P125642481956313003od_c_a,Xs: list_P125642481956313003od_c_a] :
      ( ( Y3
        = ( append8983669691956257088od_c_a @ Xs @ Y3 ) )
      = ( Xs = nil_Product_prod_c_a ) ) ).

% self_append_conv2
thf(fact_1177_Nil__is__append__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( nil_Product_prod_b_c
        = ( append2547753245680614915od_b_c @ Xs @ Ys2 ) )
      = ( ( Xs = nil_Product_prod_b_c )
        & ( Ys2 = nil_Product_prod_b_c ) ) ) ).

% Nil_is_append_conv
thf(fact_1178_Nil__is__append__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( nil_Product_prod_c_a
        = ( append8983669691956257088od_c_a @ Xs @ Ys2 ) )
      = ( ( Xs = nil_Product_prod_c_a )
        & ( Ys2 = nil_Product_prod_c_a ) ) ) ).

% Nil_is_append_conv
thf(fact_1179_append__is__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys2 )
        = nil_Product_prod_b_c )
      = ( ( Xs = nil_Product_prod_b_c )
        & ( Ys2 = nil_Product_prod_b_c ) ) ) ).

% append_is_Nil_conv
thf(fact_1180_append__is__Nil__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys2 )
        = nil_Product_prod_c_a )
      = ( ( Xs = nil_Product_prod_c_a )
        & ( Ys2 = nil_Product_prod_c_a ) ) ) ).

% append_is_Nil_conv
thf(fact_1181_append__eq__append__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,Us: list_P125642481956313003od_c_a,Vs: list_P125642481956313003od_c_a] :
      ( ( ( ( size_s2614380629626057239od_c_a @ Xs )
          = ( size_s2614380629626057239od_c_a @ Ys2 ) )
        | ( ( size_s2614380629626057239od_c_a @ Us )
          = ( size_s2614380629626057239od_c_a @ Vs ) ) )
     => ( ( ( append8983669691956257088od_c_a @ Xs @ Us )
          = ( append8983669691956257088od_c_a @ Ys2 @ Vs ) )
        = ( ( Xs = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1182_append1__eq__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,X3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Y3: product_prod_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
        = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) ) )
      = ( ( Xs = Ys2 )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_1183_append1__eq__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,X3: product_prod_c_a,Ys2: list_P125642481956313003od_c_a,Y3: product_prod_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ ( cons_P1742027962761213787od_c_a @ X3 @ nil_Product_prod_c_a ) )
        = ( append8983669691956257088od_c_a @ Ys2 @ ( cons_P1742027962761213787od_c_a @ Y3 @ nil_Product_prod_c_a ) ) )
      = ( ( Xs = Ys2 )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_1184_set__append,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( set_Product_prod_c_a2 @ ( append8983669691956257088od_c_a @ Xs @ Ys2 ) )
      = ( sup_su1776960780389684313od_c_a @ ( set_Product_prod_c_a2 @ Xs ) @ ( set_Product_prod_c_a2 @ Ys2 ) ) ) ).

% set_append
thf(fact_1185_set__append,axiom,
    ! [Xs: list_c,Ys2: list_c] :
      ( ( set_c2 @ ( append_c @ Xs @ Ys2 ) )
      = ( sup_sup_set_c @ ( set_c2 @ Xs ) @ ( set_c2 @ Ys2 ) ) ) ).

% set_append
thf(fact_1186_append__Nil,axiom,
    ! [Ys2: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_1187_append__Nil,axiom,
    ! [Ys2: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ nil_Product_prod_c_a @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_1188_append__Cons,axiom,
    ! [X3: product_prod_c_a,Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ ( cons_P1742027962761213787od_c_a @ X3 @ Xs ) @ Ys2 )
      = ( cons_P1742027962761213787od_c_a @ X3 @ ( append8983669691956257088od_c_a @ Xs @ Ys2 ) ) ) ).

% append_Cons
thf(fact_1189_append_Oleft__neutral,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ A )
      = A ) ).

% append.left_neutral
thf(fact_1190_append_Oleft__neutral,axiom,
    ! [A: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ nil_Product_prod_c_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_1191_eq__Nil__appendI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_1192_eq__Nil__appendI,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append8983669691956257088od_c_a @ nil_Product_prod_c_a @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_1193_Cons__eq__appendI,axiom,
    ! [X3: product_prod_c_a,Xs1: list_P125642481956313003od_c_a,Ys2: list_P125642481956313003od_c_a,Xs: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( ( cons_P1742027962761213787od_c_a @ X3 @ Xs1 )
        = Ys2 )
     => ( ( Xs
          = ( append8983669691956257088od_c_a @ Xs1 @ Zs ) )
       => ( ( cons_P1742027962761213787od_c_a @ X3 @ Xs )
          = ( append8983669691956257088od_c_a @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1194_UnE,axiom,
    ! [C: b,A4: set_b,B6: set_b] :
      ( ( member_b @ C @ ( sup_sup_set_b @ A4 @ B6 ) )
     => ( ~ ( member_b @ C @ A4 )
       => ( member_b @ C @ B6 ) ) ) ).

% UnE
thf(fact_1195_UnE,axiom,
    ! [C: nat,A4: set_nat,B6: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B6 ) )
     => ( ~ ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B6 ) ) ) ).

% UnE
thf(fact_1196_UnE,axiom,
    ! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B6 ) )
     => ( ~ ( member8206827879206165904at_nat @ C @ A4 )
       => ( member8206827879206165904at_nat @ C @ B6 ) ) ) ).

% UnE
thf(fact_1197_UnE,axiom,
    ! [C: c,A4: set_c,B6: set_c] :
      ( ( member_c @ C @ ( sup_sup_set_c @ A4 @ B6 ) )
     => ( ~ ( member_c @ C @ A4 )
       => ( member_c @ C @ B6 ) ) ) ).

% UnE
thf(fact_1198_UnI1,axiom,
    ! [C: b,A4: set_b,B6: set_b] :
      ( ( member_b @ C @ A4 )
     => ( member_b @ C @ ( sup_sup_set_b @ A4 @ B6 ) ) ) ).

% UnI1
thf(fact_1199_UnI1,axiom,
    ! [C: nat,A4: set_nat,B6: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B6 ) ) ) ).

% UnI1
thf(fact_1200_UnI1,axiom,
    ! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ A4 )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B6 ) ) ) ).

% UnI1
thf(fact_1201_UnI1,axiom,
    ! [C: c,A4: set_c,B6: set_c] :
      ( ( member_c @ C @ A4 )
     => ( member_c @ C @ ( sup_sup_set_c @ A4 @ B6 ) ) ) ).

% UnI1
thf(fact_1202_UnI2,axiom,
    ! [C: b,B6: set_b,A4: set_b] :
      ( ( member_b @ C @ B6 )
     => ( member_b @ C @ ( sup_sup_set_b @ A4 @ B6 ) ) ) ).

% UnI2
thf(fact_1203_UnI2,axiom,
    ! [C: nat,B6: set_nat,A4: set_nat] :
      ( ( member_nat @ C @ B6 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B6 ) ) ) ).

% UnI2
thf(fact_1204_UnI2,axiom,
    ! [C: produc859450856879609959at_nat,B6: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ B6 )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B6 ) ) ) ).

% UnI2
thf(fact_1205_UnI2,axiom,
    ! [C: c,B6: set_c,A4: set_c] :
      ( ( member_c @ C @ B6 )
     => ( member_c @ C @ ( sup_sup_set_c @ A4 @ B6 ) ) ) ).

% UnI2
thf(fact_1206_Un__def,axiom,
    ( sup_sup_set_b
    = ( ^ [A7: set_b,B4: set_b] :
          ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A7 )
              | ( member_b @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_1207_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A7: set_nat,B4: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              | ( member_nat @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_1208_s_H__mmap__map_I3_J,axiom,
    ! [Q: b] :
      ( case_option_o_o @ $true
      @ ( ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
        @ ( accept @ Q ) )
      @ ( lookup_b_o @ ac @ Q ) ) ).

% s'_mmap_map(3)
thf(fact_1209_valid__before_I4_J,axiom,
    ! [Q: b] :
      ( case_option_o_o @ $true
      @ ( ^ [Y6: $o,Z2: $o] : ( Y6 = Z2 )
        @ ( accept @ Q ) )
      @ ( lookup_b_o @ ac2 @ Q ) ) ).

% valid_before(4)
thf(fact_1210_e_H__fold__sup__st_H_H_I2_J,axiom,
    ! [Q: b,Bs: a] :
      ( case_option_o_b @ $true
      @ ( ^ [Y6: b,Z2: b] : ( Y6 = Z2 )
        @ ( step @ Q @ Bs ) )
      @ ( lookup7095594596153002965_b_a_b @ st @ ( product_Pair_b_a @ Q @ Bs ) ) ) ).

% e'_fold_sup_st''(2)
thf(fact_1211_valid__before_I3_J,axiom,
    ! [Q: b,Bs: a] :
      ( case_option_o_b @ $true
      @ ( ^ [Y6: b,Z2: b] : ( Y6 = Z2 )
        @ ( step @ Q @ Bs ) )
      @ ( lookup7095594596153002965_b_a_b @ st3 @ ( product_Pair_b_a @ Q @ Bs ) ) ) ).

% valid_before(3)
thf(fact_1212_s_H__mmap__map_I2_J,axiom,
    ! [Q: b,Bs: a] :
      ( case_option_o_b @ $true
      @ ( ^ [Y6: b,Z2: b] : ( Y6 = Z2 )
        @ ( step @ Q @ Bs ) )
      @ ( lookup7095594596153002965_b_a_b @ st2 @ ( product_Pair_b_a @ Q @ Bs ) ) ) ).

% s'_mmap_map(2)
thf(fact_1213_local_Oinv__def,axiom,
    ( inv
    = ( ^ [St4: mappin8597647756751374250_b_a_b] :
        ! [Q2: b,Bs2: a] :
          ( case_option_o_b @ $true
          @ ( ^ [Y6: b,Z2: b] : ( Y6 = Z2 )
            @ ( step @ Q2 @ Bs2 ) )
          @ ( lookup7095594596153002965_b_a_b @ St4 @ ( product_Pair_b_a @ Q2 @ Bs2 ) ) ) ) ) ).

% local.inv_def
thf(fact_1214_distinct__before_I1_J,axiom,
    distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ s2 ) ).

% distinct_before(1)
thf(fact_1215_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1216_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1217_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1218_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1219_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1220_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1221_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1222_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1223_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M6: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1224_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1225_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% gr0_implies_Suc
thf(fact_1226_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1227_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1228_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1229_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1230_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1231_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1232_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1233_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N2 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_1234_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1235_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X3
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_1236_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_1237_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1238_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1239_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1240_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1241_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1242_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X4: nat,Y4: nat] :
              ( ( P @ X4 @ Y4 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
         => ( P @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_1243_zero__induct,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1244_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1245_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_1246_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_1247_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% not0_implies_Suc
thf(fact_1248_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1249_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1250_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1251_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1252_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ~ ( P @ I4 ) )
            & ( P @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_1253_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_nat @ M6 @ N )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1254_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_nat @ M6 @ N )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_1255_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_1256_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_1257_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1258_valid__before_I6_J,axiom,
    distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ e2 ) ).

% valid_before(6)
thf(fact_1259_assms_I4_J,axiom,
    ! [T4: c] :
      ( ( member_c @ T4 @ ( set_c2 @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ rho ) ) )
     => ( ord_less_eq_c @ T4 @ t ) ) ).

% assms(4)
thf(fact_1260_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size_char_eq_0

% Helper facts (7)
thf(help_If_2_1_If_001t__Option__Ooption_Itf__c_J_T,axiom,
    ! [X3: option_c,Y3: option_c] :
      ( ( if_option_c @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__c_J_T,axiom,
    ! [X3: option_c,Y3: option_c] :
      ( ( if_option_c @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: option7520157102916957007_c_nat,Y3: option7520157102916957007_c_nat] :
      ( ( if_opt8655011569862983689_c_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: option7520157102916957007_c_nat,Y3: option7520157102916957007_c_nat] :
      ( ( if_opt8655011569862983689_c_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( if_opt6109864365331422477at_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( if_opt6109864365331422477at_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( z
    = ( collect_b
      @ ^ [X: b] :
        ? [L: nat] :
          ( ( ord_less_nat @ L @ i )
          & ( ( steps_b_a_c @ step @ rho2 @ init @ ( product_Pair_nat_nat @ L @ j ) )
            = X )
          & ( ( step @ X @ bs )
            = q ) ) ) ) ).

%------------------------------------------------------------------------------