TPTP Problem File: SLH0012^1.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    : FO_Theory_Rewriting/0055_NF/prob_00248_010739__18428358_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1846 ( 502 unt; 544 typ;   0 def)
%            Number of atoms       : 3588 (1143 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 12873 ( 707   ~;  60   |; 161   &;10166   @)
%                                         (   0 <=>;1779  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   8 avg)
%            Number of types       :  104 ( 103 usr)
%            Number of type conns  : 1131 (1131   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  444 ( 441 usr;  49 con; 0-5 aty)
%            Number of variables   : 4001 (  80   ^;3829   !;  92   ?;4001   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:01:17.577
%------------------------------------------------------------------------------
% Could-be-implicit typings (103)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    set_Pr825864233230045127term_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc441850963132823271term_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_Mt__Nat__Onat_J,type,
    produc7218439583208144722_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    produc6821386223223610744term_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J_J,type,
    produc709930994605831291rm_a_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J_Mt__Nat__Onat_J,type,
    produc5520244230421620859_o_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    fset_P7293260781336502521term_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    set_Pr7374270731007213939term_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
    fset_P7393603901796714959rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc2056175833795995411term_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    produc2633411654673805033rm_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    fset_P6640269656128473term_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    set_Pr3965413743549523539term_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    produc1197155033112071543rm_a_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    fset_P6525107853456574705rm_a_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    fset_P3577670608877728751rm_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    set_Pr7334014819586186091rm_a_c: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    set_Pr4386577575007340137rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    produc7033602110846336755term_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Tree____Automata__Ota____rule_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__a_J_J,type,
    fset_T7482312105790613833rm_a_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Nat__Onat_J_J,type,
    fset_P4615588779303864480_a_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__c_J_J,type,
    fset_P7911707877349473372rm_a_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__b_J_J,type,
    fset_P7840673837303127387rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc1095327201564732427rm_a_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc357393685978478089rm_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Tree____Automata__Ota____rule_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__a_J_J,type,
    fset_T4690922579222374526_a_b_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    fset_P1023298988895795115rm_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_Mt__Nat__Onat_J,type,
    produc3631645709340582396_c_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    produc1685657453158572540rm_a_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J_J,type,
    produc5656680841319006879_a_c_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J_Mt__Nat__Onat_J,type,
    produc8681781744089100281_o_nat: $tType ).

thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    subter720051973439326747term_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Nat__Onat_J,type,
    produc8106822358505058234_a_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__a_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    fset_t4263150265408568880term_a: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    set_te6354304872801232554term_a: $tType ).

thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    subter2881564190832396198rm_a_b: $tType ).

thf(ty_n_t__Tree____Automata__Oreg_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__a_J,type,
    tree_r6075998776338966420rm_a_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__c_J,type,
    produc2491715938940346412rm_a_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__b_J,type,
    produc2491715934637117611rm_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    fset_l235587408706331465term_a: $tType ).

thf(ty_n_t__Tree____Automata__Ota_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__a_J,type,
    tree_t5222909361115825503rm_a_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc1234881154892807749rm_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__a_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    fset_term_a_term_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fset_P5670320511379867111at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Tree____Automata__Ota____rule_Itf__c_Mtf__a_J_J,type,
    fset_T7970579319148954716le_c_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Tree____Automata__Ota____rule_Itf__b_Mtf__a_J_J,type,
    fset_T8606228319753941405le_b_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    fset_P7886154461505542575_c_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
    fset_P2289935846317411886_b_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    fset_P5917089267984057005_a_nat: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__c_J_J,type,
    fset_P5318063784554775253_nat_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
    fset_P5247029744508429268_nat_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    fset_fset_term_a_b: $tType ).

thf(ty_n_t__Tree____Automata__Ota_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__a_J,type,
    tree_ta_term_a_b_a: $tType ).

thf(ty_n_t__Term__Oterm_Itf__a_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    term_a4617543059560371146term_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__c_Mtf__c_J_J,type,
    fset_P8014105740199022861od_c_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__c_Mtf__b_J_J,type,
    fset_P7943071700152676876od_c_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    fset_P8649754740804009550od_b_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    fset_P8578720700757663565od_b_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__c_Mtf__c_J_J,type,
    set_Product_prod_c_c: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    set_Product_prod_b_b: $tType ).

thf(ty_n_t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    list_Bot_bot_term_a: $tType ).

thf(ty_n_t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fset_Bot_bot_term_a: $tType ).

thf(ty_n_t__Term__Oterm_Itf__a_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    term_a_term_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mtf__c_J,type,
    subterm_and_ctxt_a_c: $tType ).

thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J,type,
    subterm_and_ctxt_a_b: $tType ).

thf(ty_n_t__Set__Oset_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
    set_Ground_gterm_a: $tType ).

thf(ty_n_t__Tree____Automata__Ota____rule_Itf__c_Mtf__a_J,type,
    tree_ta_rule_c_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    product_prod_c_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    product_prod_b_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__c_J,type,
    product_prod_nat_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    product_prod_nat_b: $tType ).

thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    list_term_a_c: $tType ).

thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    list_term_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    fset_term_a_c: $tType ).

thf(ty_n_t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    fset_term_a_b: $tType ).

thf(ty_n_t__Tree____Automata__Ota_It__Nat__Onat_Mtf__a_J,type,
    tree_ta_nat_a: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    set_term_a_c: $tType ).

thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    set_term_a_b: $tType ).

thf(ty_n_t__Tree____Automata__Oreg_Itf__b_Mtf__a_J,type,
    tree_reg_b_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    product_prod_c_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__c_Mtf__b_J,type,
    product_prod_c_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    product_prod_b_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    product_prod_b_b: $tType ).

thf(ty_n_t__Tree____Automata__Ota_Itf__c_Mtf__a_J,type,
    tree_ta_c_a: $tType ).

thf(ty_n_t__Tree____Automata__Ota_Itf__b_Mtf__a_J,type,
    tree_ta_b_a: $tType ).

thf(ty_n_t__Bot____Terms__Obot____term_Itf__a_J,type,
    bot_bot_term_a: $tType ).

thf(ty_n_t__Term__Oterm_Itf__a_Mt__Nat__Onat_J,type,
    term_a_nat: $tType ).

thf(ty_n_t__Ground____Terms__Ogterm_Itf__a_J,type,
    ground_gterm_a: $tType ).

thf(ty_n_t__Term__Oterm_Itf__a_Mtf__c_J,type,
    term_a_c: $tType ).

thf(ty_n_t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_a_b: $tType ).

thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
    fset_nat: $tType ).

thf(ty_n_t__List__Olist_Itf__c_J,type,
    list_c: $tType ).

thf(ty_n_t__FSet__Ofset_Itf__c_J,type,
    fset_c: $tType ).

thf(ty_n_t__FSet__Ofset_Itf__b_J,type,
    fset_b: $tType ).

thf(ty_n_t__FSet__Ofset_Itf__a_J,type,
    fset_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__c,type,
    c: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (441)
thf(sy_c_Bot__Terms_Obless__eq_001tf__a,type,
    bot_bless_eq_a: set_Pr3965413743549523539term_a ).

thf(sy_c_Bot__Terms_Oterm__to__bot__term_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    bot_te359145989583603559term_a: term_a4617543059560371146term_a > bot_bot_term_a ).

thf(sy_c_Bot__Terms_Oterm__to__bot__term_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    bot_te522286312025294962rm_a_b: term_a_term_a_b > bot_bot_term_a ).

thf(sy_c_Bot__Terms_Oterm__to__bot__term_001tf__a_001tf__b,type,
    bot_te3394087401593457953rm_a_b: term_a_b > bot_bot_term_a ).

thf(sy_c_Bot__Terms_Oterm__to__bot__term_001tf__a_001tf__c,type,
    bot_te3394087401593457954rm_a_c: term_a_c > bot_bot_term_a ).

thf(sy_c_FSet_Ofcard_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fcard_term_a_b: fset_term_a_b > nat ).

thf(sy_c_FSet_OffUnion_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    ffUnion_term_a_b: fset_fset_term_a_b > fset_term_a_b ).

thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fimage3742054321524018237term_a: ( produc7033602110846336755term_a > produc7033602110846336755term_a ) > fset_P6640269656128473term_a > fset_P6640269656128473term_a ).

thf(sy_c_FSet_Ofimage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fimage6249664427166183207rm_a_b: ( term_a_b > term_a_b ) > fset_term_a_b > fset_term_a_b ).

thf(sy_c_FSet_Ofimage_001t__Term__Oterm_Itf__a_Mtf__c_J_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    fimage6916563907009320361rm_a_c: ( term_a_c > term_a_c ) > fset_term_a_c > fset_term_a_c ).

thf(sy_c_FSet_Ofinsert_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    finser8320498192367885249term_a: bot_bot_term_a > fset_Bot_bot_term_a > fset_Bot_bot_term_a ).

thf(sy_c_FSet_Ofinsert_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    finser5027672521447216124rm_a_b: fset_term_a_b > fset_fset_term_a_b > fset_fset_term_a_b ).

thf(sy_c_FSet_Ofinsert_001t__Nat__Onat,type,
    finsert_nat: nat > fset_nat > fset_nat ).

thf(sy_c_FSet_Ofinsert_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    finser5549460190070563387term_a: produc7033602110846336755term_a > fset_P6640269656128473term_a > fset_P6640269656128473term_a ).

thf(sy_c_FSet_Ofinsert_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    finser2611709832528954927od_b_b: product_prod_b_b > fset_P8578720700757663565od_b_b > fset_P8578720700757663565od_b_b ).

thf(sy_c_FSet_Ofinsert_001t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    finser9047626291714283503od_c_c: product_prod_c_c > fset_P8014105740199022861od_c_c > fset_P8014105740199022861od_c_c ).

thf(sy_c_FSet_Ofinsert_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    finsert_term_a_b: term_a_b > fset_term_a_b > fset_term_a_b ).

thf(sy_c_FSet_Ofinsert_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    finsert_term_a_c: term_a_c > fset_term_a_c > fset_term_a_c ).

thf(sy_c_FSet_Ofinsert_001tf__b,type,
    finsert_b: b > fset_b > fset_b ).

thf(sy_c_FSet_Ofinsert_001tf__c,type,
    finsert_c: c > fset_c > fset_c ).

thf(sy_c_FSet_Ofmember_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fmembe1418995468851910914term_a: bot_bot_term_a > fset_Bot_bot_term_a > $o ).

thf(sy_c_FSet_Ofmember_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    fmembe3754333024224722621rm_a_b: fset_term_a_b > fset_fset_term_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fmembe1891194496946673298term_a: list_Bot_bot_term_a > fset_l235587408706331465term_a > $o ).

thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
    fmember_nat: nat > fset_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fmembe4510397451897038394term_a: produc7033602110846336755term_a > fset_P6640269656128473term_a > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Nat__Onat_J,type,
    fmembe4530531765636674689_a_nat: produc8106822358505058234_a_nat > fset_P4615588779303864480_a_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__b_J,type,
    fmembe2649604070268071844rm_a_b: produc2491715934637117611rm_a_b > fset_P7840673837303127387rm_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__c_J,type,
    fmembe2649604074571300645rm_a_c: produc2491715938940346412rm_a_c > fset_P7911707877349473372rm_a_c > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    fmembe4744889074212852528rm_a_b: produc2633411654673805033rm_a_b > fset_P7393603901796714959rm_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    fmembe4453961250406969178term_a: produc2056175833795995411term_a > fset_P7293260781336502521term_a > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fmembe1449479052724974408at_nat: product_prod_nat_nat > fset_P5670320511379867111at_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    fmembe6134593446672421132rm_a_b: produc1234881154892807749rm_a_b > fset_P1023298988895795115rm_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    fmembe3856250531246888093_nat_b: product_prod_nat_b > fset_P5247029744508429268_nat_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__c_J,type,
    fmembe3856250535550116894_nat_c: product_prod_nat_c > fset_P5318063784554775253_nat_c > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    fmembe2328057640968976720rm_a_b: produc357393685978478089rm_a_b > fset_P3577670608877728751rm_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    fmembe3065991156555231058rm_a_c: produc1095327201564732427rm_a_c > fset_P6525107853456574705rm_a_c > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    fmembe1853531392176376823_b_nat: product_prod_b_nat > fset_P2289935846317411886_b_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    fmembe4950632938690628014od_b_b: product_prod_b_b > fset_P8578720700757663565od_b_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    fmembe4950632942993856815od_b_c: product_prod_b_c > fset_P8649754740804009550od_b_c > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    fmembe3088975721132771320_c_nat: product_prod_c_nat > fset_P7886154461505542575_c_nat > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__c_Mtf__b_J,type,
    fmembe2163177356717951981od_c_b: product_prod_c_b > fset_P7943071700152676876od_c_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    fmembe2163177361021180782od_c_c: product_prod_c_c > fset_P8014105740199022861od_c_c > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__a_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fmembe4236371882286196753term_a: term_a4617543059560371146term_a > fset_t4263150265408568880term_a > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__a_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    fmembe6890459216773255580rm_a_b: term_a_term_a_b > fset_term_a_term_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fmember_term_a_b: term_a_b > fset_term_a_b > $o ).

thf(sy_c_FSet_Ofmember_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    fmember_term_a_c: term_a_c > fset_term_a_c > $o ).

thf(sy_c_FSet_Ofmember_001t__Tree____Automata__Ota____rule_Itf__c_Mtf__a_J,type,
    fmembe5561272363050924325le_c_a: tree_ta_rule_c_a > fset_T7970579319148954716le_c_a > $o ).

thf(sy_c_FSet_Ofmember_001tf__b,type,
    fmember_b: b > fset_b > $o ).

thf(sy_c_FSet_Ofmember_001tf__c,type,
    fmember_c: c > fset_c > $o ).

thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fset_P6484774398675748578term_a: fset_P6640269656128473term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_FSet_Ofset_Ofset_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fset_term_a_b2: fset_term_a_b > set_term_a_b ).

thf(sy_c_FSet_Ofset_Ofset_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    fset_term_a_c2: fset_term_a_c > set_term_a_c ).

thf(sy_c_FSet_Ofset__of__list_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fset_o2064584849798326990rm_a_b: list_term_a_b > fset_term_a_b ).

thf(sy_c_FSet_Ofset__of__list_001tf__c,type,
    fset_of_list_c: list_c > fset_c ).

thf(sy_c_FSet_Ofthe__elem_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fthe_e5677522983250699724term_a: fset_Bot_bot_term_a > bot_bot_term_a ).

thf(sy_c_FSet_Ofthe__elem_001t__Nat__Onat,type,
    fthe_elem_nat: fset_nat > nat ).

thf(sy_c_FSet_Ofthe__elem_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fthe_elem_term_a_b: fset_term_a_b > term_a_b ).

thf(sy_c_FSet__Utils_OfId__on_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f5231733412705268353term_a: fset_Bot_bot_term_a > fset_P6640269656128473term_a ).

thf(sy_c_FSet__Utils_OfId__on_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    fSet_f2178757263449557692rm_a_b: fset_fset_term_a_b > fset_P7393603901796714959rm_a_b ).

thf(sy_c_FSet__Utils_OfId__on_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fSet_f2002284675934200977term_a: fset_l235587408706331465term_a > fset_P7293260781336502521term_a ).

thf(sy_c_FSet__Utils_OfId__on_001t__Nat__Onat,type,
    fSet_fId_on_nat: fset_nat > fset_P5670320511379867111at_nat ).

thf(sy_c_FSet__Utils_OfId__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_fId_on_term_a_b: fset_term_a_b > fset_P3577670608877728751rm_a_b ).

thf(sy_c_FSet__Utils_OfId__on_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    fSet_fId_on_term_a_c: fset_term_a_c > fset_P6525107853456574705rm_a_c ).

thf(sy_c_FSet__Utils_OfId__on_001tf__b,type,
    fSet_fId_on_b: fset_b > fset_P8578720700757663565od_b_b ).

thf(sy_c_FSet__Utils_OfId__on_001tf__c,type,
    fSet_fId_on_c: fset_c > fset_P8014105740199022861od_c_c ).

thf(sy_c_FSet__Utils_OfImage_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f5750617162473777234term_a: fset_P6640269656128473term_a > fset_Bot_bot_term_a > fset_Bot_bot_term_a ).

thf(sy_c_FSet__Utils_OfImage_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Nat__Onat,type,
    fSet_f3412622471138897305_a_nat: fset_P4615588779303864480_a_nat > fset_Bot_bot_term_a > fset_nat ).

thf(sy_c_FSet__Utils_OfImage_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__b,type,
    fSet_f6643166002469431670rm_a_b: fset_P7840673837303127387rm_a_b > fset_Bot_bot_term_a > fset_b ).

thf(sy_c_FSet__Utils_OfImage_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__c,type,
    fSet_f6643166002469431671rm_a_c: fset_P7911707877349473372rm_a_c > fset_Bot_bot_term_a > fset_c ).

thf(sy_c_FSet__Utils_OfImage_001t__Nat__Onat_001t__Nat__Onat,type,
    fSet_fImage_nat_nat: fset_P5670320511379867111at_nat > fset_nat > fset_nat ).

thf(sy_c_FSet__Utils_OfImage_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f8842252047629264932rm_a_b: fset_P1023298988895795115rm_a_b > fset_nat > fset_term_a_b ).

thf(sy_c_FSet__Utils_OfImage_001t__Nat__Onat_001tf__b,type,
    fSet_fImage_nat_b: fset_P5247029744508429268_nat_b > fset_nat > fset_b ).

thf(sy_c_FSet__Utils_OfImage_001t__Nat__Onat_001tf__c,type,
    fSet_fImage_nat_c: fset_P5318063784554775253_nat_c > fset_nat > fset_c ).

thf(sy_c_FSet__Utils_OfImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f8210585146631509224rm_a_b: fset_P3577670608877728751rm_a_b > fset_term_a_b > fset_term_a_b ).

thf(sy_c_FSet__Utils_OfImage_001tf__b_001t__Nat__Onat,type,
    fSet_fImage_b_nat: fset_P2289935846317411886_b_nat > fset_b > fset_nat ).

thf(sy_c_FSet__Utils_OfImage_001tf__b_001tf__b,type,
    fSet_fImage_b_b: fset_P8578720700757663565od_b_b > fset_b > fset_b ).

thf(sy_c_FSet__Utils_OfImage_001tf__b_001tf__c,type,
    fSet_fImage_b_c: fset_P8649754740804009550od_b_c > fset_b > fset_c ).

thf(sy_c_FSet__Utils_OfImage_001tf__c_001t__Nat__Onat,type,
    fSet_fImage_c_nat: fset_P7886154461505542575_c_nat > fset_c > fset_nat ).

thf(sy_c_FSet__Utils_OfImage_001tf__c_001tf__b,type,
    fSet_fImage_c_b: fset_P7943071700152676876od_c_b > fset_c > fset_b ).

thf(sy_c_FSet__Utils_OfImage_001tf__c_001tf__c,type,
    fSet_fImage_c_c: fset_P8014105740199022861od_c_c > fset_c > fset_c ).

thf(sy_c_FSet__Utils_OfSigma_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f5829356563446949852term_a: fset_Bot_bot_term_a > ( bot_bot_term_a > fset_Bot_bot_term_a ) > fset_P6640269656128473term_a ).

thf(sy_c_FSet__Utils_OfSigma_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fSet_f5193722992130926076term_a: fset_l235587408706331465term_a > ( list_Bot_bot_term_a > fset_l235587408706331465term_a ) > fset_P7293260781336502521term_a ).

thf(sy_c_FSet__Utils_OfSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    fSet_fSigma_nat_nat: fset_nat > ( nat > fset_nat ) > fset_P5670320511379867111at_nat ).

thf(sy_c_FSet__Utils_OfSigma_001t__Nat__Onat_001tf__b,type,
    fSet_fSigma_nat_b: fset_nat > ( nat > fset_b ) > fset_P5247029744508429268_nat_b ).

thf(sy_c_FSet__Utils_OfSigma_001t__Nat__Onat_001tf__c,type,
    fSet_fSigma_nat_c: fset_nat > ( nat > fset_c ) > fset_P5318063784554775253_nat_c ).

thf(sy_c_FSet__Utils_OfSigma_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f1273034697330015346rm_a_b: fset_term_a_b > ( term_a_b > fset_term_a_b ) > fset_P3577670608877728751rm_a_b ).

thf(sy_c_FSet__Utils_OfSigma_001tf__b_001t__Nat__Onat,type,
    fSet_fSigma_b_nat: fset_b > ( b > fset_nat ) > fset_P2289935846317411886_b_nat ).

thf(sy_c_FSet__Utils_OfSigma_001tf__b_001tf__b,type,
    fSet_fSigma_b_b: fset_b > ( b > fset_b ) > fset_P8578720700757663565od_b_b ).

thf(sy_c_FSet__Utils_OfSigma_001tf__b_001tf__c,type,
    fSet_fSigma_b_c: fset_b > ( b > fset_c ) > fset_P8649754740804009550od_b_c ).

thf(sy_c_FSet__Utils_OfSigma_001tf__c_001t__Nat__Onat,type,
    fSet_fSigma_c_nat: fset_c > ( c > fset_nat ) > fset_P7886154461505542575_c_nat ).

thf(sy_c_FSet__Utils_OfSigma_001tf__c_001tf__b,type,
    fSet_fSigma_c_b: fset_c > ( c > fset_b ) > fset_P7943071700152676876od_c_b ).

thf(sy_c_FSet__Utils_OfSigma_001tf__c_001tf__c,type,
    fSet_fSigma_c_c: fset_c > ( c > fset_c ) > fset_P8014105740199022861od_c_c ).

thf(sy_c_FSet__Utils_Ofconverse_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f3482253648094759588term_a: fset_P6640269656128473term_a > fset_P6640269656128473term_a ).

thf(sy_c_FSet__Utils_Ofconverse_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fSet_f52542278078447812term_a: fset_P7293260781336502521term_a > fset_P7293260781336502521term_a ).

thf(sy_c_FSet__Utils_Ofconverse_001t__Nat__Onat_001t__Nat__Onat,type,
    fSet_f5560228655207641650at_nat: fset_P5670320511379867111at_nat > fset_P5670320511379867111at_nat ).

thf(sy_c_FSet__Utils_Ofconverse_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f4591896755199508282rm_a_b: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b ).

thf(sy_c_FSet__Utils_Ofconverse_001tf__b_001tf__b,type,
    fSet_fconverse_b_b: fset_P8578720700757663565od_b_b > fset_P8578720700757663565od_b_b ).

thf(sy_c_FSet__Utils_Ofconverse_001tf__c_001tf__c,type,
    fSet_fconverse_c_c: fset_P8014105740199022861od_c_c > fset_P8014105740199022861od_c_c ).

thf(sy_c_FSet__Utils_Offunas__gterm_001tf__a,type,
    fSet_ffunas_gterm_a: ground_gterm_a > fset_P5917089267984057005_a_nat ).

thf(sy_c_FSet__Utils_Offunas__term_001tf__a_001tf__c,type,
    fSet_ffunas_term_a_c: term_a_c > fset_P5917089267984057005_a_nat ).

thf(sy_c_FSet__Utils_Offuns__term_001tf__a_001tf__b,type,
    fSet_ffuns_term_a_b: term_a_b > fset_a ).

thf(sy_c_FSet__Utils_Offuns__term_001tf__a_001tf__c,type,
    fSet_ffuns_term_a_c: term_a_c > fset_a ).

thf(sy_c_FSet__Utils_Ofinj__on_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f3581483917893148177rm_a_b: ( term_a_b > term_a_b ) > fset_term_a_b > $o ).

thf(sy_c_FSet__Utils_Ofrelcomp_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f5283962319602365193term_a: fset_P6640269656128473term_a > fset_P6640269656128473term_a > fset_P6640269656128473term_a ).

thf(sy_c_FSet__Utils_Ofrelcomp_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fSet_f4048747136859825209term_a: fset_P7293260781336502521term_a > fset_P7293260781336502521term_a > fset_P7293260781336502521term_a ).

thf(sy_c_FSet__Utils_Ofrelcomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    fSet_f5609620875662277726at_nat: fset_P5670320511379867111at_nat > fset_P5670320511379867111at_nat > fset_P5670320511379867111at_nat ).

thf(sy_c_FSet__Utils_Ofrelcomp_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f8278883550849121706rm_a_b: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b ).

thf(sy_c_FSet__Utils_Ofrelcomp_001tf__b_001tf__b_001tf__b,type,
    fSet_frelcomp_b_b_b: fset_P8578720700757663565od_b_b > fset_P8578720700757663565od_b_b > fset_P8578720700757663565od_b_b ).

thf(sy_c_FSet__Utils_Ofrelcomp_001tf__c_001tf__c_001tf__c,type,
    fSet_frelcomp_c_c_c: fset_P8014105740199022861od_c_c > fset_P8014105740199022861od_c_c > fset_P8014105740199022861od_c_c ).

thf(sy_c_FSet__Utils_Oftrancl_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f7983512403460980795term_a: fset_P6640269656128473term_a > fset_P6640269656128473term_a ).

thf(sy_c_FSet__Utils_Oftrancl_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    fSet_f445164888270409035term_a: fset_P7293260781336502521term_a > fset_P7293260781336502521term_a ).

thf(sy_c_FSet__Utils_Oftrancl_001t__Nat__Onat,type,
    fSet_ftrancl_nat: fset_P5670320511379867111at_nat > fset_P5670320511379867111at_nat ).

thf(sy_c_FSet__Utils_Oftrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f6224701657781119302rm_a_b: fset_P3577670608877728751rm_a_b > fset_P3577670608877728751rm_a_b ).

thf(sy_c_FSet__Utils_Oftrancl_001tf__b,type,
    fSet_ftrancl_b: fset_P8578720700757663565od_b_b > fset_P8578720700757663565od_b_b ).

thf(sy_c_FSet__Utils_Oftrancl_001tf__c,type,
    fSet_ftrancl_c: fset_P8014105740199022861od_c_c > fset_P8014105740199022861od_c_c ).

thf(sy_c_FSet__Utils_Ofvars__ctxt_001tf__a_001tf__c,type,
    fSet_fvars_ctxt_a_c: subterm_and_ctxt_a_c > fset_c ).

thf(sy_c_FSet__Utils_Ofvars__term_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    fSet_f2604459547217817330term_a: term_a4617543059560371146term_a > fset_Bot_bot_term_a ).

thf(sy_c_FSet__Utils_Ofvars__term_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_f7033438052906747389rm_a_b: term_a_term_a_b > fset_term_a_b ).

thf(sy_c_FSet__Utils_Ofvars__term_001tf__a_001tf__b,type,
    fSet_fvars_term_a_b: term_a_b > fset_b ).

thf(sy_c_FSet__Utils_Ofvars__term_001tf__a_001tf__c,type,
    fSet_fvars_term_a_c: term_a_c > fset_c ).

thf(sy_c_FSet__Utils_Othe__finv__into_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    fSet_t5505035234354303037rm_a_b: fset_term_a_b > ( term_a_b > term_a_b ) > term_a_b > term_a_b ).

thf(sy_c_Fun_Oid_001_Eo,type,
    id_o: $o > $o ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    id_Pro7590044230637570276term_a: produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Fun_Oid_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    id_term_a_b: term_a_b > term_a_b ).

thf(sy_c_Fun_Oid_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    id_term_a_c: term_a_c > term_a_c ).

thf(sy_c_Fun_Omap__fun_001t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_001_Eo_001_Eo,type,
    map_fu7453585163258293968_a_o_o: ( fset_P6640269656128473term_a > set_Pr3965413743549523539term_a ) > ( $o > $o ) > ( set_Pr3965413743549523539term_a > $o ) > fset_P6640269656128473term_a > $o ).

thf(sy_c_Fun_Omap__fun_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_001_Eo_001_Eo,type,
    map_fu435557360475869606_b_o_o: ( fset_term_a_b > set_term_a_b ) > ( $o > $o ) > ( set_term_a_b > $o ) > fset_term_a_b > $o ).

thf(sy_c_Fun_Omap__fun_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_001_Eo_001_Eo,type,
    map_fu5563206862702799912_c_o_o: ( fset_term_a_c > set_term_a_c ) > ( $o > $o ) > ( set_term_a_c > $o ) > fset_term_a_c > $o ).

thf(sy_c_Fun_Omap__fun_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_M_Eo_J_001_062_It__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_M_Eo_J,type,
    map_fu4212813072579612070rm_a_o: ( produc7033602110846336755term_a > produc7033602110846336755term_a ) > ( ( set_Pr3965413743549523539term_a > $o ) > fset_P6640269656128473term_a > $o ) > ( produc7033602110846336755term_a > set_Pr3965413743549523539term_a > $o ) > produc7033602110846336755term_a > fset_P6640269656128473term_a > $o ).

thf(sy_c_Fun_Omap__fun_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001_062_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J_001_062_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
    map_fu2757774360759362054_a_b_o: ( term_a_b > term_a_b ) > ( ( set_term_a_b > $o ) > fset_term_a_b > $o ) > ( term_a_b > set_term_a_b > $o ) > term_a_b > fset_term_a_b > $o ).

thf(sy_c_Fun_Omap__fun_001t__Term__Oterm_Itf__a_Mtf__c_J_001t__Term__Oterm_Itf__a_Mtf__c_J_001_062_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_M_Eo_J_001_062_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J_M_Eo_J,type,
    map_fu2492594514265942088_a_c_o: ( term_a_c > term_a_c ) > ( ( set_term_a_c > $o ) > fset_term_a_c > $o ) > ( term_a_c > set_term_a_c > $o ) > term_a_c > fset_term_a_c > $o ).

thf(sy_c_Ground__Terms_Ogterm__of__term_001tf__a_001tf__c,type,
    ground3594116404241818628rm_a_c: term_a_c > ground_gterm_a ).

thf(sy_c_Ground__Terms_Oterm__of__gterm_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    ground2947561787367041419term_a: ground_gterm_a > term_a4617543059560371146term_a ).

thf(sy_c_Ground__Terms_Oterm__of__gterm_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    ground678623728659755670rm_a_b: ground_gterm_a > term_a_term_a_b ).

thf(sy_c_Ground__Terms_Oterm__of__gterm_001tf__a_001tf__b,type,
    ground5786051259813020157rm_a_b: ground_gterm_a > term_a_b ).

thf(sy_c_Ground__Terms_Oterm__of__gterm_001tf__a_001tf__c,type,
    ground5786051259813020158rm_a_c: ground_gterm_a > term_a_c ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    minus_5522361424680520477rm_a_b: fset_term_a_b > fset_term_a_b > fset_term_a_b ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    minus_4931416064860695866term_a: set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    minus_4684887306374467492rm_a_c: set_term_a_c > set_term_a_c > set_term_a_c ).

thf(sy_c_If_001t__FSet__Ofset_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    if_fse3313186383936193710rm_a_b: $o > fset_fset_term_a_b > fset_fset_term_a_b > fset_fset_term_a_b ).

thf(sy_c_If_001t__FSet__Ofset_It__Nat__Onat_J,type,
    if_fset_nat: $o > fset_nat > fset_nat > fset_nat ).

thf(sy_c_If_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    if_fset_term_a_b: $o > fset_term_a_b > fset_term_a_b > fset_term_a_b ).

thf(sy_c_If_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    if_fset_term_a_c: $o > fset_term_a_c > fset_term_a_c > fset_term_a_c ).

thf(sy_c_If_001t__FSet__Ofset_Itf__b_J,type,
    if_fset_b: $o > fset_b > fset_b > fset_b ).

thf(sy_c_If_001t__FSet__Ofset_Itf__c_J,type,
    if_fset_c: $o > fset_c > fset_c > fset_c ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    if_term_a_b: $o > term_a_b > term_a_b > term_a_b ).

thf(sy_c_If_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    if_term_a_c: $o > term_a_c > term_a_c > term_a_c ).

thf(sy_c_If_001tf__b,type,
    if_b: $o > b > b > b ).

thf(sy_c_If_001tf__c,type,
    if_c: $o > c > c > c ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    inf_in7980079480601282006rm_a_b: fset_term_a_b > fset_term_a_b > fset_term_a_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    inf_in2157114169531486017term_a: set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    inf_inf_set_term_a_c: set_term_a_c > set_term_a_c > set_term_a_c ).

thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    sup_su1847537981033997936rm_a_b: fset_term_a_b > fset_term_a_b > fset_term_a_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_Itf__c_J,type,
    sup_sup_fset_c: fset_c > fset_c > fset_c ).

thf(sy_c_List_Olist_OCons_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    cons_Bot_bot_term_a: bot_bot_term_a > list_Bot_bot_term_a > list_Bot_bot_term_a ).

thf(sy_c_List_Olist__update_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    list_u5503441972953478683term_a: list_Bot_bot_term_a > nat > bot_bot_term_a > list_Bot_bot_term_a ).

thf(sy_c_List_Onth_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    nth_Bot_bot_term_a: list_Bot_bot_term_a > nat > bot_bot_term_a ).

thf(sy_c_List_Onth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    nth_term_a_b: list_term_a_b > nat > term_a_b ).

thf(sy_c_List_Onth_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    nth_term_a_c: list_term_a_c > nat > term_a_c ).

thf(sy_c_List_Onth_001tf__c,type,
    nth_c: list_c > nat > c ).

thf(sy_c_NF_Ofstates_001tf__a_001tf__b,type,
    fstates_a_b: fset_term_a_b > fset_Bot_bot_term_a ).

thf(sy_c_NF_Ofstates_001tf__a_001tf__c,type,
    fstates_a_c: fset_term_a_c > fset_Bot_bot_term_a ).

thf(sy_c_NF_Onf__reg_001tf__a_001tf__b,type,
    nf_reg_a_b: fset_term_a_b > fset_P5917089267984057005_a_nat > tree_r6075998776338966420rm_a_a ).

thf(sy_c_NF_Onf__reg_001tf__a_001tf__c,type,
    nf_reg_a_c: fset_term_a_c > fset_P5917089267984057005_a_nat > tree_r6075998776338966420rm_a_a ).

thf(sy_c_NF_Onf__ta_001tf__a_001tf__b,type,
    nf_ta_a_b: fset_term_a_b > fset_P5917089267984057005_a_nat > tree_t5222909361115825503rm_a_a ).

thf(sy_c_NF_Onf__ta_001tf__a_001tf__c,type,
    nf_ta_a_c: fset_term_a_c > fset_P5917089267984057005_a_nat > tree_t5222909361115825503rm_a_a ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    size_s1103687553077312429term_a: list_Bot_bot_term_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    size_s8906293707977694520rm_a_b: list_term_a_b > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    size_s8977327748024040505rm_a_c: list_term_a_c > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__c_J,type,
    size_size_list_c: list_c > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J,type,
    bot_bo104646228311450846rm_a_o: produc7033602110846336755term_a > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J,type,
    bot_bot_term_a_c_o: term_a_c > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
    bot_bot_o: $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    bot_bo4313458260509495885term_a: fset_Bot_bot_term_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    bot_bo7844603177471244936rm_a_b: fset_fset_term_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Nat__Onat_J,type,
    bot_bot_fset_nat: fset_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    bot_bo3304177514724984389term_a: fset_P6640269656128473term_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    bot_bo173352615033564763rm_a_b: fset_P3577670608877728751rm_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    bot_bo121928729148956601od_b_b: fset_P8578720700757663565od_b_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__c_Mtf__c_J_J,type,
    bot_bo8780685805445091705od_c_c: fset_P8014105740199022861od_c_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bot_bo1583650863486268376rm_a_b: fset_term_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    bot_bo1654684903532614361rm_a_c: fset_term_a_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Tree____Automata__Ota____rule_It__Bot____Terms__Obot____term_Itf__a_J_Mtf__a_J_J,type,
    bot_bo1498970239794229685rm_a_a: fset_T7482312105790613833rm_a_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Tree____Automata__Ota____rule_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__a_J_J,type,
    bot_bo3560329175948364010_a_b_a: fset_T4690922579222374526_a_b_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Tree____Automata__Ota____rule_Itf__b_Mtf__a_J_J,type,
    bot_bo6352603316032826929le_b_a: fset_T8606228319753941405le_b_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Tree____Automata__Ota____rule_Itf__c_Mtf__a_J_J,type,
    bot_bo5716954315427840240le_c_a: fset_T7970579319148954716le_c_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_Itf__a_J,type,
    bot_bot_fset_a: fset_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_Itf__b_J,type,
    bot_bot_fset_b: fset_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_Itf__c_J,type,
    bot_bot_fset_c: fset_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J_Mt__Nat__Onat_J,type,
    bot_bo2899094611167244519_o_nat: produc5520244230421620859_o_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_I_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J_Mt__Nat__Onat_J,type,
    bot_bo6296390520458004621_o_nat: produc8681781744089100281_o_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J_J,type,
    bot_bo7312153412206230759rm_a_o: produc709930994605831291rm_a_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J_J,type,
    bot_bo3271289617687911219_a_c_o: produc5656680841319006879_a_c_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bot_bo2769642828321324397at_nat: product_prod_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    bot_bo6150146448544589324term_a: produc6821386223223610744term_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    bot_bo1639525010489079912rm_a_c: produc1685657453158572540rm_a_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_Mt__Nat__Onat_J,type,
    bot_bo6547199808529123302_a_nat: produc7218439583208144722_a_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_Mt__Nat__Onat_J,type,
    bot_bo3585513266671089768_c_nat: produc3631645709340582396_c_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    bot_bo8902706747039010275rm_a_c: produc1197155033112071543rm_a_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
    bot_bo2731025238921882491term_a: set_Ground_gterm_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    bot_bo2447593420621686463term_a: set_Pr3965413743549523539term_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    bot_bo5671043144174269919term_a: set_Pr7374270731007213939term_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_J,type,
    bot_bo6119087614312989235term_a: set_Pr825864233230045127term_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
    bot_bo197521221353338581rm_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J_J,type,
    bot_bo3144958465932184535rm_a_c: set_Pr7334014819586186091rm_a_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    bot_bot_set_term_a_b: set_term_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    bot_bot_set_term_a_c: set_term_a_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
    bot_bot_set_b: set_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
    bot_bot_set_c: set_c ).

thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    ord_le3921303545160784792rm_a_b: fset_term_a_b > fset_term_a_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_le1203424502768444845at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    ord_le7216997114146882585term_a: fset_Bot_bot_term_a > fset_Bot_bot_term_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
    ord_le1241528519210447629_a_nat: fset_P5917089267984057005_a_nat > fset_P5917089267984057005_a_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    ord_le8417325568038501796rm_a_b: fset_term_a_b > fset_term_a_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_Itf__c_J,type,
    ord_less_eq_fset_c: fset_c > fset_c > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    ord_le2299786838182399475term_a: set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    ord_le2776320456296813995rm_a_c: set_term_a_c > set_term_a_c > $o ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J_001t__Nat__Onat,type,
    produc6792305905529949363_o_nat: ( produc7033602110846336755term_a > $o ) > nat > produc5520244230421620859_o_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J_001t__Nat__Onat,type,
    produc9092562165568265075_o_nat: ( term_a_c > $o ) > nat > produc8681781744089100281_o_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc780340308816795563term_a: bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Nat__Onat,type,
    produc6633979256947609458_a_nat: bot_bot_term_a > nat > produc8106822358505058234_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__b,type,
    produc5551744644817243101rm_a_b: bot_bot_term_a > b > produc2491715934637117611rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__c,type,
    produc5551744644817243102rm_a_c: bot_bot_term_a > c > produc2491715938940346412rm_a_c ).

thf(sy_c_Product__Type_OPair_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc8129827254723108897rm_a_b: fset_term_a_b > fset_term_a_b > produc2633411654673805033rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    produc1077782494051512011term_a: list_Bot_bot_term_a > list_Bot_bot_term_a > produc2056175833795995411term_a ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_M_Eo_J,type,
    produc4976332323224857011rm_a_o: nat > ( produc7033602110846336755term_a > $o ) > produc709930994605831291rm_a_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_062_It__Term__Oterm_Itf__a_Mtf__c_J_M_Eo_J,type,
    produc4559189924671168081_a_c_o: nat > ( term_a_c > $o ) > produc5656680841319006879_a_c_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc8051651767284650410term_a: nat > set_Pr3965413743549523539term_a > produc6821386223223610744term_a ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc5999293419280449332rm_a_c: nat > set_term_a_c > produc1685657453158572540rm_a_c ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc1516572978046417917rm_a_b: nat > term_a_b > produc1234881154892807749rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__b,type,
    product_Pair_nat_b: nat > b > product_prod_nat_b ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__c,type,
    product_Pair_nat_c: nat > c > product_prod_nat_c ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    produc1145950395756198999term_a: produc7033602110846336755term_a > produc7033602110846336755term_a > produc441850963132823271term_a ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J_001t__Nat__Onat,type,
    produc5627630658439173836_a_nat: set_Pr3965413743549523539term_a > nat > produc7218439583208144722_a_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_001t__Nat__Onat,type,
    produc4240701259145610164_c_nat: set_term_a_c > nat > produc3631645709340582396_c_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc3305361712476037935rm_a_c: set_term_a_c > set_term_a_c > produc1197155033112071543rm_a_c ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    produc7020197800436672577rm_a_b: term_a_b > term_a_b > produc357393685978478089rm_a_b ).

thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__c_J_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    produc7687097280279809731rm_a_c: term_a_c > term_a_c > produc1095327201564732427rm_a_c ).

thf(sy_c_Product__Type_OPair_001tf__b_001t__Nat__Onat,type,
    product_Pair_b_nat: b > nat > product_prod_b_nat ).

thf(sy_c_Product__Type_OPair_001tf__b_001tf__b,type,
    product_Pair_b_b: b > b > product_prod_b_b ).

thf(sy_c_Product__Type_OPair_001tf__b_001tf__c,type,
    product_Pair_b_c: b > c > product_prod_b_c ).

thf(sy_c_Product__Type_OPair_001tf__c_001t__Nat__Onat,type,
    product_Pair_c_nat: c > nat > product_prod_c_nat ).

thf(sy_c_Product__Type_OPair_001tf__c_001tf__b,type,
    product_Pair_c_b: c > b > product_prod_c_b ).

thf(sy_c_Product__Type_OPair_001tf__c_001tf__c,type,
    product_Pair_c_c: c > c > product_prod_c_c ).

thf(sy_c_Product__Type_Oapfst_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc285492028240800985term_a: ( bot_bot_term_a > bot_bot_term_a ) > produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_Oapsnd_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc6734519164561896987term_a: ( bot_bot_term_a > bot_bot_term_a ) > produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_Ocurry_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001_Eo,type,
    produc4484525301753677177rm_a_o: ( produc7033602110846336755term_a > $o ) > bot_bot_term_a > bot_bot_term_a > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    produc7987411809206226644term_a: ( produc7033602110846336755term_a > produc7033602110846336755term_a ) > bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_Ocurry_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc2116068065703686708term_a: ( produc7033602110846336755term_a > set_Pr3965413743549523539term_a ) > bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Product__Type_Ocurry_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc5035402668397846058rm_a_c: ( produc7033602110846336755term_a > set_term_a_c ) > bot_bot_term_a > bot_bot_term_a > set_term_a_c ).

thf(sy_c_Product__Type_Ocurry_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_001_Eo,type,
    produc3073380265971003737rm_a_o: ( produc2056175833795995411term_a > $o ) > list_Bot_bot_term_a > list_Bot_bot_term_a > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc1310100445399344235_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).

thf(sy_c_Product__Type_Ocurry_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001_Eo,type,
    produc6008550485034061667_a_b_o: ( produc357393685978478089rm_a_b > $o ) > term_a_b > term_a_b > $o ).

thf(sy_c_Product__Type_Ocurry_001tf__b_001tf__b_001_Eo,type,
    product_curry_b_b_o: ( product_prod_b_b > $o ) > b > b > $o ).

thf(sy_c_Product__Type_Ocurry_001tf__c_001tf__c_001_Eo,type,
    product_curry_c_c_o: ( product_prod_c_c > $o ) > c > c > $o ).

thf(sy_c_Product__Type_Ointernal__case__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001_Eo,type,
    produc5755667415957967490rm_a_o: ( bot_bot_term_a > bot_bot_term_a > $o ) > produc7033602110846336755term_a > $o ).

thf(sy_c_Product__Type_Ointernal__case__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    produc1035157467515367901term_a: ( bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a ) > produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_Ointernal__case__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc512496843244024125term_a: ( bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a ) > produc7033602110846336755term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Product__Type_Ointernal__case__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc7561195436278546913rm_a_c: ( bot_bot_term_a > bot_bot_term_a > set_term_a_c ) > produc7033602110846336755term_a > set_term_a_c ).

thf(sy_c_Product__Type_Omap__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc5208308128219516446term_a: ( bot_bot_term_a > bot_bot_term_a ) > ( bot_bot_term_a > bot_bot_term_a ) > produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001_Eo,type,
    produc480846774855951832rm_a_o: ( bot_bot_term_a > bot_bot_term_a > $o ) > produc7033602110846336755term_a > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    produc1985598078976903475term_a: ( bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a ) > produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc2700287094330126483term_a: ( bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a ) > produc7033602110846336755term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc5838651610459311114rm_a_b: ( bot_bot_term_a > bot_bot_term_a > set_term_a_b ) > produc7033602110846336755term_a > set_term_a_b ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc5909685650505657099rm_a_c: ( bot_bot_term_a > bot_bot_term_a > set_term_a_c ) > produc7033602110846336755term_a > set_term_a_c ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    produc2411476669527729797term_a: ( nat > nat > set_Pr3965413743549523539term_a ) > product_prod_nat_nat > set_Pr3965413743549523539term_a ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc1714188619747066776rm_a_b: ( nat > nat > set_term_a_b ) > product_prod_nat_nat > set_term_a_b ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc1785222659793412761rm_a_c: ( nat > nat > set_term_a_c ) > product_prod_nat_nat > set_term_a_c ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__b_001tf__b_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc8686635659240032694rm_a_b: ( b > b > set_term_a_b ) > product_prod_b_b > set_term_a_b ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__b_001tf__b_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc8757669699286378679rm_a_c: ( b > b > set_term_a_c ) > product_prod_b_b > set_term_a_c ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__c_001tf__c_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    produc8697811999727086582rm_a_b: ( c > c > set_term_a_b ) > product_prod_c_c > set_term_a_b ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__c_001tf__c_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    produc8768846039773432567rm_a_c: ( c > c > set_term_a_c ) > product_prod_c_c > set_term_a_c ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc3704134870435948439term_a: produc7033602110846336755term_a > bot_bot_term_a ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc7794171008584320473term_a: produc7033602110846336755term_a > bot_bot_term_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Oswap_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    produc5746675070383261003term_a: produc7033602110846336755term_a > produc7033602110846336755term_a ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    collec7536399311987840478term_a: ( produc7033602110846336755term_a > $o ) > set_Pr3965413743549523539term_a ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    collec2650751154016459518term_a: ( produc2056175833795995411term_a > $o ) > set_Pr7374270731007213939term_a ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    collec7763623725074747986term_a: ( produc441850963132823271term_a > $o ) > set_Pr825864233230045127term_a ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    collec99177395925924084rm_a_b: ( produc357393685978478089rm_a_b > $o ) > set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    collec837110911512178422rm_a_c: ( produc1095327201564732427rm_a_c > $o ) > set_Pr7334014819586186091rm_a_c ).

thf(sy_c_Set_OCollect_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    collect_term_a_b: ( term_a_b > $o ) > set_term_a_b ).

thf(sy_c_Set_OCollect_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    collect_term_a_c: ( term_a_c > $o ) > set_term_a_c ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    insert7072317625840424131term_a: produc7033602110846336755term_a > set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Set_Oinsert_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    insert_term_a_c: term_a_c > set_term_a_c > set_term_a_c ).

thf(sy_c_Subterm__and__Context_Octxt_OHole_001tf__a_001tf__b,type,
    subterm_and_Hole_a_b: subterm_and_ctxt_a_b ).

thf(sy_c_Subterm__and__Context_Octxt_OHole_001tf__a_001tf__c,type,
    subterm_and_Hole_a_c: subterm_and_ctxt_a_c ).

thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    subter8974050253859094258term_a: subter720051973439326747term_a > term_a4617543059560371146term_a > term_a4617543059560371146term_a ).

thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    subter7263177522250274301rm_a_b: subter2881564190832396198rm_a_b > term_a_term_a_b > term_a_term_a_b ).

thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001tf__b,type,
    subter2376574525758040790rm_a_b: subterm_and_ctxt_a_b > term_a_b > term_a_b ).

thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001tf__c,type,
    subter2376574525758040791rm_a_c: subterm_and_ctxt_a_c > term_a_c > term_a_c ).

thf(sy_c_Subterm__and__Context_Osubst__apply__ctxt_001tf__a_001tf__b_001tf__b,type,
    subter4086027020339385547_a_b_b: subterm_and_ctxt_a_b > ( b > term_a_b ) > subterm_and_ctxt_a_b ).

thf(sy_c_Subterm__and__Context_Osubst__apply__ctxt_001tf__a_001tf__b_001tf__c,type,
    subter4086027020339385548_a_b_c: subterm_and_ctxt_a_b > ( b > term_a_c ) > subterm_and_ctxt_a_c ).

thf(sy_c_Subterm__and__Context_Osubst__apply__ctxt_001tf__a_001tf__c_001tf__b,type,
    subter1298571438366709514_a_c_b: subterm_and_ctxt_a_c > ( c > term_a_b ) > subterm_and_ctxt_a_b ).

thf(sy_c_Subterm__and__Context_Osubst__apply__ctxt_001tf__a_001tf__c_001tf__c,type,
    subter1298571438366709515_a_c_c: subterm_and_ctxt_a_c > ( c > term_a_c ) > subterm_and_ctxt_a_c ).

thf(sy_c_Subterm__and__Context_Osupt_001tf__a_001tf__b,type,
    subterm_and_supt_a_b: set_Pr4386577575007340137rm_a_b ).

thf(sy_c_Subterm__and__Context_Osupt_001tf__a_001tf__c,type,
    subterm_and_supt_a_c: set_Pr7334014819586186091rm_a_c ).

thf(sy_c_Term_Orestrict__subst__domain_001tf__b_001tf__a,type,
    restri22458263168500592in_b_a: set_b > ( b > term_a_b ) > b > term_a_b ).

thf(sy_c_Term_Orestrict__subst__domain_001tf__c_001tf__a,type,
    restri6458374718050600367in_c_a: set_c > ( c > term_a_c ) > c > term_a_c ).

thf(sy_c_Term_Osubst_001tf__b_001tf__a,type,
    subst_b_a: b > term_a_b > b > term_a_b ).

thf(sy_c_Term_Osubst_001tf__c_001tf__a,type,
    subst_c_a: c > term_a_c > c > term_a_c ).

thf(sy_c_Term_Osubst__apply__term_001tf__a_001tf__b_001tf__b,type,
    subst_7999470309526761004_a_b_b: term_a_b > ( b > term_a_b ) > term_a_b ).

thf(sy_c_Term_Osubst__apply__term_001tf__a_001tf__b_001tf__c,type,
    subst_7999470309526761005_a_b_c: term_a_b > ( b > term_a_c ) > term_a_c ).

thf(sy_c_Term_Osubst__apply__term_001tf__a_001tf__c_001tf__b,type,
    subst_5212014727554084971_a_c_b: term_a_c > ( c > term_a_b ) > term_a_b ).

thf(sy_c_Term_Osubst__apply__term_001tf__a_001tf__c_001tf__c,type,
    subst_5212014727554084972_a_c_c: term_a_c > ( c > term_a_c ) > term_a_c ).

thf(sy_c_Term_Oterm_OFun_001tf__a_001tf__b,type,
    fun_a_b: a > list_term_a_b > term_a_b ).

thf(sy_c_Term_Oterm_OFun_001tf__a_001tf__c,type,
    fun_a_c: a > list_term_a_c > term_a_c ).

thf(sy_c_Term_Oterm_OVar_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    var_Bot_bot_term_a_a: bot_bot_term_a > term_a4617543059560371146term_a ).

thf(sy_c_Term_Oterm_OVar_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    var_term_a_b_a: term_a_b > term_a_term_a_b ).

thf(sy_c_Term_Oterm_OVar_001tf__b_001tf__a,type,
    var_b_a: b > term_a_b ).

thf(sy_c_Term_Oterm_OVar_001tf__c_001tf__a,type,
    var_c_a: c > term_a_c ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_a6346539342315119866term_a: term_a4617543059560371146term_a > term_a4617543059560371146term_a ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__b,type,
    term_a5382035503001191118rm_a_b: term_a4617543059560371146term_a > term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__c,type,
    term_a5382035503001191119rm_a_c: term_a4617543059560371146term_a > term_a_c ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Nat__Onat_001tf__b,type,
    term_a4040953500496759367_nat_b: term_a_nat > term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Nat__Onat_001tf__c,type,
    term_a4040953500496759368_nat_c: term_a_nat > term_a_c ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_a6651074359774737541term_a: term_a_term_a_b > term_a4617543059560371146term_a ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_a2564936574044254608rm_a_b: term_a_term_a_b > term_a_term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__b,type,
    term_a3222098444227397763_a_b_b: term_a_term_a_b > term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__c,type,
    term_a3222098444227397764_a_b_c: term_a_term_a_b > term_a_c ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__b_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_a6990854600579303330term_a: term_a_b > term_a4617543059560371146term_a ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__b_001tf__b,type,
    term_a2780309480206857766_a_b_b: term_a_b > term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__b_001tf__c,type,
    term_a2780309480206857767_a_b_c: term_a_b > term_a_c ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__c_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_a3900797222385290531term_a: term_a_c > term_a4617543059560371146term_a ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__c_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_a4269149868823119406rm_a_b: term_a_c > term_a_term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__c_001tf__b,type,
    term_a9216225935088957541_a_c_b: term_a_c > term_a_b ).

thf(sy_c_Term__Context_Oadapt__vars_001tf__a_001tf__c_001tf__c,type,
    term_a9216225935088957542_a_c_c: term_a_c > term_a_c ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__c,type,
    term_a8854742828266083164rm_a_c: subter720051973439326747term_a > subterm_and_ctxt_a_c ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__c,type,
    term_a7029920194893525777_a_b_c: subter2881564190832396198rm_a_b > subterm_and_ctxt_a_c ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001tf__b_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_a1240189888989419567term_a: subterm_and_ctxt_a_b > subter720051973439326747term_a ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001tf__b_001tf__b,type,
    term_a1758706544955394649_a_b_b: subterm_and_ctxt_a_b > subterm_and_ctxt_a_b ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001tf__b_001tf__c,type,
    term_a1758706544955394650_a_b_c: subterm_and_ctxt_a_b > subterm_and_ctxt_a_c ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001tf__c_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_a7373504547650182576term_a: subterm_and_ctxt_a_c > subter720051973439326747term_a ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001tf__c_001tf__b,type,
    term_a8194622999837494424_a_c_b: subterm_and_ctxt_a_c > subterm_and_ctxt_a_b ).

thf(sy_c_Term__Context_Oadapt__vars__ctxt_001tf__a_001tf__c_001tf__c,type,
    term_a8194622999837494425_a_c_c: subterm_and_ctxt_a_c > subterm_and_ctxt_a_c ).

thf(sy_c_Term__Context_Oground_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_g4884896889520838057term_a: term_a4617543059560371146term_a > $o ).

thf(sy_c_Term__Context_Oground_001tf__a_001t__Nat__Onat,type,
    term_ground_a_nat: term_a_nat > $o ).

thf(sy_c_Term__Context_Oground_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_g2255532262831151796rm_a_b: term_a_term_a_b > $o ).

thf(sy_c_Term__Context_Oground_001tf__a_001tf__b,type,
    term_ground_a_b: term_a_b > $o ).

thf(sy_c_Term__Context_Oground_001tf__a_001tf__c,type,
    term_ground_a_c: term_a_c > $o ).

thf(sy_c_Term__Context_Oground__ctxt_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    term_g5220492397686563420term_a: subter720051973439326747term_a > $o ).

thf(sy_c_Term__Context_Oground__ctxt_001tf__a_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    term_g161376099170733415rm_a_b: subter2881564190832396198rm_a_b > $o ).

thf(sy_c_Term__Context_Oground__ctxt_001tf__a_001tf__b,type,
    term_ground_ctxt_a_b: subterm_and_ctxt_a_b > $o ).

thf(sy_c_Term__Context_Oground__ctxt_001tf__a_001tf__c,type,
    term_ground_ctxt_a_c: subterm_and_ctxt_a_c > $o ).

thf(sy_c_Transitive__Closure_Oacyclic_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    transi4548540958286804730term_a: set_Pr3965413743549523539term_a > $o ).

thf(sy_c_Transitive__Closure_Oacyclic_001t__Nat__Onat,type,
    transi3139058337892597825ic_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Transitive__Closure_Oacyclic_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    transi5314701259734769157rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_Transitive__Closure_Oacyclic_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    transi5314701264037997958rm_a_c: set_Pr7334014819586186091rm_a_c > $o ).

thf(sy_c_Transitive__Closure_Oacyclic_001tf__b,type,
    transitive_acyclic_b: set_Product_prod_b_b > $o ).

thf(sy_c_Transitive__Closure_Oacyclic_001tf__c,type,
    transitive_acyclic_c: set_Product_prod_c_c > $o ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    transi2774326411154761896term_a: set_Pr3965413743549523539term_a > set_Pr3965413743549523539term_a ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    transi306844429749061140term_a: set_Pr825864233230045127term_a > set_Pr825864233230045127term_a ).

thf(sy_c_Tree__Automata_O_092_060L_062_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_L8125518851412548553rm_a_a: tree_r6075998776338966420rm_a_a > set_Ground_gterm_a ).

thf(sy_c_Tree__Automata_O_092_060L_062_001tf__b_001tf__a,type,
    tree_L_b_a: tree_reg_b_a > set_Ground_gterm_a ).

thf(sy_c_Tree__Automata_O_092_060Q_062_001tf__c_001tf__a,type,
    tree_Q_c_a: tree_ta_c_a > fset_c ).

thf(sy_c_Tree__Automata_Oeps__states_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    tree_e551949735975627802term_a: fset_P6640269656128473term_a > fset_Bot_bot_term_a ).

thf(sy_c_Tree__Automata_Oeps__states_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    tree_e8828569120810636757rm_a_b: fset_P7393603901796714959rm_a_b > fset_fset_term_a_b ).

thf(sy_c_Tree__Automata_Oeps__states_001t__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J,type,
    tree_e5611726204267773354term_a: fset_P7293260781336502521term_a > fset_l235587408706331465term_a ).

thf(sy_c_Tree__Automata_Oeps__states_001t__Nat__Onat,type,
    tree_eps_states_nat: fset_P5670320511379867111at_nat > fset_nat ).

thf(sy_c_Tree__Automata_Oeps__states_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    tree_e4197689029844849445rm_a_b: fset_P3577670608877728751rm_a_b > fset_term_a_b ).

thf(sy_c_Tree__Automata_Oeps__states_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    tree_e4197689034148078246rm_a_c: fset_P6525107853456574705rm_a_c > fset_term_a_c ).

thf(sy_c_Tree__Automata_Oeps__states_001tf__b,type,
    tree_eps_states_b: fset_P8578720700757663565od_b_b > fset_b ).

thf(sy_c_Tree__Automata_Oeps__states_001tf__c,type,
    tree_eps_states_c: fset_P8014105740199022861od_c_c > fset_c ).

thf(sy_c_Tree__Automata_Ogta__lang_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_g6148166148305676299rm_a_a: fset_Bot_bot_term_a > tree_t5222909361115825503rm_a_a > set_Ground_gterm_a ).

thf(sy_c_Tree__Automata_Ois__ta__eps__free_001tf__b_001tf__a,type,
    tree_i6598266728802621794ee_b_a: tree_ta_b_a > $o ).

thf(sy_c_Tree__Automata_Ois__ta__eps__free_001tf__c_001tf__a,type,
    tree_i3810811146829945761ee_c_a: tree_ta_c_a > $o ).

thf(sy_c_Tree__Automata_Oreg_OReg_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_R2766333946944686633rm_a_a: fset_Bot_bot_term_a > tree_t5222909361115825503rm_a_a > tree_r6075998776338966420rm_a_a ).

thf(sy_c_Tree__Automata_Oreg_OReg_001tf__b_001tf__a,type,
    tree_Reg_b_a: fset_b > tree_ta_b_a > tree_reg_b_a ).

thf(sy_c_Tree__Automata_Ota_OTA_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_T877157127653060943rm_a_a: fset_T7482312105790613833rm_a_a > fset_P6640269656128473term_a > tree_t5222909361115825503rm_a_a ).

thf(sy_c_Tree__Automata_Ota_OTA_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    tree_TA_term_a_b_a: fset_T4690922579222374526_a_b_a > fset_P3577670608877728751rm_a_b > tree_ta_term_a_b_a ).

thf(sy_c_Tree__Automata_Ota_OTA_001tf__b_001tf__a,type,
    tree_TA_b_a: fset_T8606228319753941405le_b_a > fset_P8578720700757663565od_b_b > tree_ta_b_a ).

thf(sy_c_Tree__Automata_Ota_OTA_001tf__c_001tf__a,type,
    tree_TA_c_a: fset_T7970579319148954716le_c_a > fset_P8014105740199022861od_c_c > tree_ta_c_a ).

thf(sy_c_Tree__Automata_Ota_Oeps_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_e5393235435095066050rm_a_a: tree_t5222909361115825503rm_a_a > fset_P6640269656128473term_a ).

thf(sy_c_Tree__Automata_Ota_Oeps_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    tree_eps_term_a_b_a: tree_ta_term_a_b_a > fset_P3577670608877728751rm_a_b ).

thf(sy_c_Tree__Automata_Ota_Oeps_001tf__b_001tf__a,type,
    tree_eps_b_a: tree_ta_b_a > fset_P8578720700757663565od_b_b ).

thf(sy_c_Tree__Automata_Ota_Oeps_001tf__c_001tf__a,type,
    tree_eps_c_a: tree_ta_c_a > fset_P8014105740199022861od_c_c ).

thf(sy_c_Tree__Automata_Ota_Orules_001tf__c_001tf__a,type,
    tree_rules_c_a: tree_ta_c_a > fset_T7970579319148954716le_c_a ).

thf(sy_c_Tree__Automata_Ota__der_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t7856459924191630025rm_a_a: tree_t5222909361115825503rm_a_a > term_a4617543059560371146term_a > fset_Bot_bot_term_a ).

thf(sy_c_Tree__Automata_Ota__der_001t__Nat__Onat_001tf__a,type,
    tree_ta_der_nat_a: tree_ta_nat_a > term_a_nat > fset_nat ).

thf(sy_c_Tree__Automata_Ota__der_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    tree_t8390983252668960894_a_b_a: tree_ta_term_a_b_a > term_a_term_a_b > fset_term_a_b ).

thf(sy_c_Tree__Automata_Ota__der_001tf__b_001tf__a,type,
    tree_ta_der_b_a: tree_ta_b_a > term_a_b > fset_b ).

thf(sy_c_Tree__Automata_Ota__der_001tf__c_001tf__a,type,
    tree_ta_der_c_a: tree_ta_c_a > term_a_c > fset_c ).

thf(sy_c_Tree__Automata_Ota__der_H_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t5754163232548924128rm_a_a: tree_t5222909361115825503rm_a_a > term_a4617543059560371146term_a > fset_t4263150265408568880term_a ).

thf(sy_c_Tree__Automata_Ota__der_H_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    tree_t5237966800154138261_a_b_a: tree_ta_term_a_b_a > term_a_term_a_b > fset_term_a_term_a_b ).

thf(sy_c_Tree__Automata_Ota__der_H_001tf__b_001tf__a,type,
    tree_ta_der_b_a2: tree_ta_b_a > term_a_b > fset_term_a_b ).

thf(sy_c_Tree__Automata_Ota__der_H_001tf__c_001tf__a,type,
    tree_ta_der_c_a2: tree_ta_c_a > term_a_c > fset_term_a_c ).

thf(sy_c_Tree__Automata_Ota__det_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t1838897014857239623rm_a_a: tree_t5222909361115825503rm_a_a > $o ).

thf(sy_c_Tree__Automata_Ota__det_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    tree_t3137586869393622524_a_b_a: tree_ta_term_a_b_a > $o ).

thf(sy_c_Tree__Automata_Ota__det_001tf__b_001tf__a,type,
    tree_ta_det_b_a: tree_ta_b_a > $o ).

thf(sy_c_Tree__Automata_Ota__det_001tf__c_001tf__a,type,
    tree_ta_det_c_a: tree_ta_c_a > $o ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a_001tf__b,type,
    tree_t1675334710359824800_a_a_b: fset_Bot_bot_term_a > tree_t5222909361115825503rm_a_a > set_term_a_b ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a_001tf__c,type,
    tree_t1675334710359824801_a_a_c: fset_Bot_bot_term_a > tree_t5222909361115825503rm_a_a > set_term_a_c ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Nat__Onat_001tf__a_001tf__b,type,
    tree_ta_lang_nat_a_b: fset_nat > tree_ta_nat_a > set_term_a_b ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Nat__Onat_001tf__a_001tf__c,type,
    tree_ta_lang_nat_a_c: fset_nat > tree_ta_nat_a > set_term_a_c ).

thf(sy_c_Tree__Automata_Ota__lang_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a_001tf__c,type,
    tree_t726036196298098988_b_a_c: fset_term_a_b > tree_ta_term_a_b_a > set_term_a_c ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__b_001tf__a_001tf__b,type,
    tree_ta_lang_b_a_b: fset_b > tree_ta_b_a > set_term_a_b ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__b_001tf__a_001tf__c,type,
    tree_ta_lang_b_a_c: fset_b > tree_ta_b_a > set_term_a_c ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__c_001tf__a_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    tree_t1904341562405769245term_a: fset_c > tree_ta_c_a > set_te6354304872801232554term_a ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__c_001tf__a_001tf__b,type,
    tree_ta_lang_c_a_b: fset_c > tree_ta_c_a > set_term_a_b ).

thf(sy_c_Tree__Automata_Ota__lang_001tf__c_001tf__a_001tf__c,type,
    tree_ta_lang_c_a_c: fset_c > tree_ta_c_a > set_term_a_c ).

thf(sy_c_Tree__Automata_Ota__productive_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t7469959674368944611rm_a_a: fset_Bot_bot_term_a > tree_t5222909361115825503rm_a_a > fset_Bot_bot_term_a ).

thf(sy_c_Tree__Automata_Ota__productive_001tf__c_001tf__a,type,
    tree_t299928733079354318ve_c_a: fset_c > tree_ta_c_a > fset_c ).

thf(sy_c_Tree__Automata_Ota__reachable_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t3813786739961944781rm_a_a: tree_t5222909361115825503rm_a_a > fset_Bot_bot_term_a ).

thf(sy_c_Tree__Automata_Ota__reachable_001t__Term__Oterm_Itf__a_Mtf__b_J_001tf__a,type,
    tree_t1746638234261971074_a_b_a: tree_ta_term_a_b_a > fset_term_a_b ).

thf(sy_c_Tree__Automata_Ota__reachable_001tf__b_001tf__a,type,
    tree_t1956540421582741029le_b_a: tree_ta_b_a > fset_b ).

thf(sy_c_Tree__Automata_Ota__reachable_001tf__c_001tf__a,type,
    tree_t8392456876464840804le_c_a: tree_ta_c_a > fset_c ).

thf(sy_c_Tree__Automata_Ota__restrict_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t4763908311024172656rm_a_a: tree_t5222909361115825503rm_a_a > fset_Bot_bot_term_a > tree_t5222909361115825503rm_a_a ).

thf(sy_c_Tree__Automata_Ota__rule_OTA__rule_001tf__a_001tf__c,type,
    tree_ta_TA_rule_a_c: a > list_c > c > tree_ta_rule_c_a ).

thf(sy_c_Tree__Automata_Ota__sig_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t7303415228429699401rm_a_a: tree_t5222909361115825503rm_a_a > fset_P5917089267984057005_a_nat ).

thf(sy_c_Tree__Automata_Ota__subset_001t__Bot____Terms__Obot____term_Itf__a_J_001tf__a,type,
    tree_t2904061461305401546rm_a_a: tree_t5222909361115825503rm_a_a > tree_t5222909361115825503rm_a_a > $o ).

thf(sy_c_Tree__Automata_Ota__union_001tf__b_001tf__a,type,
    tree_ta_union_b_a: tree_ta_b_a > tree_ta_b_a > tree_ta_b_a ).

thf(sy_c_Tree__Automata_Ota__union_001tf__c_001tf__a,type,
    tree_ta_union_c_a: tree_ta_c_a > tree_ta_c_a > tree_ta_c_a ).

thf(sy_c_Utils_Olinear__term_001tf__a_001tf__b,type,
    linear_term_a_b: term_a_b > $o ).

thf(sy_c_Utils_Olinear__term_001tf__a_001tf__c,type,
    linear_term_a_c: term_a_c > $o ).

thf(sy_c_Utils_Otrancl__list_001t__Bot____Terms__Obot____term_Itf__a_J,type,
    trancl6444742763886629761term_a: set_Pr3965413743549523539term_a > set_Pr7374270731007213939term_a ).

thf(sy_c_member_001t__Ground____Terms__Ogterm_Itf__a_J,type,
    member5261315023172567896term_a: ground_gterm_a > set_Ground_gterm_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    member2247295376459944092term_a: produc7033602110846336755term_a > set_Pr3965413743549523539term_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_Mt__List__Olist_It__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    member4272934500791052220term_a: produc2056175833795995411term_a > set_Pr7374270731007213939term_a > $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__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_Mt__Product____Type__Oprod_It__Bot____Terms__Obot____term_Itf__a_J_Mt__Bot____Terms__Obot____term_Itf__a_J_J_J,type,
    member7618331267207336464term_a: produc441850963132823271term_a > set_Pr825864233230045127term_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
    member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__c_J_Mt__Term__Oterm_Itf__a_Mtf__c_J_J,type,
    member6607649026611388852rm_a_c: produc1095327201564732427rm_a_c > set_Pr7334014819586186091rm_a_c > $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__c_Mtf__c_J,type,
    member5074992359041316560od_c_c: product_prod_c_c > set_Product_prod_c_c > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mt__Bot____Terms__Obot____term_Itf__a_J_J,type,
    member7590767121479010419term_a: term_a4617543059560371146term_a > set_te6354304872801232554term_a > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
    member_term_a_b: term_a_b > set_term_a_b > $o ).

thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__c_J,type,
    member_term_a_c: term_a_c > set_term_a_c > $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_C,type,
    c2: subterm_and_ctxt_a_c ).

thf(sy_v_R,type,
    r: fset_term_a_b ).

thf(sy_v__092_060F_062,type,
    f: fset_P5917089267984057005_a_nat ).

thf(sy_v__092_060sigma_062,type,
    sigma: b > term_a_c ).

thf(sy_v_l,type,
    l: term_a_b ).

% Relevant facts (1278)
thf(fact_0_gr,axiom,
    term_ground_a_c @ ( subst_7999470309526761005_a_b_c @ l @ sigma ) ).

% gr
thf(fact_1_lift__total_Otrans,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,Z: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ bot_bless_eq_a )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y @ Z ) @ bot_bless_eq_a )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Z ) @ bot_bless_eq_a ) ) ) ).

% lift_total.trans
thf(fact_2_lift__total_Oanti__sym,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ bot_bless_eq_a )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y @ X ) @ bot_bless_eq_a )
       => ( X = Y ) ) ) ).

% lift_total.anti_sym
thf(fact_3_instance__to__bless__eq,axiom,
    ! [S: term_a_b,Sigma: b > term_a_b,T: term_a_b] :
      ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
        = T )
     => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a ) ) ).

% instance_to_bless_eq
thf(fact_4_instance__to__bless__eq,axiom,
    ! [S: term_a_b,Sigma: b > term_a_c,T: term_a_c] :
      ( ( ( subst_7999470309526761005_a_b_c @ S @ Sigma )
        = T )
     => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a ) ) ).

% instance_to_bless_eq
thf(fact_5_instance__to__bless__eq,axiom,
    ! [S: term_a_c,Sigma: c > term_a_b,T: term_a_b] :
      ( ( ( subst_5212014727554084971_a_c_b @ S @ Sigma )
        = T )
     => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a ) ) ).

% instance_to_bless_eq
thf(fact_6_instance__to__bless__eq,axiom,
    ! [S: term_a_c,Sigma: c > term_a_c,T: term_a_c] :
      ( ( ( subst_5212014727554084972_a_c_c @ S @ Sigma )
        = T )
     => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a ) ) ).

% instance_to_bless_eq
thf(fact_7_assms,axiom,
    fmember_term_a_b @ l @ r ).

% assms
thf(fact_8_prod_Oinject,axiom,
    ! [X1: term_a_b,X2: term_a_b,Y1: term_a_b,Y2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ X1 @ X2 )
        = ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: list_Bot_bot_term_a,X2: list_Bot_bot_term_a,Y1: list_Bot_bot_term_a,Y2: list_Bot_bot_term_a] :
      ( ( ( produc1077782494051512011term_a @ X1 @ X2 )
        = ( produc1077782494051512011term_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_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_11_prod_Oinject,axiom,
    ! [X1: c,X2: c,Y1: c,Y2: c] :
      ( ( ( product_Pair_c_c @ X1 @ X2 )
        = ( product_Pair_c_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

% prod.inject
thf(fact_13_prod_Oinject,axiom,
    ! [X1: bot_bot_term_a,X2: bot_bot_term_a,Y1: bot_bot_term_a,Y2: bot_bot_term_a] :
      ( ( ( produc780340308816795563term_a @ X1 @ X2 )
        = ( produc780340308816795563term_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,A2: list_Bot_bot_term_a,B2: list_Bot_bot_term_a] :
      ( ( ( produc1077782494051512011term_a @ A @ B )
        = ( produc1077782494051512011term_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_16_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_17_old_Oprod_Oinject,axiom,
    ! [A: c,B: c,A2: c,B2: c] :
      ( ( ( product_Pair_c_c @ A @ B )
        = ( product_Pair_c_c @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

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

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A2: bot_bot_term_a,B2: bot_bot_term_a] :
      ( ( ( produc780340308816795563term_a @ A @ B )
        = ( produc780340308816795563term_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_bless__eq__refl,axiom,
    ! [S: bot_bot_term_a] : ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ S @ S ) @ bot_bless_eq_a ) ).

% bless_eq_refl
thf(fact_21_bless__eq__trans,axiom,
    ! [S: bot_bot_term_a,T: bot_bot_term_a,U: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ S @ T ) @ bot_bless_eq_a )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ T @ U ) @ bot_bless_eq_a )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ S @ U ) @ bot_bless_eq_a ) ) ) ).

% bless_eq_trans
thf(fact_22_bless__eq__anti__sym,axiom,
    ! [S: bot_bot_term_a,T: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ S @ T ) @ bot_bless_eq_a )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ T @ S ) @ bot_bless_eq_a )
       => ( S = T ) ) ) ).

% bless_eq_anti_sym
thf(fact_23_cgr,axiom,
    term_ground_a_c @ ( subter2376574525758040791rm_a_c @ c2 @ ( subst_7999470309526761005_a_b_c @ l @ sigma ) ) ).

% cgr
thf(fact_24_bless__eq__to__instance,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a )
     => ( ( linear_term_a_b @ S )
       => ? [Sigma2: b > term_a_b] :
            ( ( subst_7999470309526761004_a_b_b @ S @ Sigma2 )
            = T ) ) ) ).

% bless_eq_to_instance
thf(fact_25_bless__eq__to__instance,axiom,
    ! [S: term_a_b,T: term_a_c] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a )
     => ( ( linear_term_a_b @ S )
       => ? [Sigma2: b > term_a_c] :
            ( ( subst_7999470309526761005_a_b_c @ S @ Sigma2 )
            = T ) ) ) ).

% bless_eq_to_instance
thf(fact_26_bless__eq__to__instance,axiom,
    ! [S: term_a_c,T: term_a_b] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a )
     => ( ( linear_term_a_c @ S )
       => ? [Sigma2: c > term_a_b] :
            ( ( subst_5212014727554084971_a_c_b @ S @ Sigma2 )
            = T ) ) ) ).

% bless_eq_to_instance
thf(fact_27_bless__eq__to__instance,axiom,
    ! [S: term_a_c,T: term_a_c] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a )
     => ( ( linear_term_a_c @ S )
       => ? [Sigma2: c > term_a_c] :
            ( ( subst_5212014727554084972_a_c_c @ S @ Sigma2 )
            = T ) ) ) ).

% bless_eq_to_instance
thf(fact_28_subst__term__eqI,axiom,
    ! [Sigma: b > term_a_c,Tau: b > term_a_c] :
      ( ! [T2: term_a_b] :
          ( ( subst_7999470309526761005_a_b_c @ T2 @ Sigma )
          = ( subst_7999470309526761005_a_b_c @ T2 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_29_subst__term__eqI,axiom,
    ! [Sigma: c > term_a_c,Tau: c > term_a_c] :
      ( ! [T2: term_a_c] :
          ( ( subst_5212014727554084972_a_c_c @ T2 @ Sigma )
          = ( subst_5212014727554084972_a_c_c @ T2 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_30_subst__term__eqI,axiom,
    ! [Sigma: c > term_a_b,Tau: c > term_a_b] :
      ( ! [T2: term_a_c] :
          ( ( subst_5212014727554084971_a_c_b @ T2 @ Sigma )
          = ( subst_5212014727554084971_a_c_b @ T2 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_31_subst__term__eqI,axiom,
    ! [Sigma: b > term_a_b,Tau: b > term_a_b] :
      ( ! [T2: term_a_b] :
          ( ( subst_7999470309526761004_a_b_b @ T2 @ Sigma )
          = ( subst_7999470309526761004_a_b_b @ T2 @ Tau ) )
     => ( Sigma = Tau ) ) ).

% subst_term_eqI
thf(fact_32_old_Oprod_Oexhaust,axiom,
    ! [Y: produc357393685978478089rm_a_b] :
      ~ ! [A3: term_a_b,B3: term_a_b] :
          ( Y
         != ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_33_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2056175833795995411term_a] :
      ~ ! [A3: list_Bot_bot_term_a,B3: list_Bot_bot_term_a] :
          ( Y
         != ( produc1077782494051512011term_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_34_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_35_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_c_c] :
      ~ ! [A3: c,B3: c] :
          ( Y
         != ( product_Pair_c_c @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_36_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_b_b] :
      ~ ! [A3: b,B3: b] :
          ( Y
         != ( product_Pair_b_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_37_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7033602110846336755term_a] :
      ~ ! [A3: bot_bot_term_a,B3: bot_bot_term_a] :
          ( Y
         != ( produc780340308816795563term_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_38_Pair__inject,axiom,
    ! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
      ( ( ( produc7020197800436672577rm_a_b @ A @ B )
        = ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_39_Pair__inject,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,A2: list_Bot_bot_term_a,B2: list_Bot_bot_term_a] :
      ( ( ( produc1077782494051512011term_a @ A @ B )
        = ( produc1077782494051512011term_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_40_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_41_Pair__inject,axiom,
    ! [A: c,B: c,A2: c,B2: c] :
      ( ( ( product_Pair_c_c @ A @ B )
        = ( product_Pair_c_c @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_42_Pair__inject,axiom,
    ! [A: b,B: b,A2: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_43_Pair__inject,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A2: bot_bot_term_a,B2: bot_bot_term_a] :
      ( ( ( produc780340308816795563term_a @ A @ B )
        = ( produc780340308816795563term_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_44_prod__cases,axiom,
    ! [P: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
      ( ! [A3: term_a_b,B3: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_45_prod__cases,axiom,
    ! [P: produc2056175833795995411term_a > $o,P2: produc2056175833795995411term_a] :
      ( ! [A3: list_Bot_bot_term_a,B3: list_Bot_bot_term_a] : ( P @ ( produc1077782494051512011term_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_46_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_47_prod__cases,axiom,
    ! [P: product_prod_c_c > $o,P2: product_prod_c_c] :
      ( ! [A3: c,B3: c] : ( P @ ( product_Pair_c_c @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_48_prod__cases,axiom,
    ! [P: product_prod_b_b > $o,P2: product_prod_b_b] :
      ( ! [A3: b,B3: b] : ( P @ ( product_Pair_b_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_49_prod__cases,axiom,
    ! [P: produc7033602110846336755term_a > $o,P2: produc7033602110846336755term_a] :
      ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a] : ( P @ ( produc780340308816795563term_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_50_surj__pair,axiom,
    ! [P2: produc357393685978478089rm_a_b] :
    ? [X3: term_a_b,Y3: term_a_b] :
      ( P2
      = ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_51_surj__pair,axiom,
    ! [P2: produc2056175833795995411term_a] :
    ? [X3: list_Bot_bot_term_a,Y3: list_Bot_bot_term_a] :
      ( P2
      = ( produc1077782494051512011term_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_52_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_53_surj__pair,axiom,
    ! [P2: product_prod_c_c] :
    ? [X3: c,Y3: c] :
      ( P2
      = ( product_Pair_c_c @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_54_surj__pair,axiom,
    ! [P2: product_prod_b_b] :
    ? [X3: b,Y3: b] :
      ( P2
      = ( product_Pair_b_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_55_surj__pair,axiom,
    ! [P2: produc7033602110846336755term_a] :
    ? [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
      ( P2
      = ( produc780340308816795563term_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_56__C_K_C,axiom,
    member_term_a_c @ ( subter2376574525758040791rm_a_c @ c2 @ ( subst_7999470309526761005_a_b_c @ l @ sigma ) ) @ ( tree_t1675334710359824801_a_a_c @ ( fstates_a_b @ r ) @ ( nf_ta_a_b @ r @ f ) ) ).

% "*"
thf(fact_57_ctxt__eq,axiom,
    ! [C: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( ( subter2376574525758040790rm_a_b @ C @ S )
        = ( subter2376574525758040790rm_a_b @ C @ T ) )
      = ( S = T ) ) ).

% ctxt_eq
thf(fact_58_ctxt__eq,axiom,
    ! [C: subterm_and_ctxt_a_c,S: term_a_c,T: term_a_c] :
      ( ( ( subter2376574525758040791rm_a_c @ C @ S )
        = ( subter2376574525758040791rm_a_c @ C @ T ) )
      = ( S = T ) ) ).

% ctxt_eq
thf(fact_59_fset__eqI,axiom,
    ! [A4: fset_term_a_c,B4: fset_term_a_c] :
      ( ! [X3: term_a_c] :
          ( ( fmember_term_a_c @ X3 @ A4 )
          = ( fmember_term_a_c @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_60_fset__eqI,axiom,
    ! [A4: fset_fset_term_a_b,B4: fset_fset_term_a_b] :
      ( ! [X3: fset_term_a_b] :
          ( ( fmembe3754333024224722621rm_a_b @ X3 @ A4 )
          = ( fmembe3754333024224722621rm_a_b @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_61_fset__eqI,axiom,
    ! [A4: fset_nat,B4: fset_nat] :
      ( ! [X3: nat] :
          ( ( fmember_nat @ X3 @ A4 )
          = ( fmember_nat @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_62_fset__eqI,axiom,
    ! [A4: fset_c,B4: fset_c] :
      ( ! [X3: c] :
          ( ( fmember_c @ X3 @ A4 )
          = ( fmember_c @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_63_fset__eqI,axiom,
    ! [A4: fset_b,B4: fset_b] :
      ( ! [X3: b] :
          ( ( fmember_b @ X3 @ A4 )
          = ( fmember_b @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_64_fset__eqI,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ! [X3: term_a_b] :
          ( ( fmember_term_a_b @ X3 @ A4 )
          = ( fmember_term_a_b @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_65_fequalityCE,axiom,
    ! [A4: fset_term_a_c,B4: fset_term_a_c,C2: term_a_c] :
      ( ( A4 = B4 )
     => ( ( ( fmember_term_a_c @ C2 @ A4 )
         => ~ ( fmember_term_a_c @ C2 @ B4 ) )
       => ~ ( ~ ( fmember_term_a_c @ C2 @ A4 )
           => ( fmember_term_a_c @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_66_fequalityCE,axiom,
    ! [A4: fset_fset_term_a_b,B4: fset_fset_term_a_b,C2: fset_term_a_b] :
      ( ( A4 = B4 )
     => ( ( ( fmembe3754333024224722621rm_a_b @ C2 @ A4 )
         => ~ ( fmembe3754333024224722621rm_a_b @ C2 @ B4 ) )
       => ~ ( ~ ( fmembe3754333024224722621rm_a_b @ C2 @ A4 )
           => ( fmembe3754333024224722621rm_a_b @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_67_fequalityCE,axiom,
    ! [A4: fset_nat,B4: fset_nat,C2: nat] :
      ( ( A4 = B4 )
     => ( ( ( fmember_nat @ C2 @ A4 )
         => ~ ( fmember_nat @ C2 @ B4 ) )
       => ~ ( ~ ( fmember_nat @ C2 @ A4 )
           => ( fmember_nat @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_68_fequalityCE,axiom,
    ! [A4: fset_c,B4: fset_c,C2: c] :
      ( ( A4 = B4 )
     => ( ( ( fmember_c @ C2 @ A4 )
         => ~ ( fmember_c @ C2 @ B4 ) )
       => ~ ( ~ ( fmember_c @ C2 @ A4 )
           => ( fmember_c @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_69_fequalityCE,axiom,
    ! [A4: fset_b,B4: fset_b,C2: b] :
      ( ( A4 = B4 )
     => ( ( ( fmember_b @ C2 @ A4 )
         => ~ ( fmember_b @ C2 @ B4 ) )
       => ~ ( ~ ( fmember_b @ C2 @ A4 )
           => ( fmember_b @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_70_fequalityCE,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b,C2: term_a_b] :
      ( ( A4 = B4 )
     => ( ( ( fmember_term_a_b @ C2 @ A4 )
         => ~ ( fmember_term_a_b @ C2 @ B4 ) )
       => ~ ( ~ ( fmember_term_a_b @ C2 @ A4 )
           => ( fmember_term_a_b @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_71_eq__fmem__trans,axiom,
    ! [A: term_a_c,B: term_a_c,A4: fset_term_a_c] :
      ( ( A = B )
     => ( ( fmember_term_a_c @ B @ A4 )
       => ( fmember_term_a_c @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_72_eq__fmem__trans,axiom,
    ! [A: fset_term_a_b,B: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( A = B )
     => ( ( fmembe3754333024224722621rm_a_b @ B @ A4 )
       => ( fmembe3754333024224722621rm_a_b @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_73_eq__fmem__trans,axiom,
    ! [A: nat,B: nat,A4: fset_nat] :
      ( ( A = B )
     => ( ( fmember_nat @ B @ A4 )
       => ( fmember_nat @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_74_eq__fmem__trans,axiom,
    ! [A: c,B: c,A4: fset_c] :
      ( ( A = B )
     => ( ( fmember_c @ B @ A4 )
       => ( fmember_c @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_75_eq__fmem__trans,axiom,
    ! [A: b,B: b,A4: fset_b] :
      ( ( A = B )
     => ( ( fmember_b @ B @ A4 )
       => ( fmember_b @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_76_eq__fmem__trans,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b] :
      ( ( A = B )
     => ( ( fmember_term_a_b @ B @ A4 )
       => ( fmember_term_a_b @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_77_eqfset__imp__iff,axiom,
    ! [A4: fset_term_a_c,B4: fset_term_a_c,X: term_a_c] :
      ( ( A4 = B4 )
     => ( ( fmember_term_a_c @ X @ A4 )
        = ( fmember_term_a_c @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_78_eqfset__imp__iff,axiom,
    ! [A4: fset_fset_term_a_b,B4: fset_fset_term_a_b,X: fset_term_a_b] :
      ( ( A4 = B4 )
     => ( ( fmembe3754333024224722621rm_a_b @ X @ A4 )
        = ( fmembe3754333024224722621rm_a_b @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_79_eqfset__imp__iff,axiom,
    ! [A4: fset_nat,B4: fset_nat,X: nat] :
      ( ( A4 = B4 )
     => ( ( fmember_nat @ X @ A4 )
        = ( fmember_nat @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_80_eqfset__imp__iff,axiom,
    ! [A4: fset_c,B4: fset_c,X: c] :
      ( ( A4 = B4 )
     => ( ( fmember_c @ X @ A4 )
        = ( fmember_c @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_81_eqfset__imp__iff,axiom,
    ! [A4: fset_b,B4: fset_b,X: b] :
      ( ( A4 = B4 )
     => ( ( fmember_b @ X @ A4 )
        = ( fmember_b @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_82_eqfset__imp__iff,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b,X: term_a_b] :
      ( ( A4 = B4 )
     => ( ( fmember_term_a_b @ X @ A4 )
        = ( fmember_term_a_b @ X @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_83_if__split__fmem1,axiom,
    ! [Q: $o,X: term_a_c,Y: term_a_c,B: fset_term_a_c] :
      ( ( fmember_term_a_c @ ( if_term_a_c @ Q @ X @ Y ) @ B )
      = ( ( Q
         => ( fmember_term_a_c @ X @ B ) )
        & ( ~ Q
         => ( fmember_term_a_c @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_84_if__split__fmem1,axiom,
    ! [Q: $o,X: fset_term_a_b,Y: fset_term_a_b,B: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ ( if_fset_term_a_b @ Q @ X @ Y ) @ B )
      = ( ( Q
         => ( fmembe3754333024224722621rm_a_b @ X @ B ) )
        & ( ~ Q
         => ( fmembe3754333024224722621rm_a_b @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_85_if__split__fmem1,axiom,
    ! [Q: $o,X: nat,Y: nat,B: fset_nat] :
      ( ( fmember_nat @ ( if_nat @ Q @ X @ Y ) @ B )
      = ( ( Q
         => ( fmember_nat @ X @ B ) )
        & ( ~ Q
         => ( fmember_nat @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_86_if__split__fmem1,axiom,
    ! [Q: $o,X: c,Y: c,B: fset_c] :
      ( ( fmember_c @ ( if_c @ Q @ X @ Y ) @ B )
      = ( ( Q
         => ( fmember_c @ X @ B ) )
        & ( ~ Q
         => ( fmember_c @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_87_if__split__fmem1,axiom,
    ! [Q: $o,X: b,Y: b,B: fset_b] :
      ( ( fmember_b @ ( if_b @ Q @ X @ Y ) @ B )
      = ( ( Q
         => ( fmember_b @ X @ B ) )
        & ( ~ Q
         => ( fmember_b @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_88_if__split__fmem1,axiom,
    ! [Q: $o,X: term_a_b,Y: term_a_b,B: fset_term_a_b] :
      ( ( fmember_term_a_b @ ( if_term_a_b @ Q @ X @ Y ) @ B )
      = ( ( Q
         => ( fmember_term_a_b @ X @ B ) )
        & ( ~ Q
         => ( fmember_term_a_b @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_89_if__split__fmem2,axiom,
    ! [A: term_a_c,Q: $o,X: fset_term_a_c,Y: fset_term_a_c] :
      ( ( fmember_term_a_c @ A @ ( if_fset_term_a_c @ Q @ X @ Y ) )
      = ( ( Q
         => ( fmember_term_a_c @ A @ X ) )
        & ( ~ Q
         => ( fmember_term_a_c @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_90_if__split__fmem2,axiom,
    ! [A: fset_term_a_b,Q: $o,X: fset_fset_term_a_b,Y: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ ( if_fse3313186383936193710rm_a_b @ Q @ X @ Y ) )
      = ( ( Q
         => ( fmembe3754333024224722621rm_a_b @ A @ X ) )
        & ( ~ Q
         => ( fmembe3754333024224722621rm_a_b @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_91_if__split__fmem2,axiom,
    ! [A: nat,Q: $o,X: fset_nat,Y: fset_nat] :
      ( ( fmember_nat @ A @ ( if_fset_nat @ Q @ X @ Y ) )
      = ( ( Q
         => ( fmember_nat @ A @ X ) )
        & ( ~ Q
         => ( fmember_nat @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_92_if__split__fmem2,axiom,
    ! [A: c,Q: $o,X: fset_c,Y: fset_c] :
      ( ( fmember_c @ A @ ( if_fset_c @ Q @ X @ Y ) )
      = ( ( Q
         => ( fmember_c @ A @ X ) )
        & ( ~ Q
         => ( fmember_c @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_93_if__split__fmem2,axiom,
    ! [A: b,Q: $o,X: fset_b,Y: fset_b] :
      ( ( fmember_b @ A @ ( if_fset_b @ Q @ X @ Y ) )
      = ( ( Q
         => ( fmember_b @ A @ X ) )
        & ( ~ Q
         => ( fmember_b @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_94_if__split__fmem2,axiom,
    ! [A: term_a_b,Q: $o,X: fset_term_a_b,Y: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ ( if_fset_term_a_b @ Q @ X @ Y ) )
      = ( ( Q
         => ( fmember_term_a_b @ A @ X ) )
        & ( ~ Q
         => ( fmember_term_a_b @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_95_eqfelem__imp__iff,axiom,
    ! [X: term_a_c,Y: term_a_c,A4: fset_term_a_c] :
      ( ( X = Y )
     => ( ( fmember_term_a_c @ X @ A4 )
        = ( fmember_term_a_c @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_96_eqfelem__imp__iff,axiom,
    ! [X: fset_term_a_b,Y: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( X = Y )
     => ( ( fmembe3754333024224722621rm_a_b @ X @ A4 )
        = ( fmembe3754333024224722621rm_a_b @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_97_eqfelem__imp__iff,axiom,
    ! [X: nat,Y: nat,A4: fset_nat] :
      ( ( X = Y )
     => ( ( fmember_nat @ X @ A4 )
        = ( fmember_nat @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_98_eqfelem__imp__iff,axiom,
    ! [X: c,Y: c,A4: fset_c] :
      ( ( X = Y )
     => ( ( fmember_c @ X @ A4 )
        = ( fmember_c @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_99_eqfelem__imp__iff,axiom,
    ! [X: b,Y: b,A4: fset_b] :
      ( ( X = Y )
     => ( ( fmember_b @ X @ A4 )
        = ( fmember_b @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_100_eqfelem__imp__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,A4: fset_term_a_b] :
      ( ( X = Y )
     => ( ( fmember_term_a_b @ X @ A4 )
        = ( fmember_term_a_b @ Y @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_101_subst__apply__term__ctxt__apply__distrib,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b,Mu: b > term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ ( subter2376574525758040790rm_a_b @ C @ T ) @ Mu )
      = ( subter2376574525758040790rm_a_b @ ( subter4086027020339385547_a_b_b @ C @ Mu ) @ ( subst_7999470309526761004_a_b_b @ T @ Mu ) ) ) ).

% subst_apply_term_ctxt_apply_distrib
thf(fact_102_subst__apply__term__ctxt__apply__distrib,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b,Mu: b > term_a_c] :
      ( ( subst_7999470309526761005_a_b_c @ ( subter2376574525758040790rm_a_b @ C @ T ) @ Mu )
      = ( subter2376574525758040791rm_a_c @ ( subter4086027020339385548_a_b_c @ C @ Mu ) @ ( subst_7999470309526761005_a_b_c @ T @ Mu ) ) ) ).

% subst_apply_term_ctxt_apply_distrib
thf(fact_103_subst__apply__term__ctxt__apply__distrib,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c,Mu: c > term_a_b] :
      ( ( subst_5212014727554084971_a_c_b @ ( subter2376574525758040791rm_a_c @ C @ T ) @ Mu )
      = ( subter2376574525758040790rm_a_b @ ( subter1298571438366709514_a_c_b @ C @ Mu ) @ ( subst_5212014727554084971_a_c_b @ T @ Mu ) ) ) ).

% subst_apply_term_ctxt_apply_distrib
thf(fact_104_subst__apply__term__ctxt__apply__distrib,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c,Mu: c > term_a_c] :
      ( ( subst_5212014727554084972_a_c_c @ ( subter2376574525758040791rm_a_c @ C @ T ) @ Mu )
      = ( subter2376574525758040791rm_a_c @ ( subter1298571438366709515_a_c_c @ C @ Mu ) @ ( subst_5212014727554084972_a_c_c @ T @ Mu ) ) ) ).

% subst_apply_term_ctxt_apply_distrib
thf(fact_105_ssubst__Pair__rhs,axiom,
    ! [R: produc7033602110846336755term_a,S: produc7033602110846336755term_a,R2: set_Pr825864233230045127term_a,S2: produc7033602110846336755term_a] :
      ( ( member7618331267207336464term_a @ ( produc1145950395756198999term_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7618331267207336464term_a @ ( produc1145950395756198999term_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_106_ssubst__Pair__rhs,axiom,
    ! [R: term_a_c,S: term_a_c,R2: set_Pr7334014819586186091rm_a_c,S2: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_107_ssubst__Pair__rhs,axiom,
    ! [R: term_a_b,S: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_108_ssubst__Pair__rhs,axiom,
    ! [R: list_Bot_bot_term_a,S: list_Bot_bot_term_a,R2: set_Pr7374270731007213939term_a,S2: list_Bot_bot_term_a] :
      ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_109_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_110_ssubst__Pair__rhs,axiom,
    ! [R: c,S: c,R2: set_Product_prod_c_c,S2: c] :
      ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_111_ssubst__Pair__rhs,axiom,
    ! [R: b,S: b,R2: set_Product_prod_b_b,S2: b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_112_ssubst__Pair__rhs,axiom,
    ! [R: bot_bot_term_a,S: bot_bot_term_a,R2: set_Pr3965413743549523539term_a,S2: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_113_curry__conv,axiom,
    ( produc4484525301753677177rm_a_o
    = ( ^ [F: produc7033602110846336755term_a > $o,A5: bot_bot_term_a,B5: bot_bot_term_a] : ( F @ ( produc780340308816795563term_a @ A5 @ B5 ) ) ) ) ).

% curry_conv
thf(fact_114_subst__apply__left__idemp,axiom,
    ! [Sigma: b > term_a_c,X: b,T: term_a_b,S: term_a_b] :
      ( ( ( Sigma @ X )
        = ( subst_7999470309526761005_a_b_c @ T @ Sigma ) )
     => ( ( subst_7999470309526761005_a_b_c @ ( subst_7999470309526761004_a_b_b @ S @ ( subst_b_a @ X @ T ) ) @ Sigma )
        = ( subst_7999470309526761005_a_b_c @ S @ Sigma ) ) ) ).

% subst_apply_left_idemp
thf(fact_115_subst__apply__left__idemp,axiom,
    ! [Sigma: c > term_a_c,X: c,T: term_a_c,S: term_a_c] :
      ( ( ( Sigma @ X )
        = ( subst_5212014727554084972_a_c_c @ T @ Sigma ) )
     => ( ( subst_5212014727554084972_a_c_c @ ( subst_5212014727554084972_a_c_c @ S @ ( subst_c_a @ X @ T ) ) @ Sigma )
        = ( subst_5212014727554084972_a_c_c @ S @ Sigma ) ) ) ).

% subst_apply_left_idemp
thf(fact_116_subst__apply__left__idemp,axiom,
    ! [Sigma: c > term_a_b,X: c,T: term_a_c,S: term_a_c] :
      ( ( ( Sigma @ X )
        = ( subst_5212014727554084971_a_c_b @ T @ Sigma ) )
     => ( ( subst_5212014727554084971_a_c_b @ ( subst_5212014727554084972_a_c_c @ S @ ( subst_c_a @ X @ T ) ) @ Sigma )
        = ( subst_5212014727554084971_a_c_b @ S @ Sigma ) ) ) ).

% subst_apply_left_idemp
thf(fact_117_subst__apply__left__idemp,axiom,
    ! [Sigma: b > term_a_b,X: b,T: term_a_b,S: term_a_b] :
      ( ( ( Sigma @ X )
        = ( subst_7999470309526761004_a_b_b @ T @ Sigma ) )
     => ( ( subst_7999470309526761004_a_b_b @ ( subst_7999470309526761004_a_b_b @ S @ ( subst_b_a @ X @ T ) ) @ Sigma )
        = ( subst_7999470309526761004_a_b_b @ S @ Sigma ) ) ) ).

% subst_apply_left_idemp
thf(fact_118_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat,T: term_a4617543059560371146term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) ) )
     => ( ( term_g4884896889520838057term_a @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te359145989583603559term_a @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_119_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat,T: term_a4617543059560371146term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) ) )
     => ( ( term_g4884896889520838057term_a @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te359145989583603559term_a @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_120_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat,T: term_a_term_a_b] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) ) )
     => ( ( term_g2255532262831151796rm_a_b @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te522286312025294962rm_a_b @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_121_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat,T: term_a_term_a_b] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) ) )
     => ( ( term_g2255532262831151796rm_a_b @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te522286312025294962rm_a_b @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_122_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat,T: term_a_b] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) ) )
     => ( ( term_ground_a_b @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_123_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat,T: term_a_c] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) ) )
     => ( ( term_ground_a_c @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_124_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat,T: term_a_b] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) ) )
     => ( ( term_ground_a_b @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_125_term__instance__of__reach__state,axiom,
    ! [Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat,T: term_a_c] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) ) )
     => ( ( term_ground_a_c @ T )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Q2 @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a ) ) ) ).

% term_instance_of_reach_state
thf(fact_126_mem__Collect__eq,axiom,
    ! [A: produc441850963132823271term_a,P: produc441850963132823271term_a > $o] :
      ( ( member7618331267207336464term_a @ A @ ( collec7763623725074747986term_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
    ! [A: produc1095327201564732427rm_a_c,P: produc1095327201564732427rm_a_c > $o] :
      ( ( member6607649026611388852rm_a_c @ A @ ( collec837110911512178422rm_a_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
    ! [A: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
      ( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
    ! [A: produc2056175833795995411term_a,P: produc2056175833795995411term_a > $o] :
      ( ( member4272934500791052220term_a @ A @ ( collec2650751154016459518term_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
    ! [A: term_a_b,P: term_a_b > $o] :
      ( ( member_term_a_b @ A @ ( collect_term_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
    ! [A: produc7033602110846336755term_a,P: produc7033602110846336755term_a > $o] :
      ( ( member2247295376459944092term_a @ A @ ( collec7536399311987840478term_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
    ! [A: term_a_c,P: term_a_c > $o] :
      ( ( member_term_a_c @ A @ ( collect_term_a_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_133_Collect__mem__eq,axiom,
    ! [A4: set_Pr825864233230045127term_a] :
      ( ( collec7763623725074747986term_a
        @ ^ [X4: produc441850963132823271term_a] : ( member7618331267207336464term_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
    ! [A4: set_Pr7334014819586186091rm_a_c] :
      ( ( collec837110911512178422rm_a_c
        @ ^ [X4: produc1095327201564732427rm_a_c] : ( member6607649026611388852rm_a_c @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( collec99177395925924084rm_a_b
        @ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
    ! [A4: set_Pr7374270731007213939term_a] :
      ( ( collec2650751154016459518term_a
        @ ^ [X4: produc2056175833795995411term_a] : ( member4272934500791052220term_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
    ! [A4: set_term_a_b] :
      ( ( collect_term_a_b
        @ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
    ! [A4: set_Pr3965413743549523539term_a] :
      ( ( collec7536399311987840478term_a
        @ ^ [X4: produc7033602110846336755term_a] : ( member2247295376459944092term_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
    ! [A4: set_term_a_c] :
      ( ( collect_term_a_c
        @ ^ [X4: term_a_c] : ( member_term_a_c @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_140_Collect__cong,axiom,
    ! [P: term_a_c > $o,Q: term_a_c > $o] :
      ( ! [X3: term_a_c] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_term_a_c @ P )
        = ( collect_term_a_c @ Q ) ) ) ).

% Collect_cong
thf(fact_141_Collect__cong,axiom,
    ! [P: produc7033602110846336755term_a > $o,Q: produc7033602110846336755term_a > $o] :
      ( ! [X3: produc7033602110846336755term_a] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec7536399311987840478term_a @ P )
        = ( collec7536399311987840478term_a @ Q ) ) ) ).

% Collect_cong
thf(fact_142_nf__ta__det,axiom,
    ! [R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] : ( tree_t1838897014857239623rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) ) ).

% nf_ta_det
thf(fact_143_nf__ta__det,axiom,
    ! [R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] : ( tree_t1838897014857239623rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) ) ).

% nf_ta_det
thf(fact_144_nf__reg__def,axiom,
    ( nf_reg_a_c
    = ( ^ [R3: fset_term_a_c,F3: fset_P5917089267984057005_a_nat] : ( tree_R2766333946944686633rm_a_a @ ( fstates_a_c @ R3 ) @ ( nf_ta_a_c @ R3 @ F3 ) ) ) ) ).

% nf_reg_def
thf(fact_145_nf__reg__def,axiom,
    ( nf_reg_a_b
    = ( ^ [R3: fset_term_a_b,F3: fset_P5917089267984057005_a_nat] : ( tree_R2766333946944686633rm_a_a @ ( fstates_a_b @ R3 ) @ ( nf_ta_a_b @ R3 @ F3 ) ) ) ) ).

% nf_reg_def
thf(fact_146_curryI,axiom,
    ! [F4: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
      ( ( F4 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
     => ( produc6008550485034061667_a_b_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_147_curryI,axiom,
    ! [F4: produc2056175833795995411term_a > $o,A: list_Bot_bot_term_a,B: list_Bot_bot_term_a] :
      ( ( F4 @ ( produc1077782494051512011term_a @ A @ B ) )
     => ( produc3073380265971003737rm_a_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_148_curryI,axiom,
    ! [F4: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( F4 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( produc1310100445399344235_nat_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_149_curryI,axiom,
    ! [F4: product_prod_c_c > $o,A: c,B: c] :
      ( ( F4 @ ( product_Pair_c_c @ A @ B ) )
     => ( product_curry_c_c_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_150_curryI,axiom,
    ! [F4: product_prod_b_b > $o,A: b,B: b] :
      ( ( F4 @ ( product_Pair_b_b @ A @ B ) )
     => ( product_curry_b_b_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_151_curryI,axiom,
    ! [F4: produc7033602110846336755term_a > $o,A: bot_bot_term_a,B: bot_bot_term_a] :
      ( ( F4 @ ( produc780340308816795563term_a @ A @ B ) )
     => ( produc4484525301753677177rm_a_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_152_internal__case__prod__def,axiom,
    produc7561195436278546913rm_a_c = produc5909685650505657099rm_a_c ).

% internal_case_prod_def
thf(fact_153_internal__case__prod__def,axiom,
    produc512496843244024125term_a = produc2700287094330126483term_a ).

% internal_case_prod_def
thf(fact_154_internal__case__prod__def,axiom,
    produc5755667415957967490rm_a_o = produc480846774855951832rm_a_o ).

% internal_case_prod_def
thf(fact_155_internal__case__prod__def,axiom,
    produc1035157467515367901term_a = produc1985598078976903475term_a ).

% internal_case_prod_def
thf(fact_156_fId__onI,axiom,
    ! [A: term_a_c,A4: fset_term_a_c] :
      ( ( fmember_term_a_c @ A @ A4 )
     => ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ A ) @ ( fSet_fId_on_term_a_c @ A4 ) ) ) ).

% fId_onI
thf(fact_157_fId__onI,axiom,
    ! [A: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ A4 )
     => ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ A @ A ) @ ( fSet_f2178757263449557692rm_a_b @ A4 ) ) ) ).

% fId_onI
thf(fact_158_fId__onI,axiom,
    ! [A: list_Bot_bot_term_a,A4: fset_l235587408706331465term_a] :
      ( ( fmembe1891194496946673298term_a @ A @ A4 )
     => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ A ) @ ( fSet_f2002284675934200977term_a @ A4 ) ) ) ).

% fId_onI
thf(fact_159_fId__onI,axiom,
    ! [A: nat,A4: fset_nat] :
      ( ( fmember_nat @ A @ A4 )
     => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( fSet_fId_on_nat @ A4 ) ) ) ).

% fId_onI
thf(fact_160_fId__onI,axiom,
    ! [A: c,A4: fset_c] :
      ( ( fmember_c @ A @ A4 )
     => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ A ) @ ( fSet_fId_on_c @ A4 ) ) ) ).

% fId_onI
thf(fact_161_fId__onI,axiom,
    ! [A: b,A4: fset_b] :
      ( ( fmember_b @ A @ A4 )
     => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ A ) @ ( fSet_fId_on_b @ A4 ) ) ) ).

% fId_onI
thf(fact_162_fId__onI,axiom,
    ! [A: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( fSet_fId_on_term_a_b @ A4 ) ) ) ).

% fId_onI
thf(fact_163_fId__onI,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ A4 )
     => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ A ) @ ( fSet_f5231733412705268353term_a @ A4 ) ) ) ).

% fId_onI
thf(fact_164_subst__simps_I1_J,axiom,
    ! [X: b,T: term_a_b] :
      ( ( subst_b_a @ X @ T @ X )
      = T ) ).

% subst_simps(1)
thf(fact_165_subst__simps_I1_J,axiom,
    ! [X: c,T: term_a_c] :
      ( ( subst_c_a @ X @ T @ X )
      = T ) ).

% subst_simps(1)
thf(fact_166_curry__case__prod,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > set_term_a_c] :
      ( ( produc5035402668397846058rm_a_c @ ( produc5909685650505657099rm_a_c @ F4 ) )
      = F4 ) ).

% curry_case_prod
thf(fact_167_curry__case__prod,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a] :
      ( ( produc2116068065703686708term_a @ ( produc2700287094330126483term_a @ F4 ) )
      = F4 ) ).

% curry_case_prod
thf(fact_168_curry__case__prod,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > $o] :
      ( ( produc4484525301753677177rm_a_o @ ( produc480846774855951832rm_a_o @ F4 ) )
      = F4 ) ).

% curry_case_prod
thf(fact_169_curry__case__prod,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a] :
      ( ( produc7987411809206226644term_a @ ( produc1985598078976903475term_a @ F4 ) )
      = F4 ) ).

% curry_case_prod
thf(fact_170_case__prod__curry,axiom,
    ! [F4: produc7033602110846336755term_a > set_term_a_c] :
      ( ( produc5909685650505657099rm_a_c @ ( produc5035402668397846058rm_a_c @ F4 ) )
      = F4 ) ).

% case_prod_curry
thf(fact_171_case__prod__curry,axiom,
    ! [F4: produc7033602110846336755term_a > set_Pr3965413743549523539term_a] :
      ( ( produc2700287094330126483term_a @ ( produc2116068065703686708term_a @ F4 ) )
      = F4 ) ).

% case_prod_curry
thf(fact_172_case__prod__curry,axiom,
    ! [F4: produc7033602110846336755term_a > $o] :
      ( ( produc480846774855951832rm_a_o @ ( produc4484525301753677177rm_a_o @ F4 ) )
      = F4 ) ).

% case_prod_curry
thf(fact_173_case__prod__curry,axiom,
    ! [F4: produc7033602110846336755term_a > produc7033602110846336755term_a] :
      ( ( produc1985598078976903475term_a @ ( produc7987411809206226644term_a @ F4 ) )
      = F4 ) ).

% case_prod_curry
thf(fact_174_mem__case__prodE,axiom,
    ! [Z: term_a_c,C2: bot_bot_term_a > bot_bot_term_a > set_term_a_c,P2: produc7033602110846336755term_a] :
      ( ( member_term_a_c @ Z @ ( produc5909685650505657099rm_a_c @ C2 @ P2 ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( P2
              = ( produc780340308816795563term_a @ X3 @ Y3 ) )
           => ~ ( member_term_a_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_175_mem__case__prodE,axiom,
    ! [Z: produc7033602110846336755term_a,C2: bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a,P2: produc7033602110846336755term_a] :
      ( ( member2247295376459944092term_a @ Z @ ( produc2700287094330126483term_a @ C2 @ P2 ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( P2
              = ( produc780340308816795563term_a @ X3 @ Y3 ) )
           => ~ ( member2247295376459944092term_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_176_mem__case__prodE,axiom,
    ! [Z: term_a_c,C2: nat > nat > set_term_a_c,P2: product_prod_nat_nat] :
      ( ( member_term_a_c @ Z @ ( produc1785222659793412761rm_a_c @ C2 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_term_a_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_177_mem__case__prodE,axiom,
    ! [Z: term_a_b,C2: nat > nat > set_term_a_b,P2: product_prod_nat_nat] :
      ( ( member_term_a_b @ Z @ ( produc1714188619747066776rm_a_b @ C2 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_term_a_b @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_178_mem__case__prodE,axiom,
    ! [Z: term_a_c,C2: c > c > set_term_a_c,P2: product_prod_c_c] :
      ( ( member_term_a_c @ Z @ ( produc8768846039773432567rm_a_c @ C2 @ P2 ) )
     => ~ ! [X3: c,Y3: c] :
            ( ( P2
              = ( product_Pair_c_c @ X3 @ Y3 ) )
           => ~ ( member_term_a_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_179_mem__case__prodE,axiom,
    ! [Z: term_a_b,C2: c > c > set_term_a_b,P2: product_prod_c_c] :
      ( ( member_term_a_b @ Z @ ( produc8697811999727086582rm_a_b @ C2 @ P2 ) )
     => ~ ! [X3: c,Y3: c] :
            ( ( P2
              = ( product_Pair_c_c @ X3 @ Y3 ) )
           => ~ ( member_term_a_b @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_180_mem__case__prodE,axiom,
    ! [Z: term_a_c,C2: b > b > set_term_a_c,P2: product_prod_b_b] :
      ( ( member_term_a_c @ Z @ ( produc8757669699286378679rm_a_c @ C2 @ P2 ) )
     => ~ ! [X3: b,Y3: b] :
            ( ( P2
              = ( product_Pair_b_b @ X3 @ Y3 ) )
           => ~ ( member_term_a_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_181_mem__case__prodE,axiom,
    ! [Z: term_a_b,C2: b > b > set_term_a_b,P2: product_prod_b_b] :
      ( ( member_term_a_b @ Z @ ( produc8686635659240032694rm_a_b @ C2 @ P2 ) )
     => ~ ! [X3: b,Y3: b] :
            ( ( P2
              = ( product_Pair_b_b @ X3 @ Y3 ) )
           => ~ ( member_term_a_b @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_182_mem__case__prodE,axiom,
    ! [Z: term_a_b,C2: bot_bot_term_a > bot_bot_term_a > set_term_a_b,P2: produc7033602110846336755term_a] :
      ( ( member_term_a_b @ Z @ ( produc5838651610459311114rm_a_b @ C2 @ P2 ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( P2
              = ( produc780340308816795563term_a @ X3 @ Y3 ) )
           => ~ ( member_term_a_b @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_183_mem__case__prodE,axiom,
    ! [Z: produc7033602110846336755term_a,C2: nat > nat > set_Pr3965413743549523539term_a,P2: product_prod_nat_nat] :
      ( ( member2247295376459944092term_a @ Z @ ( produc2411476669527729797term_a @ C2 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member2247295376459944092term_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_184_old_Oprod_Ocase,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > set_term_a_c,X1: bot_bot_term_a,X2: bot_bot_term_a] :
      ( ( produc5909685650505657099rm_a_c @ F4 @ ( produc780340308816795563term_a @ X1 @ X2 ) )
      = ( F4 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_185_old_Oprod_Ocase,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a,X1: bot_bot_term_a,X2: bot_bot_term_a] :
      ( ( produc2700287094330126483term_a @ F4 @ ( produc780340308816795563term_a @ X1 @ X2 ) )
      = ( F4 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_186_old_Oprod_Ocase,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > $o,X1: bot_bot_term_a,X2: bot_bot_term_a] :
      ( ( produc480846774855951832rm_a_o @ F4 @ ( produc780340308816795563term_a @ X1 @ X2 ) )
      = ( F4 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_187_old_Oprod_Ocase,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a,X1: bot_bot_term_a,X2: bot_bot_term_a] :
      ( ( produc1985598078976903475term_a @ F4 @ ( produc780340308816795563term_a @ X1 @ X2 ) )
      = ( F4 @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_188_curryE,axiom,
    ! [F4: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
      ( ( produc6008550485034061667_a_b_o @ F4 @ A @ B )
     => ( F4 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_189_curryE,axiom,
    ! [F4: produc2056175833795995411term_a > $o,A: list_Bot_bot_term_a,B: list_Bot_bot_term_a] :
      ( ( produc3073380265971003737rm_a_o @ F4 @ A @ B )
     => ( F4 @ ( produc1077782494051512011term_a @ A @ B ) ) ) ).

% curryE
thf(fact_190_curryE,axiom,
    ! [F4: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryE
thf(fact_191_curryE,axiom,
    ! [F4: product_prod_c_c > $o,A: c,B: c] :
      ( ( product_curry_c_c_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_c_c @ A @ B ) ) ) ).

% curryE
thf(fact_192_curryE,axiom,
    ! [F4: product_prod_b_b > $o,A: b,B: b] :
      ( ( product_curry_b_b_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_b_b @ A @ B ) ) ) ).

% curryE
thf(fact_193_curryE,axiom,
    ! [F4: produc7033602110846336755term_a > $o,A: bot_bot_term_a,B: bot_bot_term_a] :
      ( ( produc4484525301753677177rm_a_o @ F4 @ A @ B )
     => ( F4 @ ( produc780340308816795563term_a @ A @ B ) ) ) ).

% curryE
thf(fact_194_curryD,axiom,
    ! [F4: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
      ( ( produc6008550485034061667_a_b_o @ F4 @ A @ B )
     => ( F4 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_195_curryD,axiom,
    ! [F4: produc2056175833795995411term_a > $o,A: list_Bot_bot_term_a,B: list_Bot_bot_term_a] :
      ( ( produc3073380265971003737rm_a_o @ F4 @ A @ B )
     => ( F4 @ ( produc1077782494051512011term_a @ A @ B ) ) ) ).

% curryD
thf(fact_196_curryD,axiom,
    ! [F4: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryD
thf(fact_197_curryD,axiom,
    ! [F4: product_prod_c_c > $o,A: c,B: c] :
      ( ( product_curry_c_c_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_c_c @ A @ B ) ) ) ).

% curryD
thf(fact_198_curryD,axiom,
    ! [F4: product_prod_b_b > $o,A: b,B: b] :
      ( ( product_curry_b_b_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_b_b @ A @ B ) ) ) ).

% curryD
thf(fact_199_curryD,axiom,
    ! [F4: produc7033602110846336755term_a > $o,A: bot_bot_term_a,B: bot_bot_term_a] :
      ( ( produc4484525301753677177rm_a_o @ F4 @ A @ B )
     => ( F4 @ ( produc780340308816795563term_a @ A @ B ) ) ) ).

% curryD
thf(fact_200_fId__onE,axiom,
    ! [C2: produc1095327201564732427rm_a_c,A4: fset_term_a_c] :
      ( ( fmembe3065991156555231058rm_a_c @ C2 @ ( fSet_fId_on_term_a_c @ A4 ) )
     => ~ ! [X3: term_a_c] :
            ( ( fmember_term_a_c @ X3 @ A4 )
           => ( C2
             != ( produc7687097280279809731rm_a_c @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_201_fId__onE,axiom,
    ! [C2: produc2633411654673805033rm_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe4744889074212852528rm_a_b @ C2 @ ( fSet_f2178757263449557692rm_a_b @ A4 ) )
     => ~ ! [X3: fset_term_a_b] :
            ( ( fmembe3754333024224722621rm_a_b @ X3 @ A4 )
           => ( C2
             != ( produc8129827254723108897rm_a_b @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_202_fId__onE,axiom,
    ! [C2: produc2056175833795995411term_a,A4: fset_l235587408706331465term_a] :
      ( ( fmembe4453961250406969178term_a @ C2 @ ( fSet_f2002284675934200977term_a @ A4 ) )
     => ~ ! [X3: list_Bot_bot_term_a] :
            ( ( fmembe1891194496946673298term_a @ X3 @ A4 )
           => ( C2
             != ( produc1077782494051512011term_a @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_203_fId__onE,axiom,
    ! [C2: product_prod_nat_nat,A4: fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ C2 @ ( fSet_fId_on_nat @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( fmember_nat @ X3 @ A4 )
           => ( C2
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_204_fId__onE,axiom,
    ! [C2: product_prod_c_c,A4: fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ C2 @ ( fSet_fId_on_c @ A4 ) )
     => ~ ! [X3: c] :
            ( ( fmember_c @ X3 @ A4 )
           => ( C2
             != ( product_Pair_c_c @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_205_fId__onE,axiom,
    ! [C2: product_prod_b_b,A4: fset_b] :
      ( ( fmembe4950632938690628014od_b_b @ C2 @ ( fSet_fId_on_b @ A4 ) )
     => ~ ! [X3: b] :
            ( ( fmember_b @ X3 @ A4 )
           => ( C2
             != ( product_Pair_b_b @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_206_fId__onE,axiom,
    ! [C2: produc357393685978478089rm_a_b,A4: fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ C2 @ ( fSet_fId_on_term_a_b @ A4 ) )
     => ~ ! [X3: term_a_b] :
            ( ( fmember_term_a_b @ X3 @ A4 )
           => ( C2
             != ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_207_fId__onE,axiom,
    ! [C2: produc7033602110846336755term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ C2 @ ( fSet_f5231733412705268353term_a @ A4 ) )
     => ~ ! [X3: bot_bot_term_a] :
            ( ( fmembe1418995468851910914term_a @ X3 @ A4 )
           => ( C2
             != ( produc780340308816795563term_a @ X3 @ X3 ) ) ) ) ).

% fId_onE
thf(fact_208_fId__on__eqI,axiom,
    ! [A: term_a_c,B: term_a_c,A4: fset_term_a_c] :
      ( ( A = B )
     => ( ( fmember_term_a_c @ A @ A4 )
       => ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ B ) @ ( fSet_fId_on_term_a_c @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_209_fId__on__eqI,axiom,
    ! [A: fset_term_a_b,B: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( A = B )
     => ( ( fmembe3754333024224722621rm_a_b @ A @ A4 )
       => ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ A @ B ) @ ( fSet_f2178757263449557692rm_a_b @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_210_fId__on__eqI,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,A4: fset_l235587408706331465term_a] :
      ( ( A = B )
     => ( ( fmembe1891194496946673298term_a @ A @ A4 )
       => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f2002284675934200977term_a @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_211_fId__on__eqI,axiom,
    ! [A: nat,B: nat,A4: fset_nat] :
      ( ( A = B )
     => ( ( fmember_nat @ A @ A4 )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fId_on_nat @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_212_fId__on__eqI,axiom,
    ! [A: c,B: c,A4: fset_c] :
      ( ( A = B )
     => ( ( fmember_c @ A @ A4 )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fId_on_c @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_213_fId__on__eqI,axiom,
    ! [A: b,B: b,A4: fset_b] :
      ( ( A = B )
     => ( ( fmember_b @ A @ A4 )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_fId_on_b @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_214_fId__on__eqI,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b] :
      ( ( A = B )
     => ( ( fmember_term_a_b @ A @ A4 )
       => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_fId_on_term_a_b @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_215_fId__on__eqI,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( A = B )
     => ( ( fmembe1418995468851910914term_a @ A @ A4 )
       => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f5231733412705268353term_a @ A4 ) ) ) ) ).

% fId_on_eqI
thf(fact_216_fId__on__iff,axiom,
    ! [X: term_a_c,Y: term_a_c,A4: fset_term_a_c] :
      ( ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ X @ Y ) @ ( fSet_fId_on_term_a_c @ A4 ) )
      = ( ( X = Y )
        & ( fmember_term_a_c @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_217_fId__on__iff,axiom,
    ! [X: fset_term_a_b,Y: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ X @ Y ) @ ( fSet_f2178757263449557692rm_a_b @ A4 ) )
      = ( ( X = Y )
        & ( fmembe3754333024224722621rm_a_b @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_218_fId__on__iff,axiom,
    ! [X: list_Bot_bot_term_a,Y: list_Bot_bot_term_a,A4: fset_l235587408706331465term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Y ) @ ( fSet_f2002284675934200977term_a @ A4 ) )
      = ( ( X = Y )
        & ( fmembe1891194496946673298term_a @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_219_fId__on__iff,axiom,
    ! [X: nat,Y: nat,A4: fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_fId_on_nat @ A4 ) )
      = ( ( X = Y )
        & ( fmember_nat @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_220_fId__on__iff,axiom,
    ! [X: c,Y: c,A4: fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_fId_on_c @ A4 ) )
      = ( ( X = Y )
        & ( fmember_c @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_221_fId__on__iff,axiom,
    ! [X: b,Y: b,A4: fset_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_fId_on_b @ A4 ) )
      = ( ( X = Y )
        & ( fmember_b @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_222_fId__on__iff,axiom,
    ! [X: term_a_b,Y: term_a_b,A4: fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( fSet_fId_on_term_a_b @ A4 ) )
      = ( ( X = Y )
        & ( fmember_term_a_b @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_223_fId__on__iff,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( fSet_f5231733412705268353term_a @ A4 ) )
      = ( ( X = Y )
        & ( fmembe1418995468851910914term_a @ X @ A4 ) ) ) ).

% fId_on_iff
thf(fact_224_ta__nf__tr__to__state,axiom,
    ! [T: term_a4617543059560371146term_a,Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_c @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_225_ta__nf__tr__to__state,axiom,
    ! [T: term_a4617543059560371146term_a,Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_b @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_226_ta__nf__tr__to__state,axiom,
    ! [T: term_a_term_a_b,Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_c @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_227_ta__nf__tr__to__state,axiom,
    ! [T: term_a_term_a_b,Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_b @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_228_ta__nf__tr__to__state,axiom,
    ! [T: term_a_c,Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_c @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_c @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_229_ta__nf__tr__to__state,axiom,
    ! [T: term_a_b,Q2: bot_bot_term_a,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_c @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_230_ta__nf__tr__to__state,axiom,
    ! [T: term_a_b,Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_b @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_231_ta__nf__tr__to__state,axiom,
    ! [T: term_a_c,Q2: bot_bot_term_a,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_c @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( fstates_a_b @ R2 ) ) ) ) ).

% ta_nf_tr_to_state
thf(fact_232_ta__der__adapt__vars__ground,axiom,
    ! [T: term_a4617543059560371146term_a,A4: tree_t5222909361115825503rm_a_a] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( tree_t7856459924191630025rm_a_a @ A4 @ ( term_a6346539342315119866term_a @ T ) )
        = ( tree_t7856459924191630025rm_a_a @ A4 @ T ) ) ) ).

% ta_der_adapt_vars_ground
thf(fact_233_ta__der__adapt__vars__ground,axiom,
    ! [T: term_a_term_a_b,A4: tree_ta_term_a_b_a] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( tree_t8390983252668960894_a_b_a @ A4 @ ( term_a2564936574044254608rm_a_b @ T ) )
        = ( tree_t8390983252668960894_a_b_a @ A4 @ T ) ) ) ).

% ta_der_adapt_vars_ground
thf(fact_234_ta__der__adapt__vars__ground,axiom,
    ! [T: term_a_b,A4: tree_ta_b_a] :
      ( ( term_ground_a_b @ T )
     => ( ( tree_ta_der_b_a @ A4 @ ( term_a2780309480206857766_a_b_b @ T ) )
        = ( tree_ta_der_b_a @ A4 @ T ) ) ) ).

% ta_der_adapt_vars_ground
thf(fact_235_ta__der__adapt__vars__ground,axiom,
    ! [T: term_a_c,A4: tree_ta_c_a] :
      ( ( term_ground_a_c @ T )
     => ( ( tree_ta_der_c_a @ A4 @ ( term_a9216225935088957542_a_c_c @ T ) )
        = ( tree_ta_der_c_a @ A4 @ T ) ) ) ).

% ta_der_adapt_vars_ground
thf(fact_236_ground__adapt__vars,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ ( term_a9216225935088957542_a_c_c @ T ) )
      = ( term_ground_a_c @ T ) ) ).

% ground_adapt_vars
thf(fact_237_ground__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_c @ ( term_a2780309480206857767_a_b_c @ T ) )
      = ( term_ground_a_b @ T ) ) ).

% ground_adapt_vars
thf(fact_238_ground__adapt__vars,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_b @ ( term_a9216225935088957541_a_c_b @ T ) )
      = ( term_ground_a_c @ T ) ) ).

% ground_adapt_vars
thf(fact_239_ground__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ ( term_a2780309480206857766_a_b_b @ T ) )
      = ( term_ground_a_b @ T ) ) ).

% ground_adapt_vars
thf(fact_240_ground__adapt__vars,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_ground_a_b @ ( term_a5382035503001191118rm_a_b @ T ) )
      = ( term_g4884896889520838057term_a @ T ) ) ).

% ground_adapt_vars
thf(fact_241_ground__adapt__vars,axiom,
    ! [T: term_a_c] :
      ( ( term_g4884896889520838057term_a @ ( term_a3900797222385290531term_a @ T ) )
      = ( term_ground_a_c @ T ) ) ).

% ground_adapt_vars
thf(fact_242_ground__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_g4884896889520838057term_a @ ( term_a6990854600579303330term_a @ T ) )
      = ( term_ground_a_b @ T ) ) ).

% ground_adapt_vars
thf(fact_243_ground__adapt__vars,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_ground_a_c @ ( term_a5382035503001191119rm_a_c @ T ) )
      = ( term_g4884896889520838057term_a @ T ) ) ).

% ground_adapt_vars
thf(fact_244_ground__adapt__vars,axiom,
    ! [T: term_a_term_a_b] :
      ( ( term_ground_a_b @ ( term_a3222098444227397763_a_b_b @ T ) )
      = ( term_g2255532262831151796rm_a_b @ T ) ) ).

% ground_adapt_vars
thf(fact_245_ground__adapt__vars,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_g4884896889520838057term_a @ ( term_a6346539342315119866term_a @ T ) )
      = ( term_g4884896889520838057term_a @ T ) ) ).

% ground_adapt_vars
thf(fact_246_adapt__vars__adapt__vars,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( term_a5382035503001191118rm_a_b @ ( term_a6990854600579303330term_a @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_247_adapt__vars__adapt__vars,axiom,
    ! [T: term_a_term_a_b] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( term_a4269149868823119406rm_a_b @ ( term_a3222098444227397764_a_b_c @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_248_adapt__vars__adapt__vars,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( term_a3900797222385290531term_a @ ( term_a5382035503001191119rm_a_c @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_249_adapt__vars__adapt__vars,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( term_a6990854600579303330term_a @ ( term_a5382035503001191118rm_a_b @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_250_adapt__vars__adapt__vars,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( term_a9216225935088957542_a_c_c @ ( term_a9216225935088957542_a_c_c @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_251_adapt__vars__adapt__vars,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( term_a5382035503001191119rm_a_c @ ( term_a3900797222385290531term_a @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_252_adapt__vars__adapt__vars,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( term_a3222098444227397764_a_b_c @ ( term_a4269149868823119406rm_a_b @ T ) )
        = T ) ) ).

% adapt_vars_adapt_vars
thf(fact_253_ta__langE,axiom,
    ! [T: term_a_c,Q: fset_c,A6: tree_ta_c_a] :
      ( ( member_term_a_c @ T @ ( tree_ta_lang_c_a_c @ Q @ A6 ) )
     => ~ ! [T3: term_a_c] :
            ( ( term_ground_a_c @ T3 )
           => ! [Q3: c] :
                ( ( fmember_c @ Q3 @ Q )
               => ( ( fmember_c @ Q3 @ ( tree_ta_der_c_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a9216225935088957542_a_c_c @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_254_ta__langE,axiom,
    ! [T: term_a_c,Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( member_term_a_c @ T @ ( tree_t1675334710359824801_a_a_c @ Q @ A6 ) )
     => ~ ! [T3: term_a4617543059560371146term_a] :
            ( ( term_g4884896889520838057term_a @ T3 )
           => ! [Q3: bot_bot_term_a] :
                ( ( fmembe1418995468851910914term_a @ Q3 @ Q )
               => ( ( fmembe1418995468851910914term_a @ Q3 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a5382035503001191119rm_a_c @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_255_ta__langE,axiom,
    ! [T: term_a_c,Q: fset_term_a_b,A6: tree_ta_term_a_b_a] :
      ( ( member_term_a_c @ T @ ( tree_t726036196298098988_b_a_c @ Q @ A6 ) )
     => ~ ! [T3: term_a_term_a_b] :
            ( ( term_g2255532262831151796rm_a_b @ T3 )
           => ! [Q3: term_a_b] :
                ( ( fmember_term_a_b @ Q3 @ Q )
               => ( ( fmember_term_a_b @ Q3 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a3222098444227397764_a_b_c @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_256_ta__langE,axiom,
    ! [T: term_a_c,Q: fset_nat,A6: tree_ta_nat_a] :
      ( ( member_term_a_c @ T @ ( tree_ta_lang_nat_a_c @ Q @ A6 ) )
     => ~ ! [T3: term_a_nat] :
            ( ( term_ground_a_nat @ T3 )
           => ! [Q3: nat] :
                ( ( fmember_nat @ Q3 @ Q )
               => ( ( fmember_nat @ Q3 @ ( tree_ta_der_nat_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a4040953500496759368_nat_c @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_257_ta__langE,axiom,
    ! [T: term_a_b,Q: fset_nat,A6: tree_ta_nat_a] :
      ( ( member_term_a_b @ T @ ( tree_ta_lang_nat_a_b @ Q @ A6 ) )
     => ~ ! [T3: term_a_nat] :
            ( ( term_ground_a_nat @ T3 )
           => ! [Q3: nat] :
                ( ( fmember_nat @ Q3 @ Q )
               => ( ( fmember_nat @ Q3 @ ( tree_ta_der_nat_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a4040953500496759367_nat_b @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_258_ta__langE,axiom,
    ! [T: term_a_b,Q: fset_c,A6: tree_ta_c_a] :
      ( ( member_term_a_b @ T @ ( tree_ta_lang_c_a_b @ Q @ A6 ) )
     => ~ ! [T3: term_a_c] :
            ( ( term_ground_a_c @ T3 )
           => ! [Q3: c] :
                ( ( fmember_c @ Q3 @ Q )
               => ( ( fmember_c @ Q3 @ ( tree_ta_der_c_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a9216225935088957541_a_c_b @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_259_ta__langE,axiom,
    ! [T: term_a_c,Q: fset_b,A6: tree_ta_b_a] :
      ( ( member_term_a_c @ T @ ( tree_ta_lang_b_a_c @ Q @ A6 ) )
     => ~ ! [T3: term_a_b] :
            ( ( term_ground_a_b @ T3 )
           => ! [Q3: b] :
                ( ( fmember_b @ Q3 @ Q )
               => ( ( fmember_b @ Q3 @ ( tree_ta_der_b_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a2780309480206857767_a_b_c @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_260_ta__langE,axiom,
    ! [T: term_a_b,Q: fset_b,A6: tree_ta_b_a] :
      ( ( member_term_a_b @ T @ ( tree_ta_lang_b_a_b @ Q @ A6 ) )
     => ~ ! [T3: term_a_b] :
            ( ( term_ground_a_b @ T3 )
           => ! [Q3: b] :
                ( ( fmember_b @ Q3 @ Q )
               => ( ( fmember_b @ Q3 @ ( tree_ta_der_b_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a2780309480206857766_a_b_b @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_261_ta__langE,axiom,
    ! [T: term_a_b,Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( member_term_a_b @ T @ ( tree_t1675334710359824800_a_a_b @ Q @ A6 ) )
     => ~ ! [T3: term_a4617543059560371146term_a] :
            ( ( term_g4884896889520838057term_a @ T3 )
           => ! [Q3: bot_bot_term_a] :
                ( ( fmembe1418995468851910914term_a @ Q3 @ Q )
               => ( ( fmembe1418995468851910914term_a @ Q3 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a5382035503001191118rm_a_b @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_262_ta__langE,axiom,
    ! [T: term_a4617543059560371146term_a,Q: fset_c,A6: tree_ta_c_a] :
      ( ( member7590767121479010419term_a @ T @ ( tree_t1904341562405769245term_a @ Q @ A6 ) )
     => ~ ! [T3: term_a_c] :
            ( ( term_ground_a_c @ T3 )
           => ! [Q3: c] :
                ( ( fmember_c @ Q3 @ Q )
               => ( ( fmember_c @ Q3 @ ( tree_ta_der_c_a @ A6 @ T3 ) )
                 => ( T
                   != ( term_a3900797222385290531term_a @ T3 ) ) ) ) ) ) ).

% ta_langE
thf(fact_263_ta__langI,axiom,
    ! [T4: term_a_c,Q2: c,Q: fset_c,A6: tree_ta_c_a,T: term_a_c] :
      ( ( term_ground_a_c @ T4 )
     => ( ( fmember_c @ Q2 @ Q )
       => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a9216225935088957542_a_c_c @ T4 ) )
           => ( member_term_a_c @ T @ ( tree_ta_lang_c_a_c @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_264_ta__langI,axiom,
    ! [T4: term_a4617543059560371146term_a,Q2: bot_bot_term_a,Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,T: term_a_c] :
      ( ( term_g4884896889520838057term_a @ T4 )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ Q )
       => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a5382035503001191119rm_a_c @ T4 ) )
           => ( member_term_a_c @ T @ ( tree_t1675334710359824801_a_a_c @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_265_ta__langI,axiom,
    ! [T4: term_a_term_a_b,Q2: term_a_b,Q: fset_term_a_b,A6: tree_ta_term_a_b_a,T: term_a_c] :
      ( ( term_g2255532262831151796rm_a_b @ T4 )
     => ( ( fmember_term_a_b @ Q2 @ Q )
       => ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a3222098444227397764_a_b_c @ T4 ) )
           => ( member_term_a_c @ T @ ( tree_t726036196298098988_b_a_c @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_266_ta__langI,axiom,
    ! [T4: term_a_nat,Q2: nat,Q: fset_nat,A6: tree_ta_nat_a,T: term_a_c] :
      ( ( term_ground_a_nat @ T4 )
     => ( ( fmember_nat @ Q2 @ Q )
       => ( ( fmember_nat @ Q2 @ ( tree_ta_der_nat_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a4040953500496759368_nat_c @ T4 ) )
           => ( member_term_a_c @ T @ ( tree_ta_lang_nat_a_c @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_267_ta__langI,axiom,
    ! [T4: term_a_nat,Q2: nat,Q: fset_nat,A6: tree_ta_nat_a,T: term_a_b] :
      ( ( term_ground_a_nat @ T4 )
     => ( ( fmember_nat @ Q2 @ Q )
       => ( ( fmember_nat @ Q2 @ ( tree_ta_der_nat_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a4040953500496759367_nat_b @ T4 ) )
           => ( member_term_a_b @ T @ ( tree_ta_lang_nat_a_b @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_268_ta__langI,axiom,
    ! [T4: term_a_c,Q2: c,Q: fset_c,A6: tree_ta_c_a,T: term_a_b] :
      ( ( term_ground_a_c @ T4 )
     => ( ( fmember_c @ Q2 @ Q )
       => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a9216225935088957541_a_c_b @ T4 ) )
           => ( member_term_a_b @ T @ ( tree_ta_lang_c_a_b @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_269_ta__langI,axiom,
    ! [T4: term_a_b,Q2: b,Q: fset_b,A6: tree_ta_b_a,T: term_a_c] :
      ( ( term_ground_a_b @ T4 )
     => ( ( fmember_b @ Q2 @ Q )
       => ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a2780309480206857767_a_b_c @ T4 ) )
           => ( member_term_a_c @ T @ ( tree_ta_lang_b_a_c @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_270_ta__langI,axiom,
    ! [T4: term_a_b,Q2: b,Q: fset_b,A6: tree_ta_b_a,T: term_a_b] :
      ( ( term_ground_a_b @ T4 )
     => ( ( fmember_b @ Q2 @ Q )
       => ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a2780309480206857766_a_b_b @ T4 ) )
           => ( member_term_a_b @ T @ ( tree_ta_lang_b_a_b @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_271_ta__langI,axiom,
    ! [T4: term_a4617543059560371146term_a,Q2: bot_bot_term_a,Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,T: term_a_b] :
      ( ( term_g4884896889520838057term_a @ T4 )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ Q )
       => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a5382035503001191118rm_a_b @ T4 ) )
           => ( member_term_a_b @ T @ ( tree_t1675334710359824800_a_a_b @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_272_ta__langI,axiom,
    ! [T4: term_a_c,Q2: c,Q: fset_c,A6: tree_ta_c_a,T: term_a4617543059560371146term_a] :
      ( ( term_ground_a_c @ T4 )
     => ( ( fmember_c @ Q2 @ Q )
       => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T4 ) )
         => ( ( T
              = ( term_a3900797222385290531term_a @ T4 ) )
           => ( member7590767121479010419term_a @ T @ ( tree_t1904341562405769245term_a @ Q @ A6 ) ) ) ) ) ) ).

% ta_langI
thf(fact_273_ta__detE,axiom,
    ! [A6: tree_ta_c_a,Q2: c,T: term_a_c,Q4: c] :
      ( ( tree_ta_det_c_a @ A6 )
     => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T ) )
       => ( ( fmember_c @ Q4 @ ( tree_ta_der_c_a @ A6 @ T ) )
         => ( Q2 = Q4 ) ) ) ) ).

% ta_detE
thf(fact_274_ta__detE,axiom,
    ! [A6: tree_ta_term_a_b_a,Q2: term_a_b,T: term_a_term_a_b,Q4: term_a_b] :
      ( ( tree_t3137586869393622524_a_b_a @ A6 )
     => ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
       => ( ( fmember_term_a_b @ Q4 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
         => ( Q2 = Q4 ) ) ) ) ).

% ta_detE
thf(fact_275_ta__detE,axiom,
    ! [A6: tree_ta_b_a,Q2: b,T: term_a_b,Q4: b] :
      ( ( tree_ta_det_b_a @ A6 )
     => ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T ) )
       => ( ( fmember_b @ Q4 @ ( tree_ta_der_b_a @ A6 @ T ) )
         => ( Q2 = Q4 ) ) ) ) ).

% ta_detE
thf(fact_276_ta__detE,axiom,
    ! [A6: tree_t5222909361115825503rm_a_a,Q2: bot_bot_term_a,T: term_a4617543059560371146term_a,Q4: bot_bot_term_a] :
      ( ( tree_t1838897014857239623rm_a_a @ A6 )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
       => ( ( fmembe1418995468851910914term_a @ Q4 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
         => ( Q2 = Q4 ) ) ) ) ).

% ta_detE
thf(fact_277_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a4617543059560371146term_a,L: term_a_b,R2: fset_term_a_b,S: term_a_b,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ S ) @ subterm_and_supt_a_b )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te359145989583603559term_a @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_278_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a_term_a_b,L: term_a_b,R2: fset_term_a_b,S: term_a_b,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ S ) @ subterm_and_supt_a_b )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te522286312025294962rm_a_b @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_279_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a4617543059560371146term_a,L: term_a_c,R2: fset_term_a_c,S: term_a_c,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ L @ S ) @ subterm_and_supt_a_c )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te359145989583603559term_a @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_280_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a_term_a_b,L: term_a_c,R2: fset_term_a_c,S: term_a_c,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ L @ S ) @ subterm_and_supt_a_c )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te522286312025294962rm_a_b @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_281_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a_b,L: term_a_b,R2: fset_term_a_b,S: term_a_b,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ S ) @ subterm_and_supt_a_b )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_282_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a_c,L: term_a_b,R2: fset_term_a_b,S: term_a_b,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_c @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ S ) @ subterm_and_supt_a_b )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_283_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a_b,L: term_a_c,R2: fset_term_a_c,S: term_a_c,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ L @ S ) @ subterm_and_supt_a_c )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_284_subt__less__eq__res__less__eq,axiom,
    ! [T: term_a_c,L: term_a_c,R2: fset_term_a_c,S: term_a_c,Q2: bot_bot_term_a,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_c @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ L @ S ) @ subterm_and_supt_a_c )
         => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a )
           => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) ) )
             => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ S ) @ Q2 ) @ bot_bless_eq_a ) ) ) ) ) ) ).

% subt_less_eq_res_less_eq
thf(fact_285_ta__nf__sound1,axiom,
    ! [T: term_a4617543059560371146term_a,L: term_a_b,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ L ) @ ( bot_te359145989583603559term_a @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_286_ta__nf__sound1,axiom,
    ! [T: term_a_term_a_b,L: term_a_b,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ L ) @ ( bot_te522286312025294962rm_a_b @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_287_ta__nf__sound1,axiom,
    ! [T: term_a4617543059560371146term_a,L: term_a_c,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ L ) @ ( bot_te359145989583603559term_a @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6346539342315119866term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_288_ta__nf__sound1,axiom,
    ! [T: term_a_term_a_b,L: term_a_c,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ L ) @ ( bot_te522286312025294962rm_a_b @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6651074359774737541term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_289_ta__nf__sound1,axiom,
    ! [T: term_a_b,L: term_a_b,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ L ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_290_ta__nf__sound1,axiom,
    ! [T: term_a_c,L: term_a_b,R2: fset_term_a_b,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_c @ T )
     => ( ( fmember_term_a_b @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ L ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_b @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_291_ta__nf__sound1,axiom,
    ! [T: term_a_b,L: term_a_c,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_b @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ L ) @ ( bot_te3394087401593457953rm_a_b @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a6990854600579303330term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_292_ta__nf__sound1,axiom,
    ! [T: term_a_c,L: term_a_c,R2: fset_term_a_c,F2: fset_P5917089267984057005_a_nat] :
      ( ( term_ground_a_c @ T )
     => ( ( fmember_term_a_c @ L @ R2 )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457954rm_a_c @ L ) @ ( bot_te3394087401593457954rm_a_c @ T ) ) @ bot_bless_eq_a )
         => ( ( tree_t7856459924191630025rm_a_a @ ( nf_ta_a_c @ R2 @ F2 ) @ ( term_a3900797222385290531term_a @ T ) )
            = bot_bo4313458260509495885term_a ) ) ) ) ).

% ta_nf_sound1
thf(fact_293_reg_Oinject,axiom,
    ! [X1: fset_Bot_bot_term_a,X2: tree_t5222909361115825503rm_a_a,Y1: fset_Bot_bot_term_a,Y2: tree_t5222909361115825503rm_a_a] :
      ( ( ( tree_R2766333946944686633rm_a_a @ X1 @ X2 )
        = ( tree_R2766333946944686633rm_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% reg.inject
thf(fact_294_adapt__vars__inj,axiom,
    ! [X: term_a_c,Y: term_a_c] :
      ( ( ( term_a3900797222385290531term_a @ X )
        = ( term_a3900797222385290531term_a @ Y ) )
     => ( ( term_ground_a_c @ X )
       => ( ( term_ground_a_c @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_295_adapt__vars__inj,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( ( term_a6990854600579303330term_a @ X )
        = ( term_a6990854600579303330term_a @ Y ) )
     => ( ( term_ground_a_b @ X )
       => ( ( term_ground_a_b @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_296_adapt__vars__inj,axiom,
    ! [X: term_a_c,Y: term_a_c] :
      ( ( ( term_a9216225935088957542_a_c_c @ X )
        = ( term_a9216225935088957542_a_c_c @ Y ) )
     => ( ( term_ground_a_c @ X )
       => ( ( term_ground_a_c @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_297_adapt__vars__inj,axiom,
    ! [X: term_a4617543059560371146term_a,Y: term_a4617543059560371146term_a] :
      ( ( ( term_a5382035503001191119rm_a_c @ X )
        = ( term_a5382035503001191119rm_a_c @ Y ) )
     => ( ( term_g4884896889520838057term_a @ X )
       => ( ( term_g4884896889520838057term_a @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_298_adapt__vars__inj,axiom,
    ! [X: term_a_term_a_b,Y: term_a_term_a_b] :
      ( ( ( term_a3222098444227397764_a_b_c @ X )
        = ( term_a3222098444227397764_a_b_c @ Y ) )
     => ( ( term_g2255532262831151796rm_a_b @ X )
       => ( ( term_g2255532262831151796rm_a_b @ Y )
         => ( X = Y ) ) ) ) ).

% adapt_vars_inj
thf(fact_299_adapt__vars__reverse,axiom,
    ! [T: term_a_b,T4: term_a4617543059560371146term_a] :
      ( ( term_ground_a_b @ T )
     => ( ( ( term_a5382035503001191118rm_a_b @ T4 )
          = T )
       => ( ( term_a6990854600579303330term_a @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_300_adapt__vars__reverse,axiom,
    ! [T: term_a_term_a_b,T4: term_a_c] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( ( term_a4269149868823119406rm_a_b @ T4 )
          = T )
       => ( ( term_a3222098444227397764_a_b_c @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_301_adapt__vars__reverse,axiom,
    ! [T: term_a4617543059560371146term_a,T4: term_a_c] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( ( term_a3900797222385290531term_a @ T4 )
          = T )
       => ( ( term_a5382035503001191119rm_a_c @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_302_adapt__vars__reverse,axiom,
    ! [T: term_a4617543059560371146term_a,T4: term_a_b] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( ( term_a6990854600579303330term_a @ T4 )
          = T )
       => ( ( term_a5382035503001191118rm_a_b @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_303_adapt__vars__reverse,axiom,
    ! [T: term_a_c,T4: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( ( term_a9216225935088957542_a_c_c @ T4 )
          = T )
       => ( ( term_a9216225935088957542_a_c_c @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_304_adapt__vars__reverse,axiom,
    ! [T: term_a_c,T4: term_a4617543059560371146term_a] :
      ( ( term_ground_a_c @ T )
     => ( ( ( term_a5382035503001191119rm_a_c @ T4 )
          = T )
       => ( ( term_a3900797222385290531term_a @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_305_adapt__vars__reverse,axiom,
    ! [T: term_a_c,T4: term_a_term_a_b] :
      ( ( term_ground_a_c @ T )
     => ( ( ( term_a3222098444227397764_a_b_c @ T4 )
          = T )
       => ( ( term_a4269149868823119406rm_a_b @ T )
          = T4 ) ) ) ).

% adapt_vars_reverse
thf(fact_306_all__not__fin__conv,axiom,
    ! [A4: fset_term_a_c] :
      ( ( ! [X4: term_a_c] :
            ~ ( fmember_term_a_c @ X4 @ A4 ) )
      = ( A4 = bot_bo1654684903532614361rm_a_c ) ) ).

% all_not_fin_conv
thf(fact_307_all__not__fin__conv,axiom,
    ! [A4: fset_fset_term_a_b] :
      ( ( ! [X4: fset_term_a_b] :
            ~ ( fmembe3754333024224722621rm_a_b @ X4 @ A4 ) )
      = ( A4 = bot_bo7844603177471244936rm_a_b ) ) ).

% all_not_fin_conv
thf(fact_308_all__not__fin__conv,axiom,
    ! [A4: fset_nat] :
      ( ( ! [X4: nat] :
            ~ ( fmember_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_fset_nat ) ) ).

% all_not_fin_conv
thf(fact_309_all__not__fin__conv,axiom,
    ! [A4: fset_c] :
      ( ( ! [X4: c] :
            ~ ( fmember_c @ X4 @ A4 ) )
      = ( A4 = bot_bot_fset_c ) ) ).

% all_not_fin_conv
thf(fact_310_all__not__fin__conv,axiom,
    ! [A4: fset_b] :
      ( ( ! [X4: b] :
            ~ ( fmember_b @ X4 @ A4 ) )
      = ( A4 = bot_bot_fset_b ) ) ).

% all_not_fin_conv
thf(fact_311_all__not__fin__conv,axiom,
    ! [A4: fset_term_a_b] :
      ( ( ! [X4: term_a_b] :
            ~ ( fmember_term_a_b @ X4 @ A4 ) )
      = ( A4 = bot_bo1583650863486268376rm_a_b ) ) ).

% all_not_fin_conv
thf(fact_312_fempty__iff,axiom,
    ! [C2: term_a_c] :
      ~ ( fmember_term_a_c @ C2 @ bot_bo1654684903532614361rm_a_c ) ).

% fempty_iff
thf(fact_313_fempty__iff,axiom,
    ! [C2: fset_term_a_b] :
      ~ ( fmembe3754333024224722621rm_a_b @ C2 @ bot_bo7844603177471244936rm_a_b ) ).

% fempty_iff
thf(fact_314_fempty__iff,axiom,
    ! [C2: nat] :
      ~ ( fmember_nat @ C2 @ bot_bot_fset_nat ) ).

% fempty_iff
thf(fact_315_fempty__iff,axiom,
    ! [C2: c] :
      ~ ( fmember_c @ C2 @ bot_bot_fset_c ) ).

% fempty_iff
thf(fact_316_fempty__iff,axiom,
    ! [C2: b] :
      ~ ( fmember_b @ C2 @ bot_bot_fset_b ) ).

% fempty_iff
thf(fact_317_fempty__iff,axiom,
    ! [C2: term_a_b] :
      ~ ( fmember_term_a_b @ C2 @ bot_bo1583650863486268376rm_a_b ) ).

% fempty_iff
thf(fact_318_fId__on__empty,axiom,
    ( ( fSet_f5231733412705268353term_a @ bot_bo4313458260509495885term_a )
    = bot_bo3304177514724984389term_a ) ).

% fId_on_empty
thf(fact_319_fId__on__empty,axiom,
    ( ( fSet_fId_on_term_a_b @ bot_bo1583650863486268376rm_a_b )
    = bot_bo173352615033564763rm_a_b ) ).

% fId_on_empty
thf(fact_320_supt__subst,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_c] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( subst_7999470309526761005_a_b_c @ S @ Sigma ) @ ( subst_7999470309526761005_a_b_c @ T @ Sigma ) ) @ subterm_and_supt_a_c ) ) ).

% supt_subst
thf(fact_321_supt__subst,axiom,
    ! [S: term_a_c,T: term_a_c,Sigma: c > term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
     => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( subst_5212014727554084972_a_c_c @ S @ Sigma ) @ ( subst_5212014727554084972_a_c_c @ T @ Sigma ) ) @ subterm_and_supt_a_c ) ) ).

% supt_subst
thf(fact_322_supt__subst,axiom,
    ! [S: term_a_c,T: term_a_c,Sigma: c > term_a_b] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_5212014727554084971_a_c_b @ S @ Sigma ) @ ( subst_5212014727554084971_a_c_b @ T @ Sigma ) ) @ subterm_and_supt_a_b ) ) ).

% supt_subst
thf(fact_323_supt__subst,axiom,
    ! [S: term_a_b,T: term_a_b,Sigma: b > term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ subterm_and_supt_a_b ) ) ).

% supt_subst
thf(fact_324_subterm__induct,axiom,
    ! [P: term_a_c > $o,T: term_a_c] :
      ( ! [T2: term_a_c] :
          ( ! [S3: term_a_c] :
              ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ T2 @ S3 ) @ subterm_and_supt_a_c )
             => ( P @ S3 ) )
         => ( P @ T2 ) )
     => ( P @ T ) ) ).

% subterm_induct
thf(fact_325_subterm__induct,axiom,
    ! [P: term_a_b > $o,T: term_a_b] :
      ( ! [T2: term_a_b] :
          ( ! [S3: term_a_b] :
              ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ S3 ) @ subterm_and_supt_a_b )
             => ( P @ S3 ) )
         => ( P @ T2 ) )
     => ( P @ T ) ) ).

% subterm_induct
thf(fact_326_supt__not__refl,axiom,
    ! [T: term_a_c] :
      ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ T @ T ) @ subterm_and_supt_a_c ) ).

% supt_not_refl
thf(fact_327_supt__not__refl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_not_refl
thf(fact_328_supt__not__sym,axiom,
    ! [S: term_a_c,T: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ T @ S ) @ subterm_and_supt_a_c ) ) ).

% supt_not_sym
thf(fact_329_supt__not__sym,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subterm_and_supt_a_b ) ) ).

% supt_not_sym
thf(fact_330_supt__irrefl,axiom,
    ! [T: term_a_c] :
      ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ T @ T ) @ subterm_and_supt_a_c ) ).

% supt_irrefl
thf(fact_331_supt__irrefl,axiom,
    ! [T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).

% supt_irrefl
thf(fact_332_supt__trans,axiom,
    ! [S: term_a_c,T: term_a_c,U: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
     => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ T @ U ) @ subterm_and_supt_a_c )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ U ) @ subterm_and_supt_a_c ) ) ) ).

% supt_trans
thf(fact_333_supt__trans,axiom,
    ! [S: term_a_b,T: term_a_b,U: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).

% supt_trans
thf(fact_334_supt__neqD,axiom,
    ! [S: term_a_c,T: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
     => ( S != T ) ) ).

% supt_neqD
thf(fact_335_supt__neqD,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ( S != T ) ) ).

% supt_neqD
thf(fact_336_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_a_c,B: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ B ) @ subterm_and_supt_a_c )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_337_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_338_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_a_c,A: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ A ) @ subterm_and_supt_a_c )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_339_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_340_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_a_c,B: term_a_c,C2: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ B ) @ subterm_and_supt_a_c )
     => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ C2 ) @ subterm_and_supt_a_c )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ C2 ) @ subterm_and_supt_a_c ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_341_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_342_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_a_c,A: term_a_c,C2: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ A ) @ subterm_and_supt_a_c )
     => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ C2 @ B ) @ subterm_and_supt_a_c )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ C2 @ A ) @ subterm_and_supt_a_c ) ) ) ).

% subterm.order.strict_trans
thf(fact_343_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.order.strict_trans
thf(fact_344_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_a_c,A: term_a_c,C2: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ A ) @ subterm_and_supt_a_c )
     => ( ( B = C2 )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ C2 @ A ) @ subterm_and_supt_a_c ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_345_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_a_b,A: term_a_b,C2: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ( ( B = C2 )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_346_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_a_c,B: term_a_c,C2: term_a_c] :
      ( ( A = B )
     => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ C2 @ B ) @ subterm_and_supt_a_c )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ C2 @ A ) @ subterm_and_supt_a_c ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_347_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_a_b,B: term_a_b,C2: term_a_b] :
      ( ( A = B )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_348_subterm_Oless__imp__not__less,axiom,
    ! [Y: term_a_c,X: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ X @ Y ) @ subterm_and_supt_a_c ) ) ).

% subterm.less_imp_not_less
thf(fact_349_subterm_Oless__imp__not__less,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_imp_not_less
thf(fact_350_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_a_c] :
      ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ A ) @ subterm_and_supt_a_c ) ).

% subterm.dual_order.irrefl
thf(fact_351_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subterm_and_supt_a_b ) ).

% subterm.dual_order.irrefl
thf(fact_352_subterm_Oless__imp__not__eq2,axiom,
    ! [Y: term_a_c,X: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ( Y != X ) ) ).

% subterm.less_imp_not_eq2
thf(fact_353_subterm_Oless__imp__not__eq2,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( Y != X ) ) ).

% subterm.less_imp_not_eq2
thf(fact_354_subterm_Oless__imp__not__eq,axiom,
    ! [Y: term_a_c,X: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ( X != Y ) ) ).

% subterm.less_imp_not_eq
thf(fact_355_subterm_Oless__imp__not__eq,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( X != Y ) ) ).

% subterm.less_imp_not_eq
thf(fact_356_subterm_Odual__order_Oasym,axiom,
    ! [A: term_a_c,B: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ B ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ A ) @ subterm_and_supt_a_c ) ) ).

% subterm.dual_order.asym
thf(fact_357_subterm_Odual__order_Oasym,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ).

% subterm.dual_order.asym
thf(fact_358_subterm_Oless__imp__triv,axiom,
    ! [Y: term_a_c,X: term_a_c,P: $o] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ X @ Y ) @ subterm_and_supt_a_c )
       => P ) ) ).

% subterm.less_imp_triv
thf(fact_359_subterm_Oless__imp__triv,axiom,
    ! [Y: term_a_b,X: term_a_b,P: $o] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b )
       => P ) ) ).

% subterm.less_imp_triv
thf(fact_360_subterm_Oless__not__sym,axiom,
    ! [Y: term_a_c,X: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ X @ Y ) @ subterm_and_supt_a_c ) ) ).

% subterm.less_not_sym
thf(fact_361_subterm_Oless__not__sym,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_not_sym
thf(fact_362_subterm_Oless__imp__neq,axiom,
    ! [Y: term_a_c,X: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ( X != Y ) ) ).

% subterm.less_imp_neq
thf(fact_363_subterm_Oless__imp__neq,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( X != Y ) ) ).

% subterm.less_imp_neq
thf(fact_364_subterm_Oless__irrefl,axiom,
    ! [X: term_a_c] :
      ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ X @ X ) @ subterm_and_supt_a_c ) ).

% subterm.less_irrefl
thf(fact_365_subterm_Oless__irrefl,axiom,
    ! [X: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ subterm_and_supt_a_b ) ).

% subterm.less_irrefl
thf(fact_366_subterm_Oorder_Oasym,axiom,
    ! [B: term_a_c,A: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ A ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ B ) @ subterm_and_supt_a_c ) ) ).

% subterm.order.asym
thf(fact_367_subterm_Oorder_Oasym,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.order.asym
thf(fact_368_subterm_Oless__trans,axiom,
    ! [Y: term_a_c,X: term_a_c,Z: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Z @ Y ) @ subterm_and_supt_a_c )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Z @ X ) @ subterm_and_supt_a_c ) ) ) ).

% subterm.less_trans
thf(fact_369_subterm_Oless__trans,axiom,
    ! [Y: term_a_b,X: term_a_b,Z: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subterm_and_supt_a_b )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X ) @ subterm_and_supt_a_b ) ) ) ).

% subterm.less_trans
thf(fact_370_subterm_Oless__asym_H,axiom,
    ! [B: term_a_c,A: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ B @ A ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A @ B ) @ subterm_and_supt_a_c ) ) ).

% subterm.less_asym'
thf(fact_371_subterm_Oless__asym_H,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym'
thf(fact_372_subterm_Oless__asym,axiom,
    ! [Y: term_a_c,X: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ Y @ X ) @ subterm_and_supt_a_c )
     => ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ X @ Y ) @ subterm_and_supt_a_c ) ) ).

% subterm.less_asym
thf(fact_373_subterm_Oless__asym,axiom,
    ! [Y: term_a_b,X: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
     => ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).

% subterm.less_asym
thf(fact_374_equalsffemptyI,axiom,
    ! [A4: fset_term_a_c] :
      ( ! [Y3: term_a_c] :
          ~ ( fmember_term_a_c @ Y3 @ A4 )
     => ( A4 = bot_bo1654684903532614361rm_a_c ) ) ).

% equalsffemptyI
thf(fact_375_equalsffemptyI,axiom,
    ! [A4: fset_fset_term_a_b] :
      ( ! [Y3: fset_term_a_b] :
          ~ ( fmembe3754333024224722621rm_a_b @ Y3 @ A4 )
     => ( A4 = bot_bo7844603177471244936rm_a_b ) ) ).

% equalsffemptyI
thf(fact_376_equalsffemptyI,axiom,
    ! [A4: fset_nat] :
      ( ! [Y3: nat] :
          ~ ( fmember_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_fset_nat ) ) ).

% equalsffemptyI
thf(fact_377_equalsffemptyI,axiom,
    ! [A4: fset_c] :
      ( ! [Y3: c] :
          ~ ( fmember_c @ Y3 @ A4 )
     => ( A4 = bot_bot_fset_c ) ) ).

% equalsffemptyI
thf(fact_378_equalsffemptyI,axiom,
    ! [A4: fset_b] :
      ( ! [Y3: b] :
          ~ ( fmember_b @ Y3 @ A4 )
     => ( A4 = bot_bot_fset_b ) ) ).

% equalsffemptyI
thf(fact_379_equalsffemptyI,axiom,
    ! [A4: fset_term_a_b] :
      ( ! [Y3: term_a_b] :
          ~ ( fmember_term_a_b @ Y3 @ A4 )
     => ( A4 = bot_bo1583650863486268376rm_a_b ) ) ).

% equalsffemptyI
thf(fact_380_equalsffemptyD,axiom,
    ! [A4: fset_term_a_c,A: term_a_c] :
      ( ( A4 = bot_bo1654684903532614361rm_a_c )
     => ~ ( fmember_term_a_c @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_381_equalsffemptyD,axiom,
    ! [A4: fset_fset_term_a_b,A: fset_term_a_b] :
      ( ( A4 = bot_bo7844603177471244936rm_a_b )
     => ~ ( fmembe3754333024224722621rm_a_b @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_382_equalsffemptyD,axiom,
    ! [A4: fset_nat,A: nat] :
      ( ( A4 = bot_bot_fset_nat )
     => ~ ( fmember_nat @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_383_equalsffemptyD,axiom,
    ! [A4: fset_c,A: c] :
      ( ( A4 = bot_bot_fset_c )
     => ~ ( fmember_c @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_384_equalsffemptyD,axiom,
    ! [A4: fset_b,A: b] :
      ( ( A4 = bot_bot_fset_b )
     => ~ ( fmember_b @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_385_equalsffemptyD,axiom,
    ! [A4: fset_term_a_b,A: term_a_b] :
      ( ( A4 = bot_bo1583650863486268376rm_a_b )
     => ~ ( fmember_term_a_b @ A @ A4 ) ) ).

% equalsffemptyD
thf(fact_386_ex__fin__conv,axiom,
    ! [A4: fset_term_a_c] :
      ( ( ? [X4: term_a_c] : ( fmember_term_a_c @ X4 @ A4 ) )
      = ( A4 != bot_bo1654684903532614361rm_a_c ) ) ).

% ex_fin_conv
thf(fact_387_ex__fin__conv,axiom,
    ! [A4: fset_fset_term_a_b] :
      ( ( ? [X4: fset_term_a_b] : ( fmembe3754333024224722621rm_a_b @ X4 @ A4 ) )
      = ( A4 != bot_bo7844603177471244936rm_a_b ) ) ).

% ex_fin_conv
thf(fact_388_ex__fin__conv,axiom,
    ! [A4: fset_nat] :
      ( ( ? [X4: nat] : ( fmember_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_fset_nat ) ) ).

% ex_fin_conv
thf(fact_389_ex__fin__conv,axiom,
    ! [A4: fset_c] :
      ( ( ? [X4: c] : ( fmember_c @ X4 @ A4 ) )
      = ( A4 != bot_bot_fset_c ) ) ).

% ex_fin_conv
thf(fact_390_ex__fin__conv,axiom,
    ! [A4: fset_b] :
      ( ( ? [X4: b] : ( fmember_b @ X4 @ A4 ) )
      = ( A4 != bot_bot_fset_b ) ) ).

% ex_fin_conv
thf(fact_391_ex__fin__conv,axiom,
    ! [A4: fset_term_a_b] :
      ( ( ? [X4: term_a_b] : ( fmember_term_a_b @ X4 @ A4 ) )
      = ( A4 != bot_bo1583650863486268376rm_a_b ) ) ).

% ex_fin_conv
thf(fact_392_femptyE,axiom,
    ! [A: term_a_c] :
      ~ ( fmember_term_a_c @ A @ bot_bo1654684903532614361rm_a_c ) ).

% femptyE
thf(fact_393_femptyE,axiom,
    ! [A: fset_term_a_b] :
      ~ ( fmembe3754333024224722621rm_a_b @ A @ bot_bo7844603177471244936rm_a_b ) ).

% femptyE
thf(fact_394_femptyE,axiom,
    ! [A: nat] :
      ~ ( fmember_nat @ A @ bot_bot_fset_nat ) ).

% femptyE
thf(fact_395_femptyE,axiom,
    ! [A: c] :
      ~ ( fmember_c @ A @ bot_bot_fset_c ) ).

% femptyE
thf(fact_396_femptyE,axiom,
    ! [A: b] :
      ~ ( fmember_b @ A @ bot_bot_fset_b ) ).

% femptyE
thf(fact_397_femptyE,axiom,
    ! [A: term_a_b] :
      ~ ( fmember_term_a_b @ A @ bot_bo1583650863486268376rm_a_b ) ).

% femptyE
thf(fact_398_reg_Oexhaust,axiom,
    ! [Y: tree_r6075998776338966420rm_a_a] :
      ~ ! [X12: fset_Bot_bot_term_a,X22: tree_t5222909361115825503rm_a_a] :
          ( Y
         != ( tree_R2766333946944686633rm_a_a @ X12 @ X22 ) ) ).

% reg.exhaust
thf(fact_399_adapt__vars2,axiom,
    ! [T: term_a_c] :
      ( ( term_a9216225935088957542_a_c_c @ ( term_a9216225935088957542_a_c_c @ T ) )
      = ( term_a9216225935088957542_a_c_c @ T ) ) ).

% adapt_vars2
thf(fact_400_adapt__vars2,axiom,
    ! [T: term_a_b] :
      ( ( term_a3900797222385290531term_a @ ( term_a2780309480206857767_a_b_c @ T ) )
      = ( term_a6990854600579303330term_a @ T ) ) ).

% adapt_vars2
thf(fact_401_adapt__vars2,axiom,
    ! [T: term_a_c] :
      ( ( term_a3900797222385290531term_a @ ( term_a9216225935088957542_a_c_c @ T ) )
      = ( term_a3900797222385290531term_a @ T ) ) ).

% adapt_vars2
thf(fact_402_adapt__vars2,axiom,
    ! [T: term_a_c] :
      ( ( term_a6990854600579303330term_a @ ( term_a9216225935088957541_a_c_b @ T ) )
      = ( term_a3900797222385290531term_a @ T ) ) ).

% adapt_vars2
thf(fact_403_adapt__vars2,axiom,
    ! [T: term_a_b] :
      ( ( term_a6990854600579303330term_a @ ( term_a2780309480206857766_a_b_b @ T ) )
      = ( term_a6990854600579303330term_a @ T ) ) ).

% adapt_vars2
thf(fact_404_adapt__vars2,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_a9216225935088957542_a_c_c @ ( term_a5382035503001191119rm_a_c @ T ) )
      = ( term_a5382035503001191119rm_a_c @ T ) ) ).

% adapt_vars2
thf(fact_405_adapt__vars2,axiom,
    ! [T: term_a_c] :
      ( ( term_a5382035503001191119rm_a_c @ ( term_a3900797222385290531term_a @ T ) )
      = ( term_a9216225935088957542_a_c_c @ T ) ) ).

% adapt_vars2
thf(fact_406_adapt__vars2,axiom,
    ! [T: term_a_b] :
      ( ( term_a5382035503001191119rm_a_c @ ( term_a6990854600579303330term_a @ T ) )
      = ( term_a2780309480206857767_a_b_c @ T ) ) ).

% adapt_vars2
thf(fact_407_adapt__vars2,axiom,
    ! [T: term_a_c] :
      ( ( term_a6346539342315119866term_a @ ( term_a3900797222385290531term_a @ T ) )
      = ( term_a3900797222385290531term_a @ T ) ) ).

% adapt_vars2
thf(fact_408_adapt__vars2,axiom,
    ! [T: term_a_b] :
      ( ( term_a6346539342315119866term_a @ ( term_a6990854600579303330term_a @ T ) )
      = ( term_a6990854600579303330term_a @ T ) ) ).

% adapt_vars2
thf(fact_409_adapt__vars__ctxt,axiom,
    ! [C: subter720051973439326747term_a,T: term_a4617543059560371146term_a] :
      ( ( term_a5382035503001191119rm_a_c @ ( subter8974050253859094258term_a @ C @ T ) )
      = ( subter2376574525758040791rm_a_c @ ( term_a8854742828266083164rm_a_c @ C ) @ ( term_a5382035503001191119rm_a_c @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_410_adapt__vars__ctxt,axiom,
    ! [C: subter2881564190832396198rm_a_b,T: term_a_term_a_b] :
      ( ( term_a3222098444227397764_a_b_c @ ( subter7263177522250274301rm_a_b @ C @ T ) )
      = ( subter2376574525758040791rm_a_c @ ( term_a7029920194893525777_a_b_c @ C ) @ ( term_a3222098444227397764_a_b_c @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_411_adapt__vars__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( term_a3900797222385290531term_a @ ( subter2376574525758040791rm_a_c @ C @ T ) )
      = ( subter8974050253859094258term_a @ ( term_a7373504547650182576term_a @ C ) @ ( term_a3900797222385290531term_a @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_412_adapt__vars__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( term_a9216225935088957541_a_c_b @ ( subter2376574525758040791rm_a_c @ C @ T ) )
      = ( subter2376574525758040790rm_a_b @ ( term_a8194622999837494424_a_c_b @ C ) @ ( term_a9216225935088957541_a_c_b @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_413_adapt__vars__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_a6990854600579303330term_a @ ( subter2376574525758040790rm_a_b @ C @ T ) )
      = ( subter8974050253859094258term_a @ ( term_a1240189888989419567term_a @ C ) @ ( term_a6990854600579303330term_a @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_414_adapt__vars__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_a2780309480206857767_a_b_c @ ( subter2376574525758040790rm_a_b @ C @ T ) )
      = ( subter2376574525758040791rm_a_c @ ( term_a1758706544955394650_a_b_c @ C ) @ ( term_a2780309480206857767_a_b_c @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_415_adapt__vars__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_a2780309480206857766_a_b_b @ ( subter2376574525758040790rm_a_b @ C @ T ) )
      = ( subter2376574525758040790rm_a_b @ ( term_a1758706544955394649_a_b_b @ C ) @ ( term_a2780309480206857766_a_b_b @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_416_adapt__vars__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( term_a9216225935088957542_a_c_c @ ( subter2376574525758040791rm_a_c @ C @ T ) )
      = ( subter2376574525758040791rm_a_c @ ( term_a8194622999837494425_a_c_c @ C ) @ ( term_a9216225935088957542_a_c_c @ T ) ) ) ).

% adapt_vars_ctxt
thf(fact_417_bot__apply,axiom,
    ( bot_bo104646228311450846rm_a_o
    = ( ^ [X4: produc7033602110846336755term_a] : bot_bot_o ) ) ).

% bot_apply
thf(fact_418_bot__apply,axiom,
    ( bot_bot_term_a_c_o
    = ( ^ [X4: term_a_c] : bot_bot_o ) ) ).

% bot_apply
thf(fact_419_split__cong,axiom,
    ! [Q2: produc7033602110846336755term_a,F4: bot_bot_term_a > bot_bot_term_a > set_term_a_c,G: bot_bot_term_a > bot_bot_term_a > set_term_a_c,P2: produc7033602110846336755term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( ( produc780340308816795563term_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F4 @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc5909685650505657099rm_a_c @ F4 @ P2 )
          = ( produc5909685650505657099rm_a_c @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_420_split__cong,axiom,
    ! [Q2: produc7033602110846336755term_a,F4: bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a,G: bot_bot_term_a > bot_bot_term_a > set_Pr3965413743549523539term_a,P2: produc7033602110846336755term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( ( produc780340308816795563term_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F4 @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc2700287094330126483term_a @ F4 @ P2 )
          = ( produc2700287094330126483term_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_421_split__cong,axiom,
    ! [Q2: produc7033602110846336755term_a,F4: bot_bot_term_a > bot_bot_term_a > $o,G: bot_bot_term_a > bot_bot_term_a > $o,P2: produc7033602110846336755term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( ( produc780340308816795563term_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F4 @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc480846774855951832rm_a_o @ F4 @ P2 )
          = ( produc480846774855951832rm_a_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_422_split__cong,axiom,
    ! [Q2: produc7033602110846336755term_a,F4: bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a,G: bot_bot_term_a > bot_bot_term_a > produc7033602110846336755term_a,P2: produc7033602110846336755term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( ( produc780340308816795563term_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F4 @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc1985598078976903475term_a @ F4 @ P2 )
          = ( produc1985598078976903475term_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_423_eps__statesI_I1_J,axiom,
    ! [P2: term_a_c,Q2: term_a_c,Delta: fset_P6525107853456574705rm_a_c] :
      ( ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ P2 @ Q2 ) @ Delta )
     => ( fmember_term_a_c @ P2 @ ( tree_e4197689034148078246rm_a_c @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_424_eps__statesI_I1_J,axiom,
    ! [P2: fset_term_a_b,Q2: fset_term_a_b,Delta: fset_P7393603901796714959rm_a_b] :
      ( ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ P2 @ Q2 ) @ Delta )
     => ( fmembe3754333024224722621rm_a_b @ P2 @ ( tree_e8828569120810636757rm_a_b @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_425_eps__statesI_I1_J,axiom,
    ! [P2: list_Bot_bot_term_a,Q2: list_Bot_bot_term_a,Delta: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ P2 @ Q2 ) @ Delta )
     => ( fmembe1891194496946673298term_a @ P2 @ ( tree_e5611726204267773354term_a @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_426_eps__statesI_I1_J,axiom,
    ! [P2: nat,Q2: nat,Delta: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) @ Delta )
     => ( fmember_nat @ P2 @ ( tree_eps_states_nat @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_427_eps__statesI_I1_J,axiom,
    ! [P2: c,Q2: c,Delta: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ Delta )
     => ( fmember_c @ P2 @ ( tree_eps_states_c @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_428_eps__statesI_I1_J,axiom,
    ! [P2: b,Q2: b,Delta: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ Delta )
     => ( fmember_b @ P2 @ ( tree_eps_states_b @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_429_eps__statesI_I1_J,axiom,
    ! [P2: term_a_b,Q2: term_a_b,Delta: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ P2 @ Q2 ) @ Delta )
     => ( fmember_term_a_b @ P2 @ ( tree_e4197689029844849445rm_a_b @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_430_eps__statesI_I1_J,axiom,
    ! [P2: bot_bot_term_a,Q2: bot_bot_term_a,Delta: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ P2 @ Q2 ) @ Delta )
     => ( fmembe1418995468851910914term_a @ P2 @ ( tree_e551949735975627802term_a @ Delta ) ) ) ).

% eps_statesI(1)
thf(fact_431_eps__statesI_I2_J,axiom,
    ! [P2: term_a_c,Q2: term_a_c,Delta: fset_P6525107853456574705rm_a_c] :
      ( ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ P2 @ Q2 ) @ Delta )
     => ( fmember_term_a_c @ Q2 @ ( tree_e4197689034148078246rm_a_c @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_432_eps__statesI_I2_J,axiom,
    ! [P2: fset_term_a_b,Q2: fset_term_a_b,Delta: fset_P7393603901796714959rm_a_b] :
      ( ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ P2 @ Q2 ) @ Delta )
     => ( fmembe3754333024224722621rm_a_b @ Q2 @ ( tree_e8828569120810636757rm_a_b @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_433_eps__statesI_I2_J,axiom,
    ! [P2: list_Bot_bot_term_a,Q2: list_Bot_bot_term_a,Delta: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ P2 @ Q2 ) @ Delta )
     => ( fmembe1891194496946673298term_a @ Q2 @ ( tree_e5611726204267773354term_a @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_434_eps__statesI_I2_J,axiom,
    ! [P2: nat,Q2: nat,Delta: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) @ Delta )
     => ( fmember_nat @ Q2 @ ( tree_eps_states_nat @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_435_eps__statesI_I2_J,axiom,
    ! [P2: c,Q2: c,Delta: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ Delta )
     => ( fmember_c @ Q2 @ ( tree_eps_states_c @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_436_eps__statesI_I2_J,axiom,
    ! [P2: b,Q2: b,Delta: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ Delta )
     => ( fmember_b @ Q2 @ ( tree_eps_states_b @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_437_eps__statesI_I2_J,axiom,
    ! [P2: term_a_b,Q2: term_a_b,Delta: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ P2 @ Q2 ) @ Delta )
     => ( fmember_term_a_b @ Q2 @ ( tree_e4197689029844849445rm_a_b @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_438_eps__statesI_I2_J,axiom,
    ! [P2: bot_bot_term_a,Q2: bot_bot_term_a,Delta: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ P2 @ Q2 ) @ Delta )
     => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_e551949735975627802term_a @ Delta ) ) ) ).

% eps_statesI(2)
thf(fact_439_bot__prod__def,axiom,
    ( bot_bo2769642828321324397at_nat
    = ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_440_bot__prod__def,axiom,
    ( bot_bo3585513266671089768_c_nat
    = ( produc4240701259145610164_c_nat @ bot_bot_set_term_a_c @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_441_bot__prod__def,axiom,
    ( bot_bo1639525010489079912rm_a_c
    = ( produc5999293419280449332rm_a_c @ bot_bot_nat @ bot_bot_set_term_a_c ) ) ).

% bot_prod_def
thf(fact_442_bot__prod__def,axiom,
    ( bot_bo3271289617687911219_a_c_o
    = ( produc4559189924671168081_a_c_o @ bot_bot_nat @ bot_bot_term_a_c_o ) ) ).

% bot_prod_def
thf(fact_443_bot__prod__def,axiom,
    ( bot_bo6296390520458004621_o_nat
    = ( produc9092562165568265075_o_nat @ bot_bot_term_a_c_o @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_444_bot__prod__def,axiom,
    ( bot_bo6547199808529123302_a_nat
    = ( produc5627630658439173836_a_nat @ bot_bo2447593420621686463term_a @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_445_bot__prod__def,axiom,
    ( bot_bo6150146448544589324term_a
    = ( produc8051651767284650410term_a @ bot_bot_nat @ bot_bo2447593420621686463term_a ) ) ).

% bot_prod_def
thf(fact_446_bot__prod__def,axiom,
    ( bot_bo8902706747039010275rm_a_c
    = ( produc3305361712476037935rm_a_c @ bot_bot_set_term_a_c @ bot_bot_set_term_a_c ) ) ).

% bot_prod_def
thf(fact_447_bot__prod__def,axiom,
    ( bot_bo7312153412206230759rm_a_o
    = ( produc4976332323224857011rm_a_o @ bot_bot_nat @ bot_bo104646228311450846rm_a_o ) ) ).

% bot_prod_def
thf(fact_448_bot__prod__def,axiom,
    ( bot_bo2899094611167244519_o_nat
    = ( produc6792305905529949363_o_nat @ bot_bo104646228311450846rm_a_o @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_449_frelcompI,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b,C2: term_a_b,S: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ S )
       => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( fSet_f8278883550849121706rm_a_b @ R @ S ) ) ) ) ).

% frelcompI
thf(fact_450_frelcompI,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a,C2: list_Bot_bot_term_a,S: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ R )
     => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ B @ C2 ) @ S )
       => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ C2 ) @ ( fSet_f4048747136859825209term_a @ R @ S ) ) ) ) ).

% frelcompI
thf(fact_451_frelcompI,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat,C2: nat,S: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ S )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( fSet_f5609620875662277726at_nat @ R @ S ) ) ) ) ).

% frelcompI
thf(fact_452_frelcompI,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c,C2: c,S: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ B @ C2 ) @ S )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ C2 ) @ ( fSet_frelcomp_c_c_c @ R @ S ) ) ) ) ).

% frelcompI
thf(fact_453_frelcompI,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b,C2: b,S: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ B @ C2 ) @ S )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ ( fSet_frelcomp_b_b_b @ R @ S ) ) ) ) ).

% frelcompI
thf(fact_454_frelcompI,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a,C2: bot_bot_term_a,S: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ B @ C2 ) @ S )
       => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ C2 ) @ ( fSet_f5283962319602365193term_a @ R @ S ) ) ) ) ).

% frelcompI
thf(fact_455_fconverseI,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( fSet_f4591896755199508282rm_a_b @ R ) ) ) ).

% fconverseI
thf(fact_456_fconverseI,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ R )
     => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ B @ A ) @ ( fSet_f52542278078447812term_a @ R ) ) ) ).

% fconverseI
thf(fact_457_fconverseI,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( fSet_f5560228655207641650at_nat @ R ) ) ) ).

% fconverseI
thf(fact_458_fconverseI,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
     => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ B @ A ) @ ( fSet_fconverse_c_c @ R ) ) ) ).

% fconverseI
thf(fact_459_fconverseI,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
     => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( fSet_fconverse_b_b @ R ) ) ) ).

% fconverseI
thf(fact_460_fconverseI,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
     => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ B @ A ) @ ( fSet_f3482253648094759588term_a @ R ) ) ) ).

% fconverseI
thf(fact_461_fconverse__iff,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f4591896755199508282rm_a_b @ R ) )
      = ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ).

% fconverse_iff
thf(fact_462_fconverse__iff,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f52542278078447812term_a @ R ) )
      = ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ B @ A ) @ R ) ) ).

% fconverse_iff
thf(fact_463_fconverse__iff,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_f5560228655207641650at_nat @ R ) )
      = ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ).

% fconverse_iff
thf(fact_464_fconverse__iff,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fconverse_c_c @ R ) )
      = ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ B @ A ) @ R ) ) ).

% fconverse_iff
thf(fact_465_fconverse__iff,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_fconverse_b_b @ R ) )
      = ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ B @ A ) @ R ) ) ).

% fconverse_iff
thf(fact_466_fconverse__iff,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f3482253648094759588term_a @ R ) )
      = ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ B @ A ) @ R ) ) ).

% fconverse_iff
thf(fact_467_eps__statesE,axiom,
    ! [P2: term_a_c,Delta: fset_P6525107853456574705rm_a_c] :
      ( ( fmember_term_a_c @ P2 @ ( tree_e4197689034148078246rm_a_c @ Delta ) )
     => ~ ! [Q3: term_a_c] :
            ~ ( ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ P2 @ Q3 ) @ Delta )
              | ( fmembe3065991156555231058rm_a_c @ ( produc7687097280279809731rm_a_c @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_468_eps__statesE,axiom,
    ! [P2: fset_term_a_b,Delta: fset_P7393603901796714959rm_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ P2 @ ( tree_e8828569120810636757rm_a_b @ Delta ) )
     => ~ ! [Q3: fset_term_a_b] :
            ~ ( ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ P2 @ Q3 ) @ Delta )
              | ( fmembe4744889074212852528rm_a_b @ ( produc8129827254723108897rm_a_b @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_469_eps__statesE,axiom,
    ! [P2: list_Bot_bot_term_a,Delta: fset_P7293260781336502521term_a] :
      ( ( fmembe1891194496946673298term_a @ P2 @ ( tree_e5611726204267773354term_a @ Delta ) )
     => ~ ! [Q3: list_Bot_bot_term_a] :
            ~ ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ P2 @ Q3 ) @ Delta )
              | ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_470_eps__statesE,axiom,
    ! [P2: nat,Delta: fset_P5670320511379867111at_nat] :
      ( ( fmember_nat @ P2 @ ( tree_eps_states_nat @ Delta ) )
     => ~ ! [Q3: nat] :
            ~ ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ P2 @ Q3 ) @ Delta )
              | ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_471_eps__statesE,axiom,
    ! [P2: c,Delta: fset_P8014105740199022861od_c_c] :
      ( ( fmember_c @ P2 @ ( tree_eps_states_c @ Delta ) )
     => ~ ! [Q3: c] :
            ~ ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q3 ) @ Delta )
              | ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_472_eps__statesE,axiom,
    ! [P2: b,Delta: fset_P8578720700757663565od_b_b] :
      ( ( fmember_b @ P2 @ ( tree_eps_states_b @ Delta ) )
     => ~ ! [Q3: b] :
            ~ ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q3 ) @ Delta )
              | ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_473_eps__statesE,axiom,
    ! [P2: term_a_b,Delta: fset_P3577670608877728751rm_a_b] :
      ( ( fmember_term_a_b @ P2 @ ( tree_e4197689029844849445rm_a_b @ Delta ) )
     => ~ ! [Q3: term_a_b] :
            ~ ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ P2 @ Q3 ) @ Delta )
              | ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_474_eps__statesE,axiom,
    ! [P2: bot_bot_term_a,Delta: fset_P6640269656128473term_a] :
      ( ( fmembe1418995468851910914term_a @ P2 @ ( tree_e551949735975627802term_a @ Delta ) )
     => ~ ! [Q3: bot_bot_term_a] :
            ~ ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ P2 @ Q3 ) @ Delta )
              | ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Q3 @ P2 ) @ Delta ) ) ) ).

% eps_statesE
thf(fact_475_fconverse__fconverse,axiom,
    ! [R: fset_P6640269656128473term_a] :
      ( ( fSet_f3482253648094759588term_a @ ( fSet_f3482253648094759588term_a @ R ) )
      = R ) ).

% fconverse_fconverse
thf(fact_476_fconverse__empty,axiom,
    ( ( fSet_f3482253648094759588term_a @ bot_bo3304177514724984389term_a )
    = bot_bo3304177514724984389term_a ) ).

% fconverse_empty
thf(fact_477_eps__states__empty,axiom,
    ( ( tree_e551949735975627802term_a @ bot_bo3304177514724984389term_a )
    = bot_bo4313458260509495885term_a ) ).

% eps_states_empty
thf(fact_478_eps__states__empty,axiom,
    ( ( tree_e4197689029844849445rm_a_b @ bot_bo173352615033564763rm_a_b )
    = bot_bo1583650863486268376rm_a_b ) ).

% eps_states_empty
thf(fact_479_bot__fun__def,axiom,
    ( bot_bo104646228311450846rm_a_o
    = ( ^ [X4: produc7033602110846336755term_a] : bot_bot_o ) ) ).

% bot_fun_def
thf(fact_480_bot__fun__def,axiom,
    ( bot_bot_term_a_c_o
    = ( ^ [X4: term_a_c] : bot_bot_o ) ) ).

% bot_fun_def
thf(fact_481_fconverseE,axiom,
    ! [Yx: produc357393685978478089rm_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ Yx @ ( fSet_f4591896755199508282rm_a_b @ R ) )
     => ~ ! [X3: term_a_b,Y3: term_a_b] :
            ( ( Yx
              = ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) )
           => ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R ) ) ) ).

% fconverseE
thf(fact_482_fconverseE,axiom,
    ! [Yx: produc2056175833795995411term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ Yx @ ( fSet_f52542278078447812term_a @ R ) )
     => ~ ! [X3: list_Bot_bot_term_a,Y3: list_Bot_bot_term_a] :
            ( ( Yx
              = ( produc1077782494051512011term_a @ Y3 @ X3 ) )
           => ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X3 @ Y3 ) @ R ) ) ) ).

% fconverseE
thf(fact_483_fconverseE,axiom,
    ! [Yx: product_prod_nat_nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ Yx @ ( fSet_f5560228655207641650at_nat @ R ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Yx
              = ( product_Pair_nat_nat @ Y3 @ X3 ) )
           => ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ).

% fconverseE
thf(fact_484_fconverseE,axiom,
    ! [Yx: product_prod_c_c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ Yx @ ( fSet_fconverse_c_c @ R ) )
     => ~ ! [X3: c,Y3: c] :
            ( ( Yx
              = ( product_Pair_c_c @ Y3 @ X3 ) )
           => ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X3 @ Y3 ) @ R ) ) ) ).

% fconverseE
thf(fact_485_fconverseE,axiom,
    ! [Yx: product_prod_b_b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ Yx @ ( fSet_fconverse_b_b @ R ) )
     => ~ ! [X3: b,Y3: b] :
            ( ( Yx
              = ( product_Pair_b_b @ Y3 @ X3 ) )
           => ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R ) ) ) ).

% fconverseE
thf(fact_486_fconverseE,axiom,
    ! [Yx: produc7033602110846336755term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ Yx @ ( fSet_f3482253648094759588term_a @ R ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( Yx
              = ( produc780340308816795563term_a @ Y3 @ X3 ) )
           => ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R ) ) ) ).

% fconverseE
thf(fact_487_fconverseD,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f4591896755199508282rm_a_b @ R ) )
     => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ).

% fconverseD
thf(fact_488_fconverseD,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f52542278078447812term_a @ R ) )
     => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ B @ A ) @ R ) ) ).

% fconverseD
thf(fact_489_fconverseD,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_f5560228655207641650at_nat @ R ) )
     => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ).

% fconverseD
thf(fact_490_fconverseD,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fconverse_c_c @ R ) )
     => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ B @ A ) @ R ) ) ).

% fconverseD
thf(fact_491_fconverseD,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_fconverse_b_b @ R ) )
     => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ B @ A ) @ R ) ) ).

% fconverseD
thf(fact_492_fconverseD,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f3482253648094759588term_a @ R ) )
     => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ B @ A ) @ R ) ) ).

% fconverseD
thf(fact_493_frelcompE,axiom,
    ! [Xz: produc357393685978478089rm_a_b,R: fset_P3577670608877728751rm_a_b,S: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ Xz @ ( fSet_f8278883550849121706rm_a_b @ R @ S ) )
     => ~ ! [X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
            ( ( Xz
              = ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) )
           => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
             => ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_494_frelcompE,axiom,
    ! [Xz: produc2056175833795995411term_a,R: fset_P7293260781336502521term_a,S: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ Xz @ ( fSet_f4048747136859825209term_a @ R @ S ) )
     => ~ ! [X3: list_Bot_bot_term_a,Y3: list_Bot_bot_term_a,Z2: list_Bot_bot_term_a] :
            ( ( Xz
              = ( produc1077782494051512011term_a @ X3 @ Z2 ) )
           => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X3 @ Y3 ) @ R )
             => ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_495_frelcompE,axiom,
    ! [Xz: product_prod_nat_nat,R: fset_P5670320511379867111at_nat,S: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ Xz @ ( fSet_f5609620875662277726at_nat @ R @ S ) )
     => ~ ! [X3: nat,Y3: nat,Z2: nat] :
            ( ( Xz
              = ( product_Pair_nat_nat @ X3 @ Z2 ) )
           => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
             => ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_496_frelcompE,axiom,
    ! [Xz: product_prod_c_c,R: fset_P8014105740199022861od_c_c,S: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ Xz @ ( fSet_frelcomp_c_c_c @ R @ S ) )
     => ~ ! [X3: c,Y3: c,Z2: c] :
            ( ( Xz
              = ( product_Pair_c_c @ X3 @ Z2 ) )
           => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X3 @ Y3 ) @ R )
             => ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_497_frelcompE,axiom,
    ! [Xz: product_prod_b_b,R: fset_P8578720700757663565od_b_b,S: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ Xz @ ( fSet_frelcomp_b_b_b @ R @ S ) )
     => ~ ! [X3: b,Y3: b,Z2: b] :
            ( ( Xz
              = ( product_Pair_b_b @ X3 @ Z2 ) )
           => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
             => ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_498_frelcompE,axiom,
    ! [Xz: produc7033602110846336755term_a,R: fset_P6640269656128473term_a,S: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ Xz @ ( fSet_f5283962319602365193term_a @ R @ S ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a,Z2: bot_bot_term_a] :
            ( ( Xz
              = ( produc780340308816795563term_a @ X3 @ Z2 ) )
           => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R )
             => ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Y3 @ Z2 ) @ S ) ) ) ) ).

% frelcompE
thf(fact_499_fImageI,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
     => ( ( fmembe1418995468851910914term_a @ A @ A4 )
       => ( fmembe1418995468851910914term_a @ B @ ( fSet_f5750617162473777234term_a @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_500_fImageI,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b,A4: fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( ( fmember_term_a_b @ A @ A4 )
       => ( fmember_term_a_b @ B @ ( fSet_f8210585146631509224rm_a_b @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_501_fImageI,axiom,
    ! [A: nat,B: c,R: fset_P5318063784554775253_nat_c,A4: fset_nat] :
      ( ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ A @ B ) @ R )
     => ( ( fmember_nat @ A @ A4 )
       => ( fmember_c @ B @ ( fSet_fImage_nat_c @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_502_fImageI,axiom,
    ! [A: nat,B: b,R: fset_P5247029744508429268_nat_b,A4: fset_nat] :
      ( ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ A @ B ) @ R )
     => ( ( fmember_nat @ A @ A4 )
       => ( fmember_b @ B @ ( fSet_fImage_nat_b @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_503_fImageI,axiom,
    ! [A: c,B: nat,R: fset_P7886154461505542575_c_nat,A4: fset_c] :
      ( ( fmembe3088975721132771320_c_nat @ ( product_Pair_c_nat @ A @ B ) @ R )
     => ( ( fmember_c @ A @ A4 )
       => ( fmember_nat @ B @ ( fSet_fImage_c_nat @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_504_fImageI,axiom,
    ! [A: c,B: b,R: fset_P7943071700152676876od_c_b,A4: fset_c] :
      ( ( fmembe2163177356717951981od_c_b @ ( product_Pair_c_b @ A @ B ) @ R )
     => ( ( fmember_c @ A @ A4 )
       => ( fmember_b @ B @ ( fSet_fImage_c_b @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_505_fImageI,axiom,
    ! [A: b,B: nat,R: fset_P2289935846317411886_b_nat,A4: fset_b] :
      ( ( fmembe1853531392176376823_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R )
     => ( ( fmember_b @ A @ A4 )
       => ( fmember_nat @ B @ ( fSet_fImage_b_nat @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_506_fImageI,axiom,
    ! [A: b,B: c,R: fset_P8649754740804009550od_b_c,A4: fset_b] :
      ( ( fmembe4950632942993856815od_b_c @ ( product_Pair_b_c @ A @ B ) @ R )
     => ( ( fmember_b @ A @ A4 )
       => ( fmember_c @ B @ ( fSet_fImage_b_c @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_507_fImageI,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat,A4: fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( fmember_nat @ A @ A4 )
       => ( fmember_nat @ B @ ( fSet_fImage_nat_nat @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_508_fImageI,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c,A4: fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
     => ( ( fmember_c @ A @ A4 )
       => ( fmember_c @ B @ ( fSet_fImage_c_c @ R @ A4 ) ) ) ) ).

% fImageI
thf(fact_509_ground__fvars__term__empty,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ( ( fSet_fvars_term_a_b @ T )
        = bot_bot_fset_b ) ) ).

% ground_fvars_term_empty
thf(fact_510_ground__fvars__term__empty,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fSet_f2604459547217817330term_a @ T )
        = bot_bo4313458260509495885term_a ) ) ).

% ground_fvars_term_empty
thf(fact_511_ground__fvars__term__empty,axiom,
    ! [T: term_a_term_a_b] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fSet_f7033438052906747389rm_a_b @ T )
        = bot_bo1583650863486268376rm_a_b ) ) ).

% ground_fvars_term_empty
thf(fact_512_ground__fvars__term__empty,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( fSet_fvars_term_a_c @ T )
        = bot_bot_fset_c ) ) ).

% ground_fvars_term_empty
thf(fact_513_ta__der_H__ta__der,axiom,
    ! [T: term_a4617543059560371146term_a,A6: tree_t5222909361115825503rm_a_a,S: term_a4617543059560371146term_a,P2: bot_bot_term_a] :
      ( ( fmembe4236371882286196753term_a @ T @ ( tree_t5754163232548924128rm_a_a @ A6 @ S ) )
     => ( ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
       => ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ S ) ) ) ) ).

% ta_der'_ta_der
thf(fact_514_ta__der_H__ta__der,axiom,
    ! [T: term_a_c,A6: tree_ta_c_a,S: term_a_c,P2: c] :
      ( ( fmember_term_a_c @ T @ ( tree_ta_der_c_a2 @ A6 @ S ) )
     => ( ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A6 @ T ) )
       => ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A6 @ S ) ) ) ) ).

% ta_der'_ta_der
thf(fact_515_ta__der_H__ta__der,axiom,
    ! [T: term_a_term_a_b,A6: tree_ta_term_a_b_a,S: term_a_term_a_b,P2: term_a_b] :
      ( ( fmembe6890459216773255580rm_a_b @ T @ ( tree_t5237966800154138261_a_b_a @ A6 @ S ) )
     => ( ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
       => ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ S ) ) ) ) ).

% ta_der'_ta_der
thf(fact_516_ta__der_H__ta__der,axiom,
    ! [T: term_a_b,A6: tree_ta_b_a,S: term_a_b,P2: b] :
      ( ( fmember_term_a_b @ T @ ( tree_ta_der_b_a2 @ A6 @ S ) )
     => ( ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A6 @ T ) )
       => ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A6 @ S ) ) ) ) ).

% ta_der'_ta_der
thf(fact_517_ta__reachableI,axiom,
    ! [T: term_a_term_a_b,Q2: term_a_b,A6: tree_ta_term_a_b_a] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
       => ( fmember_term_a_b @ Q2 @ ( tree_t1746638234261971074_a_b_a @ A6 ) ) ) ) ).

% ta_reachableI
thf(fact_518_ta__reachableI,axiom,
    ! [T: term_a_b,Q2: b,A6: tree_ta_b_a] :
      ( ( term_ground_a_b @ T )
     => ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T ) )
       => ( fmember_b @ Q2 @ ( tree_t1956540421582741029le_b_a @ A6 ) ) ) ) ).

% ta_reachableI
thf(fact_519_ta__reachableI,axiom,
    ! [T: term_a4617543059560371146term_a,Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( term_g4884896889520838057term_a @ T )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t3813786739961944781rm_a_a @ A6 ) ) ) ) ).

% ta_reachableI
thf(fact_520_ta__reachableI,axiom,
    ! [T: term_a_c,Q2: c,A6: tree_ta_c_a] :
      ( ( term_ground_a_c @ T )
     => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T ) )
       => ( fmember_c @ Q2 @ ( tree_t8392456876464840804le_c_a @ A6 ) ) ) ) ).

% ta_reachableI
thf(fact_521_subterm_OacyclicI__order,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F4: nat > term_a_c] :
      ( ! [A3: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R )
         => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_c ) )
     => ( transi3139058337892597825ic_nat @ R ) ) ).

% subterm.acyclicI_order
thf(fact_522_subterm_OacyclicI__order,axiom,
    ! [R: set_Product_prod_c_c,F4: c > term_a_c] :
      ( ! [A3: c,B3: c] :
          ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A3 @ B3 ) @ R )
         => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_c ) )
     => ( transitive_acyclic_c @ R ) ) ).

% subterm.acyclicI_order
thf(fact_523_subterm_OacyclicI__order,axiom,
    ! [R: set_Product_prod_b_b,F4: b > term_a_c] :
      ( ! [A3: b,B3: b] :
          ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R )
         => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_c ) )
     => ( transitive_acyclic_b @ R ) ) ).

% subterm.acyclicI_order
thf(fact_524_subterm_OacyclicI__order,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F4: nat > term_a_b] :
      ( ! [A3: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_b ) )
     => ( transi3139058337892597825ic_nat @ R ) ) ).

% subterm.acyclicI_order
thf(fact_525_subterm_OacyclicI__order,axiom,
    ! [R: set_Product_prod_c_c,F4: c > term_a_b] :
      ( ! [A3: c,B3: c] :
          ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A3 @ B3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_b ) )
     => ( transitive_acyclic_c @ R ) ) ).

% subterm.acyclicI_order
thf(fact_526_subterm_OacyclicI__order,axiom,
    ! [R: set_Product_prod_b_b,F4: b > term_a_b] :
      ( ! [A3: b,B3: b] :
          ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_b ) )
     => ( transitive_acyclic_b @ R ) ) ).

% subterm.acyclicI_order
thf(fact_527_subterm_OacyclicI__order,axiom,
    ! [R: set_Pr3965413743549523539term_a,F4: bot_bot_term_a > term_a_c] :
      ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a] :
          ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A3 @ B3 ) @ R )
         => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_c ) )
     => ( transi4548540958286804730term_a @ R ) ) ).

% subterm.acyclicI_order
thf(fact_528_subterm_OacyclicI__order,axiom,
    ! [R: set_Pr3965413743549523539term_a,F4: bot_bot_term_a > term_a_b] :
      ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a] :
          ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A3 @ B3 ) @ R )
         => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_b ) )
     => ( transi4548540958286804730term_a @ R ) ) ).

% subterm.acyclicI_order
thf(fact_529_subterm_OacyclicI__order,axiom,
    ! [R: set_Pr7334014819586186091rm_a_c,F4: term_a_c > term_a_c] :
      ( ! [A3: term_a_c,B3: term_a_c] :
          ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ A3 @ B3 ) @ R )
         => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_c ) )
     => ( transi5314701264037997958rm_a_c @ R ) ) ).

% subterm.acyclicI_order
thf(fact_530_subterm_OacyclicI__order,axiom,
    ! [R: set_Pr4386577575007340137rm_a_b,F4: term_a_b > term_a_c] :
      ( ! [A3: term_a_b,B3: term_a_b] :
          ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R )
         => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( F4 @ A3 ) @ ( F4 @ B3 ) ) @ subterm_and_supt_a_c ) )
     => ( transi5314701259734769157rm_a_b @ R ) ) ).

% subterm.acyclicI_order
thf(fact_531_ground__ctxt__apply,axiom,
    ! [C: subter720051973439326747term_a,T: term_a4617543059560371146term_a] :
      ( ( term_g4884896889520838057term_a @ ( subter8974050253859094258term_a @ C @ T ) )
      = ( ( term_g5220492397686563420term_a @ C )
        & ( term_g4884896889520838057term_a @ T ) ) ) ).

% ground_ctxt_apply
thf(fact_532_ground__ctxt__apply,axiom,
    ! [C: subter2881564190832396198rm_a_b,T: term_a_term_a_b] :
      ( ( term_g2255532262831151796rm_a_b @ ( subter7263177522250274301rm_a_b @ C @ T ) )
      = ( ( term_g161376099170733415rm_a_b @ C )
        & ( term_g2255532262831151796rm_a_b @ T ) ) ) ).

% ground_ctxt_apply
thf(fact_533_ground__ctxt__apply,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( term_ground_a_b @ ( subter2376574525758040790rm_a_b @ C @ T ) )
      = ( ( term_ground_ctxt_a_b @ C )
        & ( term_ground_a_b @ T ) ) ) ).

% ground_ctxt_apply
thf(fact_534_ground__ctxt__apply,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( term_ground_a_c @ ( subter2376574525758040791rm_a_c @ C @ T ) )
      = ( ( term_ground_ctxt_a_c @ C )
        & ( term_ground_a_c @ T ) ) ) ).

% ground_ctxt_apply
thf(fact_535_ctxt__supt,axiom,
    ! [C: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
      ( ( C != subterm_and_Hole_a_b )
     => ( ( S
          = ( subter2376574525758040790rm_a_b @ C @ T ) )
       => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b ) ) ) ).

% ctxt_supt
thf(fact_536_ctxt__supt,axiom,
    ! [C: subterm_and_ctxt_a_c,S: term_a_c,T: term_a_c] :
      ( ( C != subterm_and_Hole_a_c )
     => ( ( S
          = ( subter2376574525758040791rm_a_c @ C @ T ) )
       => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c ) ) ) ).

% ctxt_supt
thf(fact_537_FSet__Utils_OImageE,axiom,
    ! [B: bot_bot_term_a,R: fset_P6640269656128473term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ B @ ( fSet_f5750617162473777234term_a @ R @ A4 ) )
     => ~ ! [X3: bot_bot_term_a] :
            ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X3 @ B ) @ R )
           => ~ ( fmembe1418995468851910914term_a @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_538_FSet__Utils_OImageE,axiom,
    ! [B: term_a_b,R: fset_P3577670608877728751rm_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ B @ ( fSet_f8210585146631509224rm_a_b @ R @ A4 ) )
     => ~ ! [X3: term_a_b] :
            ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ R )
           => ~ ( fmember_term_a_b @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_539_FSet__Utils_OImageE,axiom,
    ! [B: nat,R: fset_P7886154461505542575_c_nat,A4: fset_c] :
      ( ( fmember_nat @ B @ ( fSet_fImage_c_nat @ R @ A4 ) )
     => ~ ! [X3: c] :
            ( ( fmembe3088975721132771320_c_nat @ ( product_Pair_c_nat @ X3 @ B ) @ R )
           => ~ ( fmember_c @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_540_FSet__Utils_OImageE,axiom,
    ! [B: nat,R: fset_P2289935846317411886_b_nat,A4: fset_b] :
      ( ( fmember_nat @ B @ ( fSet_fImage_b_nat @ R @ A4 ) )
     => ~ ! [X3: b] :
            ( ( fmembe1853531392176376823_b_nat @ ( product_Pair_b_nat @ X3 @ B ) @ R )
           => ~ ( fmember_b @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_541_FSet__Utils_OImageE,axiom,
    ! [B: c,R: fset_P5318063784554775253_nat_c,A4: fset_nat] :
      ( ( fmember_c @ B @ ( fSet_fImage_nat_c @ R @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ X3 @ B ) @ R )
           => ~ ( fmember_nat @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_542_FSet__Utils_OImageE,axiom,
    ! [B: c,R: fset_P8649754740804009550od_b_c,A4: fset_b] :
      ( ( fmember_c @ B @ ( fSet_fImage_b_c @ R @ A4 ) )
     => ~ ! [X3: b] :
            ( ( fmembe4950632942993856815od_b_c @ ( product_Pair_b_c @ X3 @ B ) @ R )
           => ~ ( fmember_b @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_543_FSet__Utils_OImageE,axiom,
    ! [B: b,R: fset_P5247029744508429268_nat_b,A4: fset_nat] :
      ( ( fmember_b @ B @ ( fSet_fImage_nat_b @ R @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ X3 @ B ) @ R )
           => ~ ( fmember_nat @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_544_FSet__Utils_OImageE,axiom,
    ! [B: b,R: fset_P7943071700152676876od_c_b,A4: fset_c] :
      ( ( fmember_b @ B @ ( fSet_fImage_c_b @ R @ A4 ) )
     => ~ ! [X3: c] :
            ( ( fmembe2163177356717951981od_c_b @ ( product_Pair_c_b @ X3 @ B ) @ R )
           => ~ ( fmember_c @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_545_FSet__Utils_OImageE,axiom,
    ! [B: nat,R: fset_P5670320511379867111at_nat,A4: fset_nat] :
      ( ( fmember_nat @ B @ ( fSet_fImage_nat_nat @ R @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R )
           => ~ ( fmember_nat @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_546_FSet__Utils_OImageE,axiom,
    ! [B: c,R: fset_P8014105740199022861od_c_c,A4: fset_c] :
      ( ( fmember_c @ B @ ( fSet_fImage_c_c @ R @ A4 ) )
     => ~ ! [X3: c] :
            ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X3 @ B ) @ R )
           => ~ ( fmember_c @ X3 @ A4 ) ) ) ).

% FSet_Utils.ImageE
thf(fact_547_frev__ImageI,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ A4 )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
       => ( fmembe1418995468851910914term_a @ B @ ( fSet_f5750617162473777234term_a @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_548_frev__ImageI,axiom,
    ! [A: term_a_b,A4: fset_term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
       => ( fmember_term_a_b @ B @ ( fSet_f8210585146631509224rm_a_b @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_549_frev__ImageI,axiom,
    ! [A: nat,A4: fset_nat,B: c,R: fset_P5318063784554775253_nat_c] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ A @ B ) @ R )
       => ( fmember_c @ B @ ( fSet_fImage_nat_c @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_550_frev__ImageI,axiom,
    ! [A: nat,A4: fset_nat,B: b,R: fset_P5247029744508429268_nat_b] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ A @ B ) @ R )
       => ( fmember_b @ B @ ( fSet_fImage_nat_b @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_551_frev__ImageI,axiom,
    ! [A: c,A4: fset_c,B: nat,R: fset_P7886154461505542575_c_nat] :
      ( ( fmember_c @ A @ A4 )
     => ( ( fmembe3088975721132771320_c_nat @ ( product_Pair_c_nat @ A @ B ) @ R )
       => ( fmember_nat @ B @ ( fSet_fImage_c_nat @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_552_frev__ImageI,axiom,
    ! [A: c,A4: fset_c,B: b,R: fset_P7943071700152676876od_c_b] :
      ( ( fmember_c @ A @ A4 )
     => ( ( fmembe2163177356717951981od_c_b @ ( product_Pair_c_b @ A @ B ) @ R )
       => ( fmember_b @ B @ ( fSet_fImage_c_b @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_553_frev__ImageI,axiom,
    ! [A: b,A4: fset_b,B: nat,R: fset_P2289935846317411886_b_nat] :
      ( ( fmember_b @ A @ A4 )
     => ( ( fmembe1853531392176376823_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R )
       => ( fmember_nat @ B @ ( fSet_fImage_b_nat @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_554_frev__ImageI,axiom,
    ! [A: b,A4: fset_b,B: c,R: fset_P8649754740804009550od_b_c] :
      ( ( fmember_b @ A @ A4 )
     => ( ( fmembe4950632942993856815od_b_c @ ( product_Pair_b_c @ A @ B ) @ R )
       => ( fmember_c @ B @ ( fSet_fImage_b_c @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_555_frev__ImageI,axiom,
    ! [A: nat,A4: fset_nat,B: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( fmember_nat @ B @ ( fSet_fImage_nat_nat @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_556_frev__ImageI,axiom,
    ! [A: c,A4: fset_c,B: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmember_c @ A @ A4 )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
       => ( fmember_c @ B @ ( fSet_fImage_c_c @ R @ A4 ) ) ) ) ).

% frev_ImageI
thf(fact_557_fImage__empty2,axiom,
    ! [R2: fset_P6640269656128473term_a] :
      ( ( fSet_f5750617162473777234term_a @ R2 @ bot_bo4313458260509495885term_a )
      = bot_bo4313458260509495885term_a ) ).

% fImage_empty2
thf(fact_558_fImage__empty2,axiom,
    ! [R2: fset_P3577670608877728751rm_a_b] :
      ( ( fSet_f8210585146631509224rm_a_b @ R2 @ bot_bo1583650863486268376rm_a_b )
      = bot_bo1583650863486268376rm_a_b ) ).

% fImage_empty2
thf(fact_559_adapt__vars__ctxt__simps_I2_J,axiom,
    ( ( term_a8194622999837494425_a_c_c @ subterm_and_Hole_a_c )
    = subterm_and_Hole_a_c ) ).

% adapt_vars_ctxt_simps(2)
thf(fact_560_fImage__empty1,axiom,
    ! [X5: fset_Bot_bot_term_a] :
      ( ( fSet_f5750617162473777234term_a @ bot_bo3304177514724984389term_a @ X5 )
      = bot_bo4313458260509495885term_a ) ).

% fImage_empty1
thf(fact_561_fImage__empty1,axiom,
    ! [X5: fset_term_a_b] :
      ( ( fSet_f8210585146631509224rm_a_b @ bot_bo173352615033564763rm_a_b @ X5 )
      = bot_bo1583650863486268376rm_a_b ) ).

% fImage_empty1
thf(fact_562_ground__ctxt_Osimps_I1_J,axiom,
    term_ground_ctxt_a_c @ subterm_and_Hole_a_c ).

% ground_ctxt.simps(1)
thf(fact_563_ctxt__apply__term_Osimps_I1_J,axiom,
    ! [S: term_a_b] :
      ( ( subter2376574525758040790rm_a_b @ subterm_and_Hole_a_b @ S )
      = S ) ).

% ctxt_apply_term.simps(1)
thf(fact_564_ctxt__apply__term_Osimps_I1_J,axiom,
    ! [S: term_a_c] :
      ( ( subter2376574525758040791rm_a_c @ subterm_and_Hole_a_c @ S )
      = S ) ).

% ctxt_apply_term.simps(1)
thf(fact_565_subst__apply__ctxt_Osimps_I1_J,axiom,
    ! [Sigma: c > term_a_c] :
      ( ( subter1298571438366709515_a_c_c @ subterm_and_Hole_a_c @ Sigma )
      = subterm_and_Hole_a_c ) ).

% subst_apply_ctxt.simps(1)
thf(fact_566_subst__apply__ctxt_Osimps_I1_J,axiom,
    ! [Sigma: c > term_a_b] :
      ( ( subter1298571438366709514_a_c_b @ subterm_and_Hole_a_c @ Sigma )
      = subterm_and_Hole_a_b ) ).

% subst_apply_ctxt.simps(1)
thf(fact_567_subst__apply__ctxt_Osimps_I1_J,axiom,
    ! [Sigma: b > term_a_c] :
      ( ( subter4086027020339385548_a_b_c @ subterm_and_Hole_a_b @ Sigma )
      = subterm_and_Hole_a_c ) ).

% subst_apply_ctxt.simps(1)
thf(fact_568_subst__apply__ctxt_Osimps_I1_J,axiom,
    ! [Sigma: b > term_a_b] :
      ( ( subter4086027020339385547_a_b_b @ subterm_and_Hole_a_b @ Sigma )
      = subterm_and_Hole_a_b ) ).

% subst_apply_ctxt.simps(1)
thf(fact_569_ta__der_H__trans,axiom,
    ! [T: term_a_c,A6: tree_ta_c_a,S: term_a_c,U: term_a_c] :
      ( ( fmember_term_a_c @ T @ ( tree_ta_der_c_a2 @ A6 @ S ) )
     => ( ( fmember_term_a_c @ U @ ( tree_ta_der_c_a2 @ A6 @ T ) )
       => ( fmember_term_a_c @ U @ ( tree_ta_der_c_a2 @ A6 @ S ) ) ) ) ).

% ta_der'_trans
thf(fact_570_ta__der_H__trans,axiom,
    ! [T: term_a_b,A6: tree_ta_b_a,S: term_a_b,U: term_a_b] :
      ( ( fmember_term_a_b @ T @ ( tree_ta_der_b_a2 @ A6 @ S ) )
     => ( ( fmember_term_a_b @ U @ ( tree_ta_der_b_a2 @ A6 @ T ) )
       => ( fmember_term_a_b @ U @ ( tree_ta_der_b_a2 @ A6 @ S ) ) ) ) ).

% ta_der'_trans
thf(fact_571_ta__der_H__refl,axiom,
    ! [T: term_a_c,A6: tree_ta_c_a] : ( fmember_term_a_c @ T @ ( tree_ta_der_c_a2 @ A6 @ T ) ) ).

% ta_der'_refl
thf(fact_572_ta__der_H__refl,axiom,
    ! [T: term_a_b,A6: tree_ta_b_a] : ( fmember_term_a_b @ T @ ( tree_ta_der_b_a2 @ A6 @ T ) ) ).

% ta_der'_refl
thf(fact_573_supt__ctxtE,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
     => ~ ! [C3: subterm_and_ctxt_a_b] :
            ( ( C3 != subterm_and_Hole_a_b )
           => ( S
             != ( subter2376574525758040790rm_a_b @ C3 @ T ) ) ) ) ).

% supt_ctxtE
thf(fact_574_supt__ctxtE,axiom,
    ! [S: term_a_c,T: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
     => ~ ! [C3: subterm_and_ctxt_a_c] :
            ( ( C3 != subterm_and_Hole_a_c )
           => ( S
             != ( subter2376574525758040791rm_a_c @ C3 @ T ) ) ) ) ).

% supt_ctxtE
thf(fact_575_supt__ctxt__conv,axiom,
    ! [S: term_a_b,T: term_a_b] :
      ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
      = ( ? [C4: subterm_and_ctxt_a_b] :
            ( ( C4 != subterm_and_Hole_a_b )
            & ( S
              = ( subter2376574525758040790rm_a_b @ C4 @ T ) ) ) ) ) ).

% supt_ctxt_conv
thf(fact_576_supt__ctxt__conv,axiom,
    ! [S: term_a_c,T: term_a_c] :
      ( ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ S @ T ) @ subterm_and_supt_a_c )
      = ( ? [C4: subterm_and_ctxt_a_c] :
            ( ( C4 != subterm_and_Hole_a_c )
            & ( S
              = ( subter2376574525758040791rm_a_c @ C4 @ T ) ) ) ) ) ).

% supt_ctxt_conv
thf(fact_577_nectxt__imp__supt__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( C != subterm_and_Hole_a_b )
     => ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C @ T ) @ T ) @ subterm_and_supt_a_b ) ) ).

% nectxt_imp_supt_ctxt
thf(fact_578_nectxt__imp__supt__ctxt,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( C != subterm_and_Hole_a_c )
     => ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( subter2376574525758040791rm_a_c @ C @ T ) @ T ) @ subterm_and_supt_a_c ) ) ).

% nectxt_imp_supt_ctxt
thf(fact_579_ta__reachableE,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t1746638234261971074_a_b_a @ A6 ) )
     => ~ ! [T2: term_a_term_a_b] :
            ( ( term_g2255532262831151796rm_a_b @ T2 )
           => ~ ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T2 ) ) ) ) ).

% ta_reachableE
thf(fact_580_ta__reachableE,axiom,
    ! [Q2: b,A6: tree_ta_b_a] :
      ( ( fmember_b @ Q2 @ ( tree_t1956540421582741029le_b_a @ A6 ) )
     => ~ ! [T2: term_a_b] :
            ( ( term_ground_a_b @ T2 )
           => ~ ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T2 ) ) ) ) ).

% ta_reachableE
thf(fact_581_ta__reachableE,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t3813786739961944781rm_a_a @ A6 ) )
     => ~ ! [T2: term_a4617543059560371146term_a] :
            ( ( term_g4884896889520838057term_a @ T2 )
           => ~ ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T2 ) ) ) ) ).

% ta_reachableE
thf(fact_582_ta__reachableE,axiom,
    ! [Q2: c,A6: tree_ta_c_a] :
      ( ( fmember_c @ Q2 @ ( tree_t8392456876464840804le_c_a @ A6 ) )
     => ~ ! [T2: term_a_c] :
            ( ( term_ground_a_c @ T2 )
           => ~ ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T2 ) ) ) ) ).

% ta_reachableE
thf(fact_583_fImage__singleton__iff,axiom,
    ! [B: bot_bot_term_a,R: fset_P6640269656128473term_a,A: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ B @ ( fSet_f5750617162473777234term_a @ R @ ( finser8320498192367885249term_a @ A @ bot_bo4313458260509495885term_a ) ) )
      = ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_584_fImage__singleton__iff,axiom,
    ! [B: c,R: fset_P5318063784554775253_nat_c,A: nat] :
      ( ( fmember_c @ B @ ( fSet_fImage_nat_c @ R @ ( finsert_nat @ A @ bot_bot_fset_nat ) ) )
      = ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_585_fImage__singleton__iff,axiom,
    ! [B: b,R: fset_P5247029744508429268_nat_b,A: nat] :
      ( ( fmember_b @ B @ ( fSet_fImage_nat_b @ R @ ( finsert_nat @ A @ bot_bot_fset_nat ) ) )
      = ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_586_fImage__singleton__iff,axiom,
    ! [B: nat,R: fset_P5670320511379867111at_nat,A: nat] :
      ( ( fmember_nat @ B @ ( fSet_fImage_nat_nat @ R @ ( finsert_nat @ A @ bot_bot_fset_nat ) ) )
      = ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_587_fImage__singleton__iff,axiom,
    ! [B: c,R: fset_P8014105740199022861od_c_c,A: c] :
      ( ( fmember_c @ B @ ( fSet_fImage_c_c @ R @ ( finsert_c @ A @ bot_bot_fset_c ) ) )
      = ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_588_fImage__singleton__iff,axiom,
    ! [B: b,R: fset_P8578720700757663565od_b_b,A: b] :
      ( ( fmember_b @ B @ ( fSet_fImage_b_b @ R @ ( finsert_b @ A @ bot_bot_fset_b ) ) )
      = ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_589_fImage__singleton__iff,axiom,
    ! [B: nat,R: fset_P4615588779303864480_a_nat,A: bot_bot_term_a] :
      ( ( fmember_nat @ B @ ( fSet_f3412622471138897305_a_nat @ R @ ( finser8320498192367885249term_a @ A @ bot_bo4313458260509495885term_a ) ) )
      = ( fmembe4530531765636674689_a_nat @ ( produc6633979256947609458_a_nat @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_590_fImage__singleton__iff,axiom,
    ! [B: c,R: fset_P7911707877349473372rm_a_c,A: bot_bot_term_a] :
      ( ( fmember_c @ B @ ( fSet_f6643166002469431671rm_a_c @ R @ ( finser8320498192367885249term_a @ A @ bot_bo4313458260509495885term_a ) ) )
      = ( fmembe2649604074571300645rm_a_c @ ( produc5551744644817243102rm_a_c @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_591_fImage__singleton__iff,axiom,
    ! [B: b,R: fset_P7840673837303127387rm_a_b,A: bot_bot_term_a] :
      ( ( fmember_b @ B @ ( fSet_f6643166002469431670rm_a_b @ R @ ( finser8320498192367885249term_a @ A @ bot_bo4313458260509495885term_a ) ) )
      = ( fmembe2649604070268071844rm_a_b @ ( produc5551744644817243101rm_a_b @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_592_fImage__singleton__iff,axiom,
    ! [B: term_a_b,R: fset_P1023298988895795115rm_a_b,A: nat] :
      ( ( fmember_term_a_b @ B @ ( fSet_f8842252047629264932rm_a_b @ R @ ( finsert_nat @ A @ bot_bot_fset_nat ) ) )
      = ( fmembe6134593446672421132rm_a_b @ ( produc1516572978046417917rm_a_b @ A @ B ) @ R ) ) ).

% fImage_singleton_iff
thf(fact_593_ta__der__to__ta__der_H,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,S: term_a4617543059560371146term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ S ) )
      = ( fmembe4236371882286196753term_a @ ( var_Bot_bot_term_a_a @ Q2 ) @ ( tree_t5754163232548924128rm_a_a @ A6 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_594_ta__der__to__ta__der_H,axiom,
    ! [Q2: c,A6: tree_ta_c_a,S: term_a_c] :
      ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ S ) )
      = ( fmember_term_a_c @ ( var_c_a @ Q2 ) @ ( tree_ta_der_c_a2 @ A6 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_595_ta__der__to__ta__der_H,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a,S: term_a_term_a_b] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ S ) )
      = ( fmembe6890459216773255580rm_a_b @ ( var_term_a_b_a @ Q2 ) @ ( tree_t5237966800154138261_a_b_a @ A6 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_596_ta__der__to__ta__der_H,axiom,
    ! [Q2: b,A6: tree_ta_b_a,S: term_a_b] :
      ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ S ) )
      = ( fmember_term_a_b @ ( var_b_a @ Q2 ) @ ( tree_ta_der_b_a2 @ A6 @ S ) ) ) ).

% ta_der_to_ta_der'
thf(fact_597_ta__der_H__to__ta__der,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,S: term_a4617543059560371146term_a] :
      ( ( fmembe4236371882286196753term_a @ ( var_Bot_bot_term_a_a @ Q2 ) @ ( tree_t5754163232548924128rm_a_a @ A6 @ S ) )
     => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_598_ta__der_H__to__ta__der,axiom,
    ! [Q2: c,A6: tree_ta_c_a,S: term_a_c] :
      ( ( fmember_term_a_c @ ( var_c_a @ Q2 ) @ ( tree_ta_der_c_a2 @ A6 @ S ) )
     => ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_599_ta__der_H__to__ta__der,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a,S: term_a_term_a_b] :
      ( ( fmembe6890459216773255580rm_a_b @ ( var_term_a_b_a @ Q2 ) @ ( tree_t5237966800154138261_a_b_a @ A6 @ S ) )
     => ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_600_ta__der_H__to__ta__der,axiom,
    ! [Q2: b,A6: tree_ta_b_a,S: term_a_b] :
      ( ( fmember_term_a_b @ ( var_b_a @ Q2 ) @ ( tree_ta_der_b_a2 @ A6 @ S ) )
     => ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ S ) ) ) ).

% ta_der'_to_ta_der
thf(fact_601_ta__der_H__empty,axiom,
    ! [T: term_a_c,S: term_a_c] :
      ( ( fmember_term_a_c @ T @ ( tree_ta_der_c_a2 @ ( tree_TA_c_a @ bot_bo5716954315427840240le_c_a @ bot_bo8780685805445091705od_c_c ) @ S ) )
     => ( T = S ) ) ).

% ta_der'_empty
thf(fact_602_ta__der_H__empty,axiom,
    ! [T: term_a_b,S: term_a_b] :
      ( ( fmember_term_a_b @ T @ ( tree_ta_der_b_a2 @ ( tree_TA_b_a @ bot_bo6352603316032826929le_b_a @ bot_bo121928729148956601od_b_b ) @ S ) )
     => ( T = S ) ) ).

% ta_der'_empty
thf(fact_603_ta__reachableI__eps,axiom,
    ! [Q2: b,A6: tree_ta_b_a,Q4: b] :
      ( ( fmember_b @ Q2 @ ( tree_t1956540421582741029le_b_a @ A6 ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Q2 @ Q4 ) @ ( tree_eps_b_a @ A6 ) )
       => ( fmember_b @ Q4 @ ( tree_t1956540421582741029le_b_a @ A6 ) ) ) ) ).

% ta_reachableI_eps
thf(fact_604_ta__reachableI__eps,axiom,
    ! [Q2: c,A6: tree_ta_c_a,Q4: c] :
      ( ( fmember_c @ Q2 @ ( tree_t8392456876464840804le_c_a @ A6 ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Q2 @ Q4 ) @ ( tree_eps_c_a @ A6 ) )
       => ( fmember_c @ Q4 @ ( tree_t8392456876464840804le_c_a @ A6 ) ) ) ) ).

% ta_reachableI_eps
thf(fact_605_ta__reachableI__eps,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,Q4: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t3813786739961944781rm_a_a @ A6 ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Q2 @ Q4 ) @ ( tree_e5393235435095066050rm_a_a @ A6 ) )
       => ( fmembe1418995468851910914term_a @ Q4 @ ( tree_t3813786739961944781rm_a_a @ A6 ) ) ) ) ).

% ta_reachableI_eps
thf(fact_606_ta__reachable__gtermI,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a,T: ground_gterm_a] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( ground678623728659755670rm_a_b @ T ) ) )
     => ( fmember_term_a_b @ Q2 @ ( tree_t1746638234261971074_a_b_a @ A6 ) ) ) ).

% ta_reachable_gtermI
thf(fact_607_ta__reachable__gtermI,axiom,
    ! [Q2: b,A6: tree_ta_b_a,T: ground_gterm_a] :
      ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( ground5786051259813020157rm_a_b @ T ) ) )
     => ( fmember_b @ Q2 @ ( tree_t1956540421582741029le_b_a @ A6 ) ) ) ).

% ta_reachable_gtermI
thf(fact_608_ta__reachable__gtermI,axiom,
    ! [Q2: c,A6: tree_ta_c_a,T: ground_gterm_a] :
      ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( ground5786051259813020158rm_a_c @ T ) ) )
     => ( fmember_c @ Q2 @ ( tree_t8392456876464840804le_c_a @ A6 ) ) ) ).

% ta_reachable_gtermI
thf(fact_609_ta__reachable__gtermI,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,T: ground_gterm_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( ground2947561787367041419term_a @ T ) ) )
     => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t3813786739961944781rm_a_a @ A6 ) ) ) ).

% ta_reachable_gtermI
thf(fact_610_ta__der__eps,axiom,
    ! [P2: bot_bot_term_a,Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,T: term_a4617543059560371146term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ P2 @ Q2 ) @ ( tree_e5393235435095066050rm_a_a @ A6 ) )
     => ( ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) ) ) ) ).

% ta_der_eps
thf(fact_611_ta__der__eps,axiom,
    ! [P2: c,Q2: c,A6: tree_ta_c_a,T: term_a_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( tree_eps_c_a @ A6 ) )
     => ( ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A6 @ T ) )
       => ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T ) ) ) ) ).

% ta_der_eps
thf(fact_612_ta__der__eps,axiom,
    ! [P2: term_a_b,Q2: term_a_b,A6: tree_ta_term_a_b_a,T: term_a_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ P2 @ Q2 ) @ ( tree_eps_term_a_b_a @ A6 ) )
     => ( ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
       => ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) ) ) ) ).

% ta_der_eps
thf(fact_613_ta__der__eps,axiom,
    ! [P2: b,Q2: b,A6: tree_ta_b_a,T: term_a_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( tree_eps_b_a @ A6 ) )
     => ( ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A6 @ T ) )
       => ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T ) ) ) ) ).

% ta_der_eps
thf(fact_614_fmem__Sigma__iff,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a,B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f5829356563446949852term_a @ A4 @ B4 ) )
      = ( ( fmembe1418995468851910914term_a @ A @ A4 )
        & ( fmembe1418995468851910914term_a @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_615_fmem__Sigma__iff,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b,B4: term_a_b > fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f1273034697330015346rm_a_b @ A4 @ B4 ) )
      = ( ( fmember_term_a_b @ A @ A4 )
        & ( fmember_term_a_b @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_616_fmem__Sigma__iff,axiom,
    ! [A: nat,B: c,A4: fset_nat,B4: nat > fset_c] :
      ( ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ A @ B ) @ ( fSet_fSigma_nat_c @ A4 @ B4 ) )
      = ( ( fmember_nat @ A @ A4 )
        & ( fmember_c @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_617_fmem__Sigma__iff,axiom,
    ! [A: nat,B: b,A4: fset_nat,B4: nat > fset_b] :
      ( ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ A @ B ) @ ( fSet_fSigma_nat_b @ A4 @ B4 ) )
      = ( ( fmember_nat @ A @ A4 )
        & ( fmember_b @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_618_fmem__Sigma__iff,axiom,
    ! [A: c,B: nat,A4: fset_c,B4: c > fset_nat] :
      ( ( fmembe3088975721132771320_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( fSet_fSigma_c_nat @ A4 @ B4 ) )
      = ( ( fmember_c @ A @ A4 )
        & ( fmember_nat @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_619_fmem__Sigma__iff,axiom,
    ! [A: c,B: b,A4: fset_c,B4: c > fset_b] :
      ( ( fmembe2163177356717951981od_c_b @ ( product_Pair_c_b @ A @ B ) @ ( fSet_fSigma_c_b @ A4 @ B4 ) )
      = ( ( fmember_c @ A @ A4 )
        & ( fmember_b @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_620_fmem__Sigma__iff,axiom,
    ! [A: b,B: nat,A4: fset_b,B4: b > fset_nat] :
      ( ( fmembe1853531392176376823_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( fSet_fSigma_b_nat @ A4 @ B4 ) )
      = ( ( fmember_b @ A @ A4 )
        & ( fmember_nat @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_621_fmem__Sigma__iff,axiom,
    ! [A: b,B: c,A4: fset_b,B4: b > fset_c] :
      ( ( fmembe4950632942993856815od_b_c @ ( product_Pair_b_c @ A @ B ) @ ( fSet_fSigma_b_c @ A4 @ B4 ) )
      = ( ( fmember_b @ A @ A4 )
        & ( fmember_c @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_622_fmem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A4: fset_nat,B4: nat > fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fSigma_nat_nat @ A4 @ B4 ) )
      = ( ( fmember_nat @ A @ A4 )
        & ( fmember_nat @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_623_fmem__Sigma__iff,axiom,
    ! [A: c,B: c,A4: fset_c,B4: c > fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fSigma_c_c @ A4 @ B4 ) )
      = ( ( fmember_c @ A @ A4 )
        & ( fmember_c @ B @ ( B4 @ A ) ) ) ) ).

% fmem_Sigma_iff
thf(fact_624_fSigmaI,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a,B: bot_bot_term_a,B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ A4 )
     => ( ( fmembe1418995468851910914term_a @ B @ ( B4 @ A ) )
       => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f5829356563446949852term_a @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_625_fSigmaI,axiom,
    ! [A: term_a_b,A4: fset_term_a_b,B: term_a_b,B4: term_a_b > fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ( ( fmember_term_a_b @ B @ ( B4 @ A ) )
       => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f1273034697330015346rm_a_b @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_626_fSigmaI,axiom,
    ! [A: nat,A4: fset_nat,B: c,B4: nat > fset_c] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( fmember_c @ B @ ( B4 @ A ) )
       => ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ A @ B ) @ ( fSet_fSigma_nat_c @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_627_fSigmaI,axiom,
    ! [A: nat,A4: fset_nat,B: b,B4: nat > fset_b] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( fmember_b @ B @ ( B4 @ A ) )
       => ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ A @ B ) @ ( fSet_fSigma_nat_b @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_628_fSigmaI,axiom,
    ! [A: c,A4: fset_c,B: nat,B4: c > fset_nat] :
      ( ( fmember_c @ A @ A4 )
     => ( ( fmember_nat @ B @ ( B4 @ A ) )
       => ( fmembe3088975721132771320_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( fSet_fSigma_c_nat @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_629_fSigmaI,axiom,
    ! [A: c,A4: fset_c,B: b,B4: c > fset_b] :
      ( ( fmember_c @ A @ A4 )
     => ( ( fmember_b @ B @ ( B4 @ A ) )
       => ( fmembe2163177356717951981od_c_b @ ( product_Pair_c_b @ A @ B ) @ ( fSet_fSigma_c_b @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_630_fSigmaI,axiom,
    ! [A: b,A4: fset_b,B: nat,B4: b > fset_nat] :
      ( ( fmember_b @ A @ A4 )
     => ( ( fmember_nat @ B @ ( B4 @ A ) )
       => ( fmembe1853531392176376823_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( fSet_fSigma_b_nat @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_631_fSigmaI,axiom,
    ! [A: b,A4: fset_b,B: c,B4: b > fset_c] :
      ( ( fmember_b @ A @ A4 )
     => ( ( fmember_c @ B @ ( B4 @ A ) )
       => ( fmembe4950632942993856815od_b_c @ ( product_Pair_b_c @ A @ B ) @ ( fSet_fSigma_b_c @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_632_fSigmaI,axiom,
    ! [A: nat,A4: fset_nat,B: nat,B4: nat > fset_nat] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( fmember_nat @ B @ ( B4 @ A ) )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fSigma_nat_nat @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_633_fSigmaI,axiom,
    ! [A: c,A4: fset_c,B: c,B4: c > fset_c] :
      ( ( fmember_c @ A @ A4 )
     => ( ( fmember_c @ B @ ( B4 @ A ) )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fSigma_c_c @ A4 @ B4 ) ) ) ) ).

% fSigmaI
thf(fact_634_term_Oinject_I1_J,axiom,
    ! [X1: b,Y1: b] :
      ( ( ( var_b_a @ X1 )
        = ( var_b_a @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% term.inject(1)
thf(fact_635_term_Oinject_I1_J,axiom,
    ! [X1: c,Y1: c] :
      ( ( ( var_c_a @ X1 )
        = ( var_c_a @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% term.inject(1)
thf(fact_636_finsert__absorb2,axiom,
    ! [X: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( finser8320498192367885249term_a @ X @ ( finser8320498192367885249term_a @ X @ A4 ) )
      = ( finser8320498192367885249term_a @ X @ A4 ) ) ).

% finsert_absorb2
thf(fact_637_finsert__absorb2,axiom,
    ! [X: term_a_b,A4: fset_term_a_b] :
      ( ( finsert_term_a_b @ X @ ( finsert_term_a_b @ X @ A4 ) )
      = ( finsert_term_a_b @ X @ A4 ) ) ).

% finsert_absorb2
thf(fact_638_finsert__absorb2,axiom,
    ! [X: nat,A4: fset_nat] :
      ( ( finsert_nat @ X @ ( finsert_nat @ X @ A4 ) )
      = ( finsert_nat @ X @ A4 ) ) ).

% finsert_absorb2
thf(fact_639_ta_Oinject,axiom,
    ! [X1: fset_T8606228319753941405le_b_a,X2: fset_P8578720700757663565od_b_b,Y1: fset_T8606228319753941405le_b_a,Y2: fset_P8578720700757663565od_b_b] :
      ( ( ( tree_TA_b_a @ X1 @ X2 )
        = ( tree_TA_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% ta.inject
thf(fact_640_finsertCI,axiom,
    ! [A: bot_bot_term_a,B4: fset_Bot_bot_term_a,B: bot_bot_term_a] :
      ( ( ~ ( fmembe1418995468851910914term_a @ A @ B4 )
       => ( A = B ) )
     => ( fmembe1418995468851910914term_a @ A @ ( finser8320498192367885249term_a @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_641_finsertCI,axiom,
    ! [A: term_a_c,B4: fset_term_a_c,B: term_a_c] :
      ( ( ~ ( fmember_term_a_c @ A @ B4 )
       => ( A = B ) )
     => ( fmember_term_a_c @ A @ ( finsert_term_a_c @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_642_finsertCI,axiom,
    ! [A: fset_term_a_b,B4: fset_fset_term_a_b,B: fset_term_a_b] :
      ( ( ~ ( fmembe3754333024224722621rm_a_b @ A @ B4 )
       => ( A = B ) )
     => ( fmembe3754333024224722621rm_a_b @ A @ ( finser5027672521447216124rm_a_b @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_643_finsertCI,axiom,
    ! [A: nat,B4: fset_nat,B: nat] :
      ( ( ~ ( fmember_nat @ A @ B4 )
       => ( A = B ) )
     => ( fmember_nat @ A @ ( finsert_nat @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_644_finsertCI,axiom,
    ! [A: c,B4: fset_c,B: c] :
      ( ( ~ ( fmember_c @ A @ B4 )
       => ( A = B ) )
     => ( fmember_c @ A @ ( finsert_c @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_645_finsertCI,axiom,
    ! [A: b,B4: fset_b,B: b] :
      ( ( ~ ( fmember_b @ A @ B4 )
       => ( A = B ) )
     => ( fmember_b @ A @ ( finsert_b @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_646_finsertCI,axiom,
    ! [A: term_a_b,B4: fset_term_a_b,B: term_a_b] :
      ( ( ~ ( fmember_term_a_b @ A @ B4 )
       => ( A = B ) )
     => ( fmember_term_a_b @ A @ ( finsert_term_a_b @ B @ B4 ) ) ) ).

% finsertCI
thf(fact_647_finsert__iff,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ ( finser8320498192367885249term_a @ B @ A4 ) )
      = ( ( A = B )
        | ( fmembe1418995468851910914term_a @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_648_finsert__iff,axiom,
    ! [A: term_a_c,B: term_a_c,A4: fset_term_a_c] :
      ( ( fmember_term_a_c @ A @ ( finsert_term_a_c @ B @ A4 ) )
      = ( ( A = B )
        | ( fmember_term_a_c @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_649_finsert__iff,axiom,
    ! [A: fset_term_a_b,B: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ ( finser5027672521447216124rm_a_b @ B @ A4 ) )
      = ( ( A = B )
        | ( fmembe3754333024224722621rm_a_b @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_650_finsert__iff,axiom,
    ! [A: nat,B: nat,A4: fset_nat] :
      ( ( fmember_nat @ A @ ( finsert_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( fmember_nat @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_651_finsert__iff,axiom,
    ! [A: c,B: c,A4: fset_c] :
      ( ( fmember_c @ A @ ( finsert_c @ B @ A4 ) )
      = ( ( A = B )
        | ( fmember_c @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_652_finsert__iff,axiom,
    ! [A: b,B: b,A4: fset_b] :
      ( ( fmember_b @ A @ ( finsert_b @ B @ A4 ) )
      = ( ( A = B )
        | ( fmember_b @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_653_finsert__iff,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ ( finsert_term_a_b @ B @ A4 ) )
      = ( ( A = B )
        | ( fmember_term_a_b @ A @ A4 ) ) ) ).

% finsert_iff
thf(fact_654_subst__apply__term__empty,axiom,
    ! [T: term_a_c] :
      ( ( subst_5212014727554084972_a_c_c @ T @ var_c_a )
      = T ) ).

% subst_apply_term_empty
thf(fact_655_subst__apply__term__empty,axiom,
    ! [T: term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ T @ var_b_a )
      = T ) ).

% subst_apply_term_empty
thf(fact_656_subst__simps_I2_J,axiom,
    ! [X: b] :
      ( ( subst_b_a @ X @ ( var_b_a @ X ) )
      = var_b_a ) ).

% subst_simps(2)
thf(fact_657_subst__simps_I2_J,axiom,
    ! [X: c] :
      ( ( subst_c_a @ X @ ( var_c_a @ X ) )
      = var_c_a ) ).

% subst_simps(2)
thf(fact_658_fSigma__empty1,axiom,
    ! [B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fSet_f5829356563446949852term_a @ bot_bo4313458260509495885term_a @ B4 )
      = bot_bo3304177514724984389term_a ) ).

% fSigma_empty1
thf(fact_659_fSigma__empty1,axiom,
    ! [B4: term_a_b > fset_term_a_b] :
      ( ( fSet_f1273034697330015346rm_a_b @ bot_bo1583650863486268376rm_a_b @ B4 )
      = bot_bo173352615033564763rm_a_b ) ).

% fSigma_empty1
thf(fact_660_fvars__term__of__gterm,axiom,
    ! [T: ground_gterm_a] :
      ( ( fSet_fvars_term_a_c @ ( ground5786051259813020158rm_a_c @ T ) )
      = bot_bot_fset_c ) ).

% fvars_term_of_gterm
thf(fact_661_fvars__term__Var,axiom,
    ! [X: b] :
      ( ( fSet_fvars_term_a_b @ ( var_b_a @ X ) )
      = ( finsert_b @ X @ bot_bot_fset_b ) ) ).

% fvars_term_Var
thf(fact_662_fvars__term__Var,axiom,
    ! [X: c] :
      ( ( fSet_fvars_term_a_c @ ( var_c_a @ X ) )
      = ( finsert_c @ X @ bot_bot_fset_c ) ) ).

% fvars_term_Var
thf(fact_663_ta_Oexhaust,axiom,
    ! [Y: tree_ta_b_a] :
      ~ ! [X12: fset_T8606228319753941405le_b_a,X22: fset_P8578720700757663565od_b_b] :
          ( Y
         != ( tree_TA_b_a @ X12 @ X22 ) ) ).

% ta.exhaust
thf(fact_664_ta_Osel_I2_J,axiom,
    ! [X1: fset_T8606228319753941405le_b_a,X2: fset_P8578720700757663565od_b_b] :
      ( ( tree_eps_b_a @ ( tree_TA_b_a @ X1 @ X2 ) )
      = X2 ) ).

% ta.sel(2)
thf(fact_665_ta_Osel_I2_J,axiom,
    ! [X1: fset_T7970579319148954716le_c_a,X2: fset_P8014105740199022861od_c_c] :
      ( ( tree_eps_c_a @ ( tree_TA_c_a @ X1 @ X2 ) )
      = X2 ) ).

% ta.sel(2)
thf(fact_666_finsert__commute,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( finser8320498192367885249term_a @ X @ ( finser8320498192367885249term_a @ Y @ A4 ) )
      = ( finser8320498192367885249term_a @ Y @ ( finser8320498192367885249term_a @ X @ A4 ) ) ) ).

% finsert_commute
thf(fact_667_finsert__commute,axiom,
    ! [X: term_a_b,Y: term_a_b,A4: fset_term_a_b] :
      ( ( finsert_term_a_b @ X @ ( finsert_term_a_b @ Y @ A4 ) )
      = ( finsert_term_a_b @ Y @ ( finsert_term_a_b @ X @ A4 ) ) ) ).

% finsert_commute
thf(fact_668_finsert__commute,axiom,
    ! [X: nat,Y: nat,A4: fset_nat] :
      ( ( finsert_nat @ X @ ( finsert_nat @ Y @ A4 ) )
      = ( finsert_nat @ Y @ ( finsert_nat @ X @ A4 ) ) ) ).

% finsert_commute
thf(fact_669_fSigma__cong,axiom,
    ! [A4: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a,C: bot_bot_term_a > fset_Bot_bot_term_a,D: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( A4 = B4 )
     => ( ! [X3: bot_bot_term_a] :
            ( ( fmembe1418995468851910914term_a @ X3 @ B4 )
           => ( ( C @ X3 )
              = ( D @ X3 ) ) )
       => ( ( fSet_f5829356563446949852term_a @ A4 @ C )
          = ( fSet_f5829356563446949852term_a @ B4 @ D ) ) ) ) ).

% fSigma_cong
thf(fact_670_fSigma__cong,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b,C: term_a_b > fset_term_a_b,D: term_a_b > fset_term_a_b] :
      ( ( A4 = B4 )
     => ( ! [X3: term_a_b] :
            ( ( fmember_term_a_b @ X3 @ B4 )
           => ( ( C @ X3 )
              = ( D @ X3 ) ) )
       => ( ( fSet_f1273034697330015346rm_a_b @ A4 @ C )
          = ( fSet_f1273034697330015346rm_a_b @ B4 @ D ) ) ) ) ).

% fSigma_cong
thf(fact_671_finsertE,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ ( finser8320498192367885249term_a @ B @ A4 ) )
     => ( ( A != B )
       => ( fmembe1418995468851910914term_a @ A @ A4 ) ) ) ).

% finsertE
thf(fact_672_finsertE,axiom,
    ! [A: term_a_c,B: term_a_c,A4: fset_term_a_c] :
      ( ( fmember_term_a_c @ A @ ( finsert_term_a_c @ B @ A4 ) )
     => ( ( A != B )
       => ( fmember_term_a_c @ A @ A4 ) ) ) ).

% finsertE
thf(fact_673_finsertE,axiom,
    ! [A: fset_term_a_b,B: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ ( finser5027672521447216124rm_a_b @ B @ A4 ) )
     => ( ( A != B )
       => ( fmembe3754333024224722621rm_a_b @ A @ A4 ) ) ) ).

% finsertE
thf(fact_674_finsertE,axiom,
    ! [A: nat,B: nat,A4: fset_nat] :
      ( ( fmember_nat @ A @ ( finsert_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( fmember_nat @ A @ A4 ) ) ) ).

% finsertE
thf(fact_675_finsertE,axiom,
    ! [A: c,B: c,A4: fset_c] :
      ( ( fmember_c @ A @ ( finsert_c @ B @ A4 ) )
     => ( ( A != B )
       => ( fmember_c @ A @ A4 ) ) ) ).

% finsertE
thf(fact_676_finsertE,axiom,
    ! [A: b,B: b,A4: fset_b] :
      ( ( fmember_b @ A @ ( finsert_b @ B @ A4 ) )
     => ( ( A != B )
       => ( fmember_b @ A @ A4 ) ) ) ).

% finsertE
thf(fact_677_finsertE,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ ( finsert_term_a_b @ B @ A4 ) )
     => ( ( A != B )
       => ( fmember_term_a_b @ A @ A4 ) ) ) ).

% finsertE
thf(fact_678_finsertI1,axiom,
    ! [A: bot_bot_term_a,B4: fset_Bot_bot_term_a] : ( fmembe1418995468851910914term_a @ A @ ( finser8320498192367885249term_a @ A @ B4 ) ) ).

% finsertI1
thf(fact_679_finsertI1,axiom,
    ! [A: term_a_c,B4: fset_term_a_c] : ( fmember_term_a_c @ A @ ( finsert_term_a_c @ A @ B4 ) ) ).

% finsertI1
thf(fact_680_finsertI1,axiom,
    ! [A: fset_term_a_b,B4: fset_fset_term_a_b] : ( fmembe3754333024224722621rm_a_b @ A @ ( finser5027672521447216124rm_a_b @ A @ B4 ) ) ).

% finsertI1
thf(fact_681_finsertI1,axiom,
    ! [A: nat,B4: fset_nat] : ( fmember_nat @ A @ ( finsert_nat @ A @ B4 ) ) ).

% finsertI1
thf(fact_682_finsertI1,axiom,
    ! [A: c,B4: fset_c] : ( fmember_c @ A @ ( finsert_c @ A @ B4 ) ) ).

% finsertI1
thf(fact_683_finsertI1,axiom,
    ! [A: b,B4: fset_b] : ( fmember_b @ A @ ( finsert_b @ A @ B4 ) ) ).

% finsertI1
thf(fact_684_finsertI1,axiom,
    ! [A: term_a_b,B4: fset_term_a_b] : ( fmember_term_a_b @ A @ ( finsert_term_a_b @ A @ B4 ) ) ).

% finsertI1
thf(fact_685_finsertI2,axiom,
    ! [A: bot_bot_term_a,B4: fset_Bot_bot_term_a,B: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ B4 )
     => ( fmembe1418995468851910914term_a @ A @ ( finser8320498192367885249term_a @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_686_finsertI2,axiom,
    ! [A: term_a_c,B4: fset_term_a_c,B: term_a_c] :
      ( ( fmember_term_a_c @ A @ B4 )
     => ( fmember_term_a_c @ A @ ( finsert_term_a_c @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_687_finsertI2,axiom,
    ! [A: fset_term_a_b,B4: fset_fset_term_a_b,B: fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ B4 )
     => ( fmembe3754333024224722621rm_a_b @ A @ ( finser5027672521447216124rm_a_b @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_688_finsertI2,axiom,
    ! [A: nat,B4: fset_nat,B: nat] :
      ( ( fmember_nat @ A @ B4 )
     => ( fmember_nat @ A @ ( finsert_nat @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_689_finsertI2,axiom,
    ! [A: c,B4: fset_c,B: c] :
      ( ( fmember_c @ A @ B4 )
     => ( fmember_c @ A @ ( finsert_c @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_690_finsertI2,axiom,
    ! [A: b,B4: fset_b,B: b] :
      ( ( fmember_b @ A @ B4 )
     => ( fmember_b @ A @ ( finsert_b @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_691_finsertI2,axiom,
    ! [A: term_a_b,B4: fset_term_a_b,B: term_a_b] :
      ( ( fmember_term_a_b @ A @ B4 )
     => ( fmember_term_a_b @ A @ ( finsert_term_a_b @ B @ B4 ) ) ) ).

% finsertI2
thf(fact_692_set__finsert,axiom,
    ! [X: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ X @ A4 )
     => ~ ! [B6: fset_Bot_bot_term_a] :
            ( ( A4
              = ( finser8320498192367885249term_a @ X @ B6 ) )
           => ( fmembe1418995468851910914term_a @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_693_set__finsert,axiom,
    ! [X: term_a_c,A4: fset_term_a_c] :
      ( ( fmember_term_a_c @ X @ A4 )
     => ~ ! [B6: fset_term_a_c] :
            ( ( A4
              = ( finsert_term_a_c @ X @ B6 ) )
           => ( fmember_term_a_c @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_694_set__finsert,axiom,
    ! [X: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ X @ A4 )
     => ~ ! [B6: fset_fset_term_a_b] :
            ( ( A4
              = ( finser5027672521447216124rm_a_b @ X @ B6 ) )
           => ( fmembe3754333024224722621rm_a_b @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_695_set__finsert,axiom,
    ! [X: nat,A4: fset_nat] :
      ( ( fmember_nat @ X @ A4 )
     => ~ ! [B6: fset_nat] :
            ( ( A4
              = ( finsert_nat @ X @ B6 ) )
           => ( fmember_nat @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_696_set__finsert,axiom,
    ! [X: c,A4: fset_c] :
      ( ( fmember_c @ X @ A4 )
     => ~ ! [B6: fset_c] :
            ( ( A4
              = ( finsert_c @ X @ B6 ) )
           => ( fmember_c @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_697_set__finsert,axiom,
    ! [X: b,A4: fset_b] :
      ( ( fmember_b @ X @ A4 )
     => ~ ! [B6: fset_b] :
            ( ( A4
              = ( finsert_b @ X @ B6 ) )
           => ( fmember_b @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_698_set__finsert,axiom,
    ! [X: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ X @ A4 )
     => ~ ! [B6: fset_term_a_b] :
            ( ( A4
              = ( finsert_term_a_b @ X @ B6 ) )
           => ( fmember_term_a_b @ X @ B6 ) ) ) ).

% set_finsert
thf(fact_699_finsert__ident,axiom,
    ! [X: bot_bot_term_a,A4: fset_Bot_bot_term_a,B4: fset_Bot_bot_term_a] :
      ( ~ ( fmembe1418995468851910914term_a @ X @ A4 )
     => ( ~ ( fmembe1418995468851910914term_a @ X @ B4 )
       => ( ( ( finser8320498192367885249term_a @ X @ A4 )
            = ( finser8320498192367885249term_a @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_700_finsert__ident,axiom,
    ! [X: term_a_c,A4: fset_term_a_c,B4: fset_term_a_c] :
      ( ~ ( fmember_term_a_c @ X @ A4 )
     => ( ~ ( fmember_term_a_c @ X @ B4 )
       => ( ( ( finsert_term_a_c @ X @ A4 )
            = ( finsert_term_a_c @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_701_finsert__ident,axiom,
    ! [X: fset_term_a_b,A4: fset_fset_term_a_b,B4: fset_fset_term_a_b] :
      ( ~ ( fmembe3754333024224722621rm_a_b @ X @ A4 )
     => ( ~ ( fmembe3754333024224722621rm_a_b @ X @ B4 )
       => ( ( ( finser5027672521447216124rm_a_b @ X @ A4 )
            = ( finser5027672521447216124rm_a_b @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_702_finsert__ident,axiom,
    ! [X: nat,A4: fset_nat,B4: fset_nat] :
      ( ~ ( fmember_nat @ X @ A4 )
     => ( ~ ( fmember_nat @ X @ B4 )
       => ( ( ( finsert_nat @ X @ A4 )
            = ( finsert_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_703_finsert__ident,axiom,
    ! [X: c,A4: fset_c,B4: fset_c] :
      ( ~ ( fmember_c @ X @ A4 )
     => ( ~ ( fmember_c @ X @ B4 )
       => ( ( ( finsert_c @ X @ A4 )
            = ( finsert_c @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_704_finsert__ident,axiom,
    ! [X: b,A4: fset_b,B4: fset_b] :
      ( ~ ( fmember_b @ X @ A4 )
     => ( ~ ( fmember_b @ X @ B4 )
       => ( ( ( finsert_b @ X @ A4 )
            = ( finsert_b @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_705_finsert__ident,axiom,
    ! [X: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ~ ( fmember_term_a_b @ X @ A4 )
     => ( ~ ( fmember_term_a_b @ X @ B4 )
       => ( ( ( finsert_term_a_b @ X @ A4 )
            = ( finsert_term_a_b @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% finsert_ident
thf(fact_706_finsert__absorb,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ A4 )
     => ( ( finser8320498192367885249term_a @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_707_finsert__absorb,axiom,
    ! [A: term_a_c,A4: fset_term_a_c] :
      ( ( fmember_term_a_c @ A @ A4 )
     => ( ( finsert_term_a_c @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_708_finsert__absorb,axiom,
    ! [A: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ A4 )
     => ( ( finser5027672521447216124rm_a_b @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_709_finsert__absorb,axiom,
    ! [A: nat,A4: fset_nat] :
      ( ( fmember_nat @ A @ A4 )
     => ( ( finsert_nat @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_710_finsert__absorb,axiom,
    ! [A: c,A4: fset_c] :
      ( ( fmember_c @ A @ A4 )
     => ( ( finsert_c @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_711_finsert__absorb,axiom,
    ! [A: b,A4: fset_b] :
      ( ( fmember_b @ A @ A4 )
     => ( ( finsert_b @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_712_finsert__absorb,axiom,
    ! [A: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ( ( finsert_term_a_b @ A @ A4 )
        = A4 ) ) ).

% finsert_absorb
thf(fact_713_finsert__eq__iff,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a,B: bot_bot_term_a,B4: fset_Bot_bot_term_a] :
      ( ~ ( fmembe1418995468851910914term_a @ A @ A4 )
     => ( ~ ( fmembe1418995468851910914term_a @ B @ B4 )
       => ( ( ( finser8320498192367885249term_a @ A @ A4 )
            = ( finser8320498192367885249term_a @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_Bot_bot_term_a] :
                  ( ( A4
                    = ( finser8320498192367885249term_a @ B @ C4 ) )
                  & ~ ( fmembe1418995468851910914term_a @ B @ C4 )
                  & ( B4
                    = ( finser8320498192367885249term_a @ A @ C4 ) )
                  & ~ ( fmembe1418995468851910914term_a @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_714_finsert__eq__iff,axiom,
    ! [A: term_a_c,A4: fset_term_a_c,B: term_a_c,B4: fset_term_a_c] :
      ( ~ ( fmember_term_a_c @ A @ A4 )
     => ( ~ ( fmember_term_a_c @ B @ B4 )
       => ( ( ( finsert_term_a_c @ A @ A4 )
            = ( finsert_term_a_c @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_term_a_c] :
                  ( ( A4
                    = ( finsert_term_a_c @ B @ C4 ) )
                  & ~ ( fmember_term_a_c @ B @ C4 )
                  & ( B4
                    = ( finsert_term_a_c @ A @ C4 ) )
                  & ~ ( fmember_term_a_c @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_715_finsert__eq__iff,axiom,
    ! [A: fset_term_a_b,A4: fset_fset_term_a_b,B: fset_term_a_b,B4: fset_fset_term_a_b] :
      ( ~ ( fmembe3754333024224722621rm_a_b @ A @ A4 )
     => ( ~ ( fmembe3754333024224722621rm_a_b @ B @ B4 )
       => ( ( ( finser5027672521447216124rm_a_b @ A @ A4 )
            = ( finser5027672521447216124rm_a_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_fset_term_a_b] :
                  ( ( A4
                    = ( finser5027672521447216124rm_a_b @ B @ C4 ) )
                  & ~ ( fmembe3754333024224722621rm_a_b @ B @ C4 )
                  & ( B4
                    = ( finser5027672521447216124rm_a_b @ A @ C4 ) )
                  & ~ ( fmembe3754333024224722621rm_a_b @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_716_finsert__eq__iff,axiom,
    ! [A: nat,A4: fset_nat,B: nat,B4: fset_nat] :
      ( ~ ( fmember_nat @ A @ A4 )
     => ( ~ ( fmember_nat @ B @ B4 )
       => ( ( ( finsert_nat @ A @ A4 )
            = ( finsert_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_nat] :
                  ( ( A4
                    = ( finsert_nat @ B @ C4 ) )
                  & ~ ( fmember_nat @ B @ C4 )
                  & ( B4
                    = ( finsert_nat @ A @ C4 ) )
                  & ~ ( fmember_nat @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_717_finsert__eq__iff,axiom,
    ! [A: c,A4: fset_c,B: c,B4: fset_c] :
      ( ~ ( fmember_c @ A @ A4 )
     => ( ~ ( fmember_c @ B @ B4 )
       => ( ( ( finsert_c @ A @ A4 )
            = ( finsert_c @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_c] :
                  ( ( A4
                    = ( finsert_c @ B @ C4 ) )
                  & ~ ( fmember_c @ B @ C4 )
                  & ( B4
                    = ( finsert_c @ A @ C4 ) )
                  & ~ ( fmember_c @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_718_finsert__eq__iff,axiom,
    ! [A: b,A4: fset_b,B: b,B4: fset_b] :
      ( ~ ( fmember_b @ A @ A4 )
     => ( ~ ( fmember_b @ B @ B4 )
       => ( ( ( finsert_b @ A @ A4 )
            = ( finsert_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_b] :
                  ( ( A4
                    = ( finsert_b @ B @ C4 ) )
                  & ~ ( fmember_b @ B @ C4 )
                  & ( B4
                    = ( finsert_b @ A @ C4 ) )
                  & ~ ( fmember_b @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_719_finsert__eq__iff,axiom,
    ! [A: term_a_b,A4: fset_term_a_b,B: term_a_b,B4: fset_term_a_b] :
      ( ~ ( fmember_term_a_b @ A @ A4 )
     => ( ~ ( fmember_term_a_b @ B @ B4 )
       => ( ( ( finsert_term_a_b @ A @ A4 )
            = ( finsert_term_a_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C4: fset_term_a_b] :
                  ( ( A4
                    = ( finsert_term_a_b @ B @ C4 ) )
                  & ~ ( fmember_term_a_b @ B @ C4 )
                  & ( B4
                    = ( finsert_term_a_b @ A @ C4 ) )
                  & ~ ( fmember_term_a_b @ A @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_720_mk__disjoint__finsert,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ A @ A4 )
     => ? [B6: fset_Bot_bot_term_a] :
          ( ( A4
            = ( finser8320498192367885249term_a @ A @ B6 ) )
          & ~ ( fmembe1418995468851910914term_a @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_721_mk__disjoint__finsert,axiom,
    ! [A: term_a_c,A4: fset_term_a_c] :
      ( ( fmember_term_a_c @ A @ A4 )
     => ? [B6: fset_term_a_c] :
          ( ( A4
            = ( finsert_term_a_c @ A @ B6 ) )
          & ~ ( fmember_term_a_c @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_722_mk__disjoint__finsert,axiom,
    ! [A: fset_term_a_b,A4: fset_fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ A @ A4 )
     => ? [B6: fset_fset_term_a_b] :
          ( ( A4
            = ( finser5027672521447216124rm_a_b @ A @ B6 ) )
          & ~ ( fmembe3754333024224722621rm_a_b @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_723_mk__disjoint__finsert,axiom,
    ! [A: nat,A4: fset_nat] :
      ( ( fmember_nat @ A @ A4 )
     => ? [B6: fset_nat] :
          ( ( A4
            = ( finsert_nat @ A @ B6 ) )
          & ~ ( fmember_nat @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_724_mk__disjoint__finsert,axiom,
    ! [A: c,A4: fset_c] :
      ( ( fmember_c @ A @ A4 )
     => ? [B6: fset_c] :
          ( ( A4
            = ( finsert_c @ A @ B6 ) )
          & ~ ( fmember_c @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_725_mk__disjoint__finsert,axiom,
    ! [A: b,A4: fset_b] :
      ( ( fmember_b @ A @ A4 )
     => ? [B6: fset_b] :
          ( ( A4
            = ( finsert_b @ A @ B6 ) )
          & ~ ( fmember_b @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_726_mk__disjoint__finsert,axiom,
    ! [A: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ? [B6: fset_term_a_b] :
          ( ( A4
            = ( finsert_term_a_b @ A @ B6 ) )
          & ~ ( fmember_term_a_b @ A @ B6 ) ) ) ).

% mk_disjoint_finsert
thf(fact_727_finsert__not__fempty,axiom,
    ! [A: bot_bot_term_a,A4: fset_Bot_bot_term_a] :
      ( ( finser8320498192367885249term_a @ A @ A4 )
     != bot_bo4313458260509495885term_a ) ).

% finsert_not_fempty
thf(fact_728_finsert__not__fempty,axiom,
    ! [A: term_a_b,A4: fset_term_a_b] :
      ( ( finsert_term_a_b @ A @ A4 )
     != bot_bo1583650863486268376rm_a_b ) ).

% finsert_not_fempty
thf(fact_729_finsert__not__fempty,axiom,
    ! [A: nat,A4: fset_nat] :
      ( ( finsert_nat @ A @ A4 )
     != bot_bot_fset_nat ) ).

% finsert_not_fempty
thf(fact_730_fsingleton__inject,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a] :
      ( ( ( finser8320498192367885249term_a @ A @ bot_bo4313458260509495885term_a )
        = ( finser8320498192367885249term_a @ B @ bot_bo4313458260509495885term_a ) )
     => ( A = B ) ) ).

% fsingleton_inject
thf(fact_731_fsingleton__inject,axiom,
    ! [A: term_a_b,B: term_a_b] :
      ( ( ( finsert_term_a_b @ A @ bot_bo1583650863486268376rm_a_b )
        = ( finsert_term_a_b @ B @ bot_bo1583650863486268376rm_a_b ) )
     => ( A = B ) ) ).

% fsingleton_inject
thf(fact_732_fsingleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( finsert_nat @ A @ bot_bot_fset_nat )
        = ( finsert_nat @ B @ bot_bot_fset_nat ) )
     => ( A = B ) ) ).

% fsingleton_inject
thf(fact_733_fdoubleton__eq__iff,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,C2: bot_bot_term_a,D2: bot_bot_term_a] :
      ( ( ( finser8320498192367885249term_a @ A @ ( finser8320498192367885249term_a @ B @ bot_bo4313458260509495885term_a ) )
        = ( finser8320498192367885249term_a @ C2 @ ( finser8320498192367885249term_a @ D2 @ bot_bo4313458260509495885term_a ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_734_fdoubleton__eq__iff,axiom,
    ! [A: term_a_b,B: term_a_b,C2: term_a_b,D2: term_a_b] :
      ( ( ( finsert_term_a_b @ A @ ( finsert_term_a_b @ B @ bot_bo1583650863486268376rm_a_b ) )
        = ( finsert_term_a_b @ C2 @ ( finsert_term_a_b @ D2 @ bot_bo1583650863486268376rm_a_b ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_735_fdoubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( finsert_nat @ A @ ( finsert_nat @ B @ bot_bot_fset_nat ) )
        = ( finsert_nat @ C2 @ ( finsert_nat @ D2 @ bot_bot_fset_nat ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_736_fset__exhaust,axiom,
    ! [S4: fset_Bot_bot_term_a] :
      ( ( S4 != bot_bo4313458260509495885term_a )
     => ~ ! [X3: bot_bot_term_a,S5: fset_Bot_bot_term_a] :
            ( S4
           != ( finser8320498192367885249term_a @ X3 @ S5 ) ) ) ).

% fset_exhaust
thf(fact_737_fset__exhaust,axiom,
    ! [S4: fset_term_a_b] :
      ( ( S4 != bot_bo1583650863486268376rm_a_b )
     => ~ ! [X3: term_a_b,S5: fset_term_a_b] :
            ( S4
           != ( finsert_term_a_b @ X3 @ S5 ) ) ) ).

% fset_exhaust
thf(fact_738_fset__exhaust,axiom,
    ! [S4: fset_nat] :
      ( ( S4 != bot_bot_fset_nat )
     => ~ ! [X3: nat,S5: fset_nat] :
            ( S4
           != ( finsert_nat @ X3 @ S5 ) ) ) ).

% fset_exhaust
thf(fact_739_FSet_Ofset__induct,axiom,
    ! [P: fset_Bot_bot_term_a > $o,S4: fset_Bot_bot_term_a] :
      ( ( P @ bot_bo4313458260509495885term_a )
     => ( ! [X3: bot_bot_term_a,S6: fset_Bot_bot_term_a] :
            ( ( P @ S6 )
           => ( P @ ( finser8320498192367885249term_a @ X3 @ S6 ) ) )
       => ( P @ S4 ) ) ) ).

% FSet.fset_induct
thf(fact_740_FSet_Ofset__induct,axiom,
    ! [P: fset_term_a_b > $o,S4: fset_term_a_b] :
      ( ( P @ bot_bo1583650863486268376rm_a_b )
     => ( ! [X3: term_a_b,S6: fset_term_a_b] :
            ( ( P @ S6 )
           => ( P @ ( finsert_term_a_b @ X3 @ S6 ) ) )
       => ( P @ S4 ) ) ) ).

% FSet.fset_induct
thf(fact_741_FSet_Ofset__induct,axiom,
    ! [P: fset_nat > $o,S4: fset_nat] :
      ( ( P @ bot_bot_fset_nat )
     => ( ! [X3: nat,S6: fset_nat] :
            ( ( P @ S6 )
           => ( P @ ( finsert_nat @ X3 @ S6 ) ) )
       => ( P @ S4 ) ) ) ).

% FSet.fset_induct
thf(fact_742_subst__apply__eq__Var,axiom,
    ! [S: term_a_b,Sigma: b > term_a_c,X: c] :
      ( ( ( subst_7999470309526761005_a_b_c @ S @ Sigma )
        = ( var_c_a @ X ) )
     => ~ ! [Y3: b] :
            ( ( S
              = ( var_b_a @ Y3 ) )
           => ( ( Sigma @ Y3 )
             != ( var_c_a @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_743_subst__apply__eq__Var,axiom,
    ! [S: term_a_c,Sigma: c > term_a_c,X: c] :
      ( ( ( subst_5212014727554084972_a_c_c @ S @ Sigma )
        = ( var_c_a @ X ) )
     => ~ ! [Y3: c] :
            ( ( S
              = ( var_c_a @ Y3 ) )
           => ( ( Sigma @ Y3 )
             != ( var_c_a @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_744_subst__apply__eq__Var,axiom,
    ! [S: term_a_c,Sigma: c > term_a_b,X: b] :
      ( ( ( subst_5212014727554084971_a_c_b @ S @ Sigma )
        = ( var_b_a @ X ) )
     => ~ ! [Y3: c] :
            ( ( S
              = ( var_c_a @ Y3 ) )
           => ( ( Sigma @ Y3 )
             != ( var_b_a @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_745_subst__apply__eq__Var,axiom,
    ! [S: term_a_b,Sigma: b > term_a_b,X: b] :
      ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
        = ( var_b_a @ X ) )
     => ~ ! [Y3: b] :
            ( ( S
              = ( var_b_a @ Y3 ) )
           => ( ( Sigma @ Y3 )
             != ( var_b_a @ X ) ) ) ) ).

% subst_apply_eq_Var
thf(fact_746_subst__apply__term_Osimps_I1_J,axiom,
    ! [X: b,Sigma: b > term_a_c] :
      ( ( subst_7999470309526761005_a_b_c @ ( var_b_a @ X ) @ Sigma )
      = ( Sigma @ X ) ) ).

% subst_apply_term.simps(1)
thf(fact_747_subst__apply__term_Osimps_I1_J,axiom,
    ! [X: c,Sigma: c > term_a_c] :
      ( ( subst_5212014727554084972_a_c_c @ ( var_c_a @ X ) @ Sigma )
      = ( Sigma @ X ) ) ).

% subst_apply_term.simps(1)
thf(fact_748_subst__apply__term_Osimps_I1_J,axiom,
    ! [X: c,Sigma: c > term_a_b] :
      ( ( subst_5212014727554084971_a_c_b @ ( var_c_a @ X ) @ Sigma )
      = ( Sigma @ X ) ) ).

% subst_apply_term.simps(1)
thf(fact_749_subst__apply__term_Osimps_I1_J,axiom,
    ! [X: b,Sigma: b > term_a_b] :
      ( ( subst_7999470309526761004_a_b_b @ ( var_b_a @ X ) @ Sigma )
      = ( Sigma @ X ) ) ).

% subst_apply_term.simps(1)
thf(fact_750_ground_Osimps_I1_J,axiom,
    ! [X: b] :
      ~ ( term_ground_a_b @ ( var_b_a @ X ) ) ).

% ground.simps(1)
thf(fact_751_ground_Osimps_I1_J,axiom,
    ! [X: bot_bot_term_a] :
      ~ ( term_g4884896889520838057term_a @ ( var_Bot_bot_term_a_a @ X ) ) ).

% ground.simps(1)
thf(fact_752_ground_Osimps_I1_J,axiom,
    ! [X: term_a_b] :
      ~ ( term_g2255532262831151796rm_a_b @ ( var_term_a_b_a @ X ) ) ).

% ground.simps(1)
thf(fact_753_ground_Osimps_I1_J,axiom,
    ! [X: c] :
      ~ ( term_ground_a_c @ ( var_c_a @ X ) ) ).

% ground.simps(1)
thf(fact_754_adapt__vars__term__of__gterm,axiom,
    ! [T: ground_gterm_a] :
      ( ( term_a6990854600579303330term_a @ ( ground5786051259813020157rm_a_b @ T ) )
      = ( ground2947561787367041419term_a @ T ) ) ).

% adapt_vars_term_of_gterm
thf(fact_755_adapt__vars__term__of__gterm,axiom,
    ! [T: ground_gterm_a] :
      ( ( term_a5382035503001191119rm_a_c @ ( ground2947561787367041419term_a @ T ) )
      = ( ground5786051259813020158rm_a_c @ T ) ) ).

% adapt_vars_term_of_gterm
thf(fact_756_adapt__vars__term__of__gterm,axiom,
    ! [T: ground_gterm_a] :
      ( ( term_a3222098444227397764_a_b_c @ ( ground678623728659755670rm_a_b @ T ) )
      = ( ground5786051259813020158rm_a_c @ T ) ) ).

% adapt_vars_term_of_gterm
thf(fact_757_adapt__vars__term__of__gterm,axiom,
    ! [T: ground_gterm_a] :
      ( ( term_a3900797222385290531term_a @ ( ground5786051259813020158rm_a_c @ T ) )
      = ( ground2947561787367041419term_a @ T ) ) ).

% adapt_vars_term_of_gterm
thf(fact_758_adapt__vars__term__of__gterm,axiom,
    ! [T: ground_gterm_a] :
      ( ( term_a9216225935088957542_a_c_c @ ( ground5786051259813020158rm_a_c @ T ) )
      = ( ground5786051259813020158rm_a_c @ T ) ) ).

% adapt_vars_term_of_gterm
thf(fact_759_fset__induct__stronger,axiom,
    ! [P: fset_Bot_bot_term_a > $o,S4: fset_Bot_bot_term_a] :
      ( ( P @ bot_bo4313458260509495885term_a )
     => ( ! [X3: bot_bot_term_a,S6: fset_Bot_bot_term_a] :
            ( ~ ( fmembe1418995468851910914term_a @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finser8320498192367885249term_a @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_760_fset__induct__stronger,axiom,
    ! [P: fset_term_a_c > $o,S4: fset_term_a_c] :
      ( ( P @ bot_bo1654684903532614361rm_a_c )
     => ( ! [X3: term_a_c,S6: fset_term_a_c] :
            ( ~ ( fmember_term_a_c @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finsert_term_a_c @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_761_fset__induct__stronger,axiom,
    ! [P: fset_fset_term_a_b > $o,S4: fset_fset_term_a_b] :
      ( ( P @ bot_bo7844603177471244936rm_a_b )
     => ( ! [X3: fset_term_a_b,S6: fset_fset_term_a_b] :
            ( ~ ( fmembe3754333024224722621rm_a_b @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finser5027672521447216124rm_a_b @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_762_fset__induct__stronger,axiom,
    ! [P: fset_nat > $o,S4: fset_nat] :
      ( ( P @ bot_bot_fset_nat )
     => ( ! [X3: nat,S6: fset_nat] :
            ( ~ ( fmember_nat @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finsert_nat @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_763_fset__induct__stronger,axiom,
    ! [P: fset_c > $o,S4: fset_c] :
      ( ( P @ bot_bot_fset_c )
     => ( ! [X3: c,S6: fset_c] :
            ( ~ ( fmember_c @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finsert_c @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_764_fset__induct__stronger,axiom,
    ! [P: fset_b > $o,S4: fset_b] :
      ( ( P @ bot_bot_fset_b )
     => ( ! [X3: b,S6: fset_b] :
            ( ~ ( fmember_b @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finsert_b @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_765_fset__induct__stronger,axiom,
    ! [P: fset_term_a_b > $o,S4: fset_term_a_b] :
      ( ( P @ bot_bo1583650863486268376rm_a_b )
     => ( ! [X3: term_a_b,S6: fset_term_a_b] :
            ( ~ ( fmember_term_a_b @ X3 @ S6 )
           => ( ( P @ S6 )
             => ( P @ ( finsert_term_a_b @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_766_fset__strong__cases,axiom,
    ! [Xs: fset_Bot_bot_term_a] :
      ( ( Xs != bot_bo4313458260509495885term_a )
     => ~ ! [Ys: fset_Bot_bot_term_a,X3: bot_bot_term_a] :
            ( ~ ( fmembe1418995468851910914term_a @ X3 @ Ys )
           => ( Xs
             != ( finser8320498192367885249term_a @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_767_fset__strong__cases,axiom,
    ! [Xs: fset_term_a_c] :
      ( ( Xs != bot_bo1654684903532614361rm_a_c )
     => ~ ! [Ys: fset_term_a_c,X3: term_a_c] :
            ( ~ ( fmember_term_a_c @ X3 @ Ys )
           => ( Xs
             != ( finsert_term_a_c @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_768_fset__strong__cases,axiom,
    ! [Xs: fset_fset_term_a_b] :
      ( ( Xs != bot_bo7844603177471244936rm_a_b )
     => ~ ! [Ys: fset_fset_term_a_b,X3: fset_term_a_b] :
            ( ~ ( fmembe3754333024224722621rm_a_b @ X3 @ Ys )
           => ( Xs
             != ( finser5027672521447216124rm_a_b @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_769_fset__strong__cases,axiom,
    ! [Xs: fset_nat] :
      ( ( Xs != bot_bot_fset_nat )
     => ~ ! [Ys: fset_nat,X3: nat] :
            ( ~ ( fmember_nat @ X3 @ Ys )
           => ( Xs
             != ( finsert_nat @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_770_fset__strong__cases,axiom,
    ! [Xs: fset_c] :
      ( ( Xs != bot_bot_fset_c )
     => ~ ! [Ys: fset_c,X3: c] :
            ( ~ ( fmember_c @ X3 @ Ys )
           => ( Xs
             != ( finsert_c @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_771_fset__strong__cases,axiom,
    ! [Xs: fset_b] :
      ( ( Xs != bot_bot_fset_b )
     => ~ ! [Ys: fset_b,X3: b] :
            ( ~ ( fmember_b @ X3 @ Ys )
           => ( Xs
             != ( finsert_b @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_772_fset__strong__cases,axiom,
    ! [Xs: fset_term_a_b] :
      ( ( Xs != bot_bo1583650863486268376rm_a_b )
     => ~ ! [Ys: fset_term_a_b,X3: term_a_b] :
            ( ~ ( fmember_term_a_b @ X3 @ Ys )
           => ( Xs
             != ( finsert_term_a_b @ X3 @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_773_fsingleton__iff,axiom,
    ! [B: bot_bot_term_a,A: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ B @ ( finser8320498192367885249term_a @ A @ bot_bo4313458260509495885term_a ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_774_fsingleton__iff,axiom,
    ! [B: term_a_c,A: term_a_c] :
      ( ( fmember_term_a_c @ B @ ( finsert_term_a_c @ A @ bot_bo1654684903532614361rm_a_c ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_775_fsingleton__iff,axiom,
    ! [B: fset_term_a_b,A: fset_term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ B @ ( finser5027672521447216124rm_a_b @ A @ bot_bo7844603177471244936rm_a_b ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_776_fsingleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( fmember_nat @ B @ ( finsert_nat @ A @ bot_bot_fset_nat ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_777_fsingleton__iff,axiom,
    ! [B: c,A: c] :
      ( ( fmember_c @ B @ ( finsert_c @ A @ bot_bot_fset_c ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_778_fsingleton__iff,axiom,
    ! [B: b,A: b] :
      ( ( fmember_b @ B @ ( finsert_b @ A @ bot_bot_fset_b ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_779_fsingleton__iff,axiom,
    ! [B: term_a_b,A: term_a_b] :
      ( ( fmember_term_a_b @ B @ ( finsert_term_a_b @ A @ bot_bo1583650863486268376rm_a_b ) )
      = ( B = A ) ) ).

% fsingleton_iff
thf(fact_780_fset__induct2,axiom,
    ! [P: fset_term_a_b > fset_term_a_b > $o,Xsa: fset_term_a_b,Ysa: fset_term_a_b] :
      ( ( P @ bot_bo1583650863486268376rm_a_b @ bot_bo1583650863486268376rm_a_b )
     => ( ! [X3: term_a_b,Xs2: fset_term_a_b] :
            ( ~ ( fmember_term_a_b @ X3 @ Xs2 )
           => ( P @ ( finsert_term_a_b @ X3 @ Xs2 ) @ bot_bo1583650863486268376rm_a_b ) )
       => ( ! [Y3: term_a_b,Ys: fset_term_a_b] :
              ( ~ ( fmember_term_a_b @ Y3 @ Ys )
             => ( P @ bot_bo1583650863486268376rm_a_b @ ( finsert_term_a_b @ Y3 @ Ys ) ) )
         => ( ! [X3: term_a_b,Xs2: fset_term_a_b,Y3: term_a_b,Ys: fset_term_a_b] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_term_a_b @ X3 @ Xs2 )
                 => ( ~ ( fmember_term_a_b @ Y3 @ Ys )
                   => ( P @ ( finsert_term_a_b @ X3 @ Xs2 ) @ ( finsert_term_a_b @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_781_fset__induct2,axiom,
    ! [P: fset_nat > fset_nat > $o,Xsa: fset_nat,Ysa: fset_nat] :
      ( ( P @ bot_bot_fset_nat @ bot_bot_fset_nat )
     => ( ! [X3: nat,Xs2: fset_nat] :
            ( ~ ( fmember_nat @ X3 @ Xs2 )
           => ( P @ ( finsert_nat @ X3 @ Xs2 ) @ bot_bot_fset_nat ) )
       => ( ! [Y3: nat,Ys: fset_nat] :
              ( ~ ( fmember_nat @ Y3 @ Ys )
             => ( P @ bot_bot_fset_nat @ ( finsert_nat @ Y3 @ Ys ) ) )
         => ( ! [X3: nat,Xs2: fset_nat,Y3: nat,Ys: fset_nat] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_nat @ X3 @ Xs2 )
                 => ( ~ ( fmember_nat @ Y3 @ Ys )
                   => ( P @ ( finsert_nat @ X3 @ Xs2 ) @ ( finsert_nat @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_782_fset__induct2,axiom,
    ! [P: fset_nat > fset_c > $o,Xsa: fset_nat,Ysa: fset_c] :
      ( ( P @ bot_bot_fset_nat @ bot_bot_fset_c )
     => ( ! [X3: nat,Xs2: fset_nat] :
            ( ~ ( fmember_nat @ X3 @ Xs2 )
           => ( P @ ( finsert_nat @ X3 @ Xs2 ) @ bot_bot_fset_c ) )
       => ( ! [Y3: c,Ys: fset_c] :
              ( ~ ( fmember_c @ Y3 @ Ys )
             => ( P @ bot_bot_fset_nat @ ( finsert_c @ Y3 @ Ys ) ) )
         => ( ! [X3: nat,Xs2: fset_nat,Y3: c,Ys: fset_c] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_nat @ X3 @ Xs2 )
                 => ( ~ ( fmember_c @ Y3 @ Ys )
                   => ( P @ ( finsert_nat @ X3 @ Xs2 ) @ ( finsert_c @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_783_fset__induct2,axiom,
    ! [P: fset_nat > fset_b > $o,Xsa: fset_nat,Ysa: fset_b] :
      ( ( P @ bot_bot_fset_nat @ bot_bot_fset_b )
     => ( ! [X3: nat,Xs2: fset_nat] :
            ( ~ ( fmember_nat @ X3 @ Xs2 )
           => ( P @ ( finsert_nat @ X3 @ Xs2 ) @ bot_bot_fset_b ) )
       => ( ! [Y3: b,Ys: fset_b] :
              ( ~ ( fmember_b @ Y3 @ Ys )
             => ( P @ bot_bot_fset_nat @ ( finsert_b @ Y3 @ Ys ) ) )
         => ( ! [X3: nat,Xs2: fset_nat,Y3: b,Ys: fset_b] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_nat @ X3 @ Xs2 )
                 => ( ~ ( fmember_b @ Y3 @ Ys )
                   => ( P @ ( finsert_nat @ X3 @ Xs2 ) @ ( finsert_b @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_784_fset__induct2,axiom,
    ! [P: fset_c > fset_nat > $o,Xsa: fset_c,Ysa: fset_nat] :
      ( ( P @ bot_bot_fset_c @ bot_bot_fset_nat )
     => ( ! [X3: c,Xs2: fset_c] :
            ( ~ ( fmember_c @ X3 @ Xs2 )
           => ( P @ ( finsert_c @ X3 @ Xs2 ) @ bot_bot_fset_nat ) )
       => ( ! [Y3: nat,Ys: fset_nat] :
              ( ~ ( fmember_nat @ Y3 @ Ys )
             => ( P @ bot_bot_fset_c @ ( finsert_nat @ Y3 @ Ys ) ) )
         => ( ! [X3: c,Xs2: fset_c,Y3: nat,Ys: fset_nat] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_c @ X3 @ Xs2 )
                 => ( ~ ( fmember_nat @ Y3 @ Ys )
                   => ( P @ ( finsert_c @ X3 @ Xs2 ) @ ( finsert_nat @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_785_fset__induct2,axiom,
    ! [P: fset_c > fset_c > $o,Xsa: fset_c,Ysa: fset_c] :
      ( ( P @ bot_bot_fset_c @ bot_bot_fset_c )
     => ( ! [X3: c,Xs2: fset_c] :
            ( ~ ( fmember_c @ X3 @ Xs2 )
           => ( P @ ( finsert_c @ X3 @ Xs2 ) @ bot_bot_fset_c ) )
       => ( ! [Y3: c,Ys: fset_c] :
              ( ~ ( fmember_c @ Y3 @ Ys )
             => ( P @ bot_bot_fset_c @ ( finsert_c @ Y3 @ Ys ) ) )
         => ( ! [X3: c,Xs2: fset_c,Y3: c,Ys: fset_c] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_c @ X3 @ Xs2 )
                 => ( ~ ( fmember_c @ Y3 @ Ys )
                   => ( P @ ( finsert_c @ X3 @ Xs2 ) @ ( finsert_c @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_786_fset__induct2,axiom,
    ! [P: fset_c > fset_b > $o,Xsa: fset_c,Ysa: fset_b] :
      ( ( P @ bot_bot_fset_c @ bot_bot_fset_b )
     => ( ! [X3: c,Xs2: fset_c] :
            ( ~ ( fmember_c @ X3 @ Xs2 )
           => ( P @ ( finsert_c @ X3 @ Xs2 ) @ bot_bot_fset_b ) )
       => ( ! [Y3: b,Ys: fset_b] :
              ( ~ ( fmember_b @ Y3 @ Ys )
             => ( P @ bot_bot_fset_c @ ( finsert_b @ Y3 @ Ys ) ) )
         => ( ! [X3: c,Xs2: fset_c,Y3: b,Ys: fset_b] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_c @ X3 @ Xs2 )
                 => ( ~ ( fmember_b @ Y3 @ Ys )
                   => ( P @ ( finsert_c @ X3 @ Xs2 ) @ ( finsert_b @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_787_fset__induct2,axiom,
    ! [P: fset_b > fset_nat > $o,Xsa: fset_b,Ysa: fset_nat] :
      ( ( P @ bot_bot_fset_b @ bot_bot_fset_nat )
     => ( ! [X3: b,Xs2: fset_b] :
            ( ~ ( fmember_b @ X3 @ Xs2 )
           => ( P @ ( finsert_b @ X3 @ Xs2 ) @ bot_bot_fset_nat ) )
       => ( ! [Y3: nat,Ys: fset_nat] :
              ( ~ ( fmember_nat @ Y3 @ Ys )
             => ( P @ bot_bot_fset_b @ ( finsert_nat @ Y3 @ Ys ) ) )
         => ( ! [X3: b,Xs2: fset_b,Y3: nat,Ys: fset_nat] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_b @ X3 @ Xs2 )
                 => ( ~ ( fmember_nat @ Y3 @ Ys )
                   => ( P @ ( finsert_b @ X3 @ Xs2 ) @ ( finsert_nat @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_788_fset__induct2,axiom,
    ! [P: fset_b > fset_c > $o,Xsa: fset_b,Ysa: fset_c] :
      ( ( P @ bot_bot_fset_b @ bot_bot_fset_c )
     => ( ! [X3: b,Xs2: fset_b] :
            ( ~ ( fmember_b @ X3 @ Xs2 )
           => ( P @ ( finsert_b @ X3 @ Xs2 ) @ bot_bot_fset_c ) )
       => ( ! [Y3: c,Ys: fset_c] :
              ( ~ ( fmember_c @ Y3 @ Ys )
             => ( P @ bot_bot_fset_b @ ( finsert_c @ Y3 @ Ys ) ) )
         => ( ! [X3: b,Xs2: fset_b,Y3: c,Ys: fset_c] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_b @ X3 @ Xs2 )
                 => ( ~ ( fmember_c @ Y3 @ Ys )
                   => ( P @ ( finsert_b @ X3 @ Xs2 ) @ ( finsert_c @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_789_fset__induct2,axiom,
    ! [P: fset_b > fset_b > $o,Xsa: fset_b,Ysa: fset_b] :
      ( ( P @ bot_bot_fset_b @ bot_bot_fset_b )
     => ( ! [X3: b,Xs2: fset_b] :
            ( ~ ( fmember_b @ X3 @ Xs2 )
           => ( P @ ( finsert_b @ X3 @ Xs2 ) @ bot_bot_fset_b ) )
       => ( ! [Y3: b,Ys: fset_b] :
              ( ~ ( fmember_b @ Y3 @ Ys )
             => ( P @ bot_bot_fset_b @ ( finsert_b @ Y3 @ Ys ) ) )
         => ( ! [X3: b,Xs2: fset_b,Y3: b,Ys: fset_b] :
                ( ( P @ Xs2 @ Ys )
               => ( ~ ( fmember_b @ X3 @ Xs2 )
                 => ( ~ ( fmember_b @ Y3 @ Ys )
                   => ( P @ ( finsert_b @ X3 @ Xs2 ) @ ( finsert_b @ Y3 @ Ys ) ) ) ) )
           => ( P @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_790_Var__supt,axiom,
    ! [X: c,T: term_a_c] :
      ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( var_c_a @ X ) @ T ) @ subterm_and_supt_a_c ) ).

% Var_supt
thf(fact_791_Var__supt,axiom,
    ! [X: b,T: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X ) @ T ) @ subterm_and_supt_a_b ) ).

% Var_supt
thf(fact_792_supt__var,axiom,
    ! [X: c,U: term_a_c] :
      ~ ( member6607649026611388852rm_a_c @ ( produc7687097280279809731rm_a_c @ ( var_c_a @ X ) @ U ) @ subterm_and_supt_a_c ) ).

% supt_var
thf(fact_793_supt__var,axiom,
    ! [X: b,U: term_a_b] :
      ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X ) @ U ) @ subterm_and_supt_a_b ) ).

% supt_var
thf(fact_794_fSigmaE,axiom,
    ! [C2: produc7033602110846336755term_a,A4: fset_Bot_bot_term_a,B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ C2 @ ( fSet_f5829356563446949852term_a @ A4 @ B4 ) )
     => ~ ! [X3: bot_bot_term_a] :
            ( ( fmembe1418995468851910914term_a @ X3 @ A4 )
           => ! [Y3: bot_bot_term_a] :
                ( ( fmembe1418995468851910914term_a @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( produc780340308816795563term_a @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_795_fSigmaE,axiom,
    ! [C2: produc357393685978478089rm_a_b,A4: fset_term_a_b,B4: term_a_b > fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ C2 @ ( fSet_f1273034697330015346rm_a_b @ A4 @ B4 ) )
     => ~ ! [X3: term_a_b] :
            ( ( fmember_term_a_b @ X3 @ A4 )
           => ! [Y3: term_a_b] :
                ( ( fmember_term_a_b @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_796_fSigmaE,axiom,
    ! [C2: product_prod_nat_c,A4: fset_nat,B4: nat > fset_c] :
      ( ( fmembe3856250535550116894_nat_c @ C2 @ ( fSet_fSigma_nat_c @ A4 @ B4 ) )
     => ~ ! [X3: nat] :
            ( ( fmember_nat @ X3 @ A4 )
           => ! [Y3: c] :
                ( ( fmember_c @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_c @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_797_fSigmaE,axiom,
    ! [C2: product_prod_nat_b,A4: fset_nat,B4: nat > fset_b] :
      ( ( fmembe3856250531246888093_nat_b @ C2 @ ( fSet_fSigma_nat_b @ A4 @ B4 ) )
     => ~ ! [X3: nat] :
            ( ( fmember_nat @ X3 @ A4 )
           => ! [Y3: b] :
                ( ( fmember_b @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_b @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_798_fSigmaE,axiom,
    ! [C2: product_prod_c_nat,A4: fset_c,B4: c > fset_nat] :
      ( ( fmembe3088975721132771320_c_nat @ C2 @ ( fSet_fSigma_c_nat @ A4 @ B4 ) )
     => ~ ! [X3: c] :
            ( ( fmember_c @ X3 @ A4 )
           => ! [Y3: nat] :
                ( ( fmember_nat @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_c_nat @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_799_fSigmaE,axiom,
    ! [C2: product_prod_c_b,A4: fset_c,B4: c > fset_b] :
      ( ( fmembe2163177356717951981od_c_b @ C2 @ ( fSet_fSigma_c_b @ A4 @ B4 ) )
     => ~ ! [X3: c] :
            ( ( fmember_c @ X3 @ A4 )
           => ! [Y3: b] :
                ( ( fmember_b @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_c_b @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_800_fSigmaE,axiom,
    ! [C2: product_prod_b_nat,A4: fset_b,B4: b > fset_nat] :
      ( ( fmembe1853531392176376823_b_nat @ C2 @ ( fSet_fSigma_b_nat @ A4 @ B4 ) )
     => ~ ! [X3: b] :
            ( ( fmember_b @ X3 @ A4 )
           => ! [Y3: nat] :
                ( ( fmember_nat @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_b_nat @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_801_fSigmaE,axiom,
    ! [C2: product_prod_b_c,A4: fset_b,B4: b > fset_c] :
      ( ( fmembe4950632942993856815od_b_c @ C2 @ ( fSet_fSigma_b_c @ A4 @ B4 ) )
     => ~ ! [X3: b] :
            ( ( fmember_b @ X3 @ A4 )
           => ! [Y3: c] :
                ( ( fmember_c @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_b_c @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_802_fSigmaE,axiom,
    ! [C2: product_prod_nat_nat,A4: fset_nat,B4: nat > fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ C2 @ ( fSet_fSigma_nat_nat @ A4 @ B4 ) )
     => ~ ! [X3: nat] :
            ( ( fmember_nat @ X3 @ A4 )
           => ! [Y3: nat] :
                ( ( fmember_nat @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_803_fSigmaE,axiom,
    ! [C2: product_prod_c_c,A4: fset_c,B4: c > fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ C2 @ ( fSet_fSigma_c_c @ A4 @ B4 ) )
     => ~ ! [X3: c] :
            ( ( fmember_c @ X3 @ A4 )
           => ! [Y3: c] :
                ( ( fmember_c @ Y3 @ ( B4 @ X3 ) )
               => ( C2
                 != ( product_Pair_c_c @ X3 @ Y3 ) ) ) ) ) ).

% fSigmaE
thf(fact_804_fSigmaD1,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b,B4: term_a_b > fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f1273034697330015346rm_a_b @ A4 @ B4 ) )
     => ( fmember_term_a_b @ A @ A4 ) ) ).

% fSigmaD1
thf(fact_805_fSigmaD1,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,A4: fset_l235587408706331465term_a,B4: list_Bot_bot_term_a > fset_l235587408706331465term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f5193722992130926076term_a @ A4 @ B4 ) )
     => ( fmembe1891194496946673298term_a @ A @ A4 ) ) ).

% fSigmaD1
thf(fact_806_fSigmaD1,axiom,
    ! [A: nat,B: nat,A4: fset_nat,B4: nat > fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fSigma_nat_nat @ A4 @ B4 ) )
     => ( fmember_nat @ A @ A4 ) ) ).

% fSigmaD1
thf(fact_807_fSigmaD1,axiom,
    ! [A: c,B: c,A4: fset_c,B4: c > fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fSigma_c_c @ A4 @ B4 ) )
     => ( fmember_c @ A @ A4 ) ) ).

% fSigmaD1
thf(fact_808_fSigmaD1,axiom,
    ! [A: b,B: b,A4: fset_b,B4: b > fset_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_fSigma_b_b @ A4 @ B4 ) )
     => ( fmember_b @ A @ A4 ) ) ).

% fSigmaD1
thf(fact_809_fSigmaD1,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a,B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f5829356563446949852term_a @ A4 @ B4 ) )
     => ( fmembe1418995468851910914term_a @ A @ A4 ) ) ).

% fSigmaD1
thf(fact_810_fSigmaD2,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b,B4: term_a_b > fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f1273034697330015346rm_a_b @ A4 @ B4 ) )
     => ( fmember_term_a_b @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_811_fSigmaD2,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,A4: fset_l235587408706331465term_a,B4: list_Bot_bot_term_a > fset_l235587408706331465term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f5193722992130926076term_a @ A4 @ B4 ) )
     => ( fmembe1891194496946673298term_a @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_812_fSigmaD2,axiom,
    ! [A: nat,B: nat,A4: fset_nat,B4: nat > fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fSigma_nat_nat @ A4 @ B4 ) )
     => ( fmember_nat @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_813_fSigmaD2,axiom,
    ! [A: c,B: c,A4: fset_c,B4: c > fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fSigma_c_c @ A4 @ B4 ) )
     => ( fmember_c @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_814_fSigmaD2,axiom,
    ! [A: b,B: b,A4: fset_b,B4: b > fset_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_fSigma_b_b @ A4 @ B4 ) )
     => ( fmember_b @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_815_fSigmaD2,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a,B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f5829356563446949852term_a @ A4 @ B4 ) )
     => ( fmembe1418995468851910914term_a @ B @ ( B4 @ A ) ) ) ).

% fSigmaD2
thf(fact_816_fSigmaE2,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,A4: fset_Bot_bot_term_a,B4: bot_bot_term_a > fset_Bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f5829356563446949852term_a @ A4 @ B4 ) )
     => ~ ( ( fmembe1418995468851910914term_a @ A @ A4 )
         => ~ ( fmembe1418995468851910914term_a @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_817_fSigmaE2,axiom,
    ! [A: term_a_b,B: term_a_b,A4: fset_term_a_b,B4: term_a_b > fset_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f1273034697330015346rm_a_b @ A4 @ B4 ) )
     => ~ ( ( fmember_term_a_b @ A @ A4 )
         => ~ ( fmember_term_a_b @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_818_fSigmaE2,axiom,
    ! [A: nat,B: c,A4: fset_nat,B4: nat > fset_c] :
      ( ( fmembe3856250535550116894_nat_c @ ( product_Pair_nat_c @ A @ B ) @ ( fSet_fSigma_nat_c @ A4 @ B4 ) )
     => ~ ( ( fmember_nat @ A @ A4 )
         => ~ ( fmember_c @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_819_fSigmaE2,axiom,
    ! [A: nat,B: b,A4: fset_nat,B4: nat > fset_b] :
      ( ( fmembe3856250531246888093_nat_b @ ( product_Pair_nat_b @ A @ B ) @ ( fSet_fSigma_nat_b @ A4 @ B4 ) )
     => ~ ( ( fmember_nat @ A @ A4 )
         => ~ ( fmember_b @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_820_fSigmaE2,axiom,
    ! [A: c,B: nat,A4: fset_c,B4: c > fset_nat] :
      ( ( fmembe3088975721132771320_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( fSet_fSigma_c_nat @ A4 @ B4 ) )
     => ~ ( ( fmember_c @ A @ A4 )
         => ~ ( fmember_nat @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_821_fSigmaE2,axiom,
    ! [A: c,B: b,A4: fset_c,B4: c > fset_b] :
      ( ( fmembe2163177356717951981od_c_b @ ( product_Pair_c_b @ A @ B ) @ ( fSet_fSigma_c_b @ A4 @ B4 ) )
     => ~ ( ( fmember_c @ A @ A4 )
         => ~ ( fmember_b @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_822_fSigmaE2,axiom,
    ! [A: b,B: nat,A4: fset_b,B4: b > fset_nat] :
      ( ( fmembe1853531392176376823_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( fSet_fSigma_b_nat @ A4 @ B4 ) )
     => ~ ( ( fmember_b @ A @ A4 )
         => ~ ( fmember_nat @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_823_fSigmaE2,axiom,
    ! [A: b,B: c,A4: fset_b,B4: b > fset_c] :
      ( ( fmembe4950632942993856815od_b_c @ ( product_Pair_b_c @ A @ B ) @ ( fSet_fSigma_b_c @ A4 @ B4 ) )
     => ~ ( ( fmember_b @ A @ A4 )
         => ~ ( fmember_c @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_824_fSigmaE2,axiom,
    ! [A: nat,B: nat,A4: fset_nat,B4: nat > fset_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_fSigma_nat_nat @ A4 @ B4 ) )
     => ~ ( ( fmember_nat @ A @ A4 )
         => ~ ( fmember_nat @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_825_fSigmaE2,axiom,
    ! [A: c,B: c,A4: fset_c,B4: c > fset_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_fSigma_c_c @ A4 @ B4 ) )
     => ~ ( ( fmember_c @ A @ A4 )
         => ~ ( fmember_c @ B @ ( B4 @ A ) ) ) ) ).

% fSigmaE2
thf(fact_826_ta__reachable__gtermE,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t1746638234261971074_a_b_a @ A6 ) )
     => ~ ! [T2: ground_gterm_a] :
            ~ ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( ground678623728659755670rm_a_b @ T2 ) ) ) ) ).

% ta_reachable_gtermE
thf(fact_827_ta__reachable__gtermE,axiom,
    ! [Q2: b,A6: tree_ta_b_a] :
      ( ( fmember_b @ Q2 @ ( tree_t1956540421582741029le_b_a @ A6 ) )
     => ~ ! [T2: ground_gterm_a] :
            ~ ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( ground5786051259813020157rm_a_b @ T2 ) ) ) ) ).

% ta_reachable_gtermE
thf(fact_828_ta__reachable__gtermE,axiom,
    ! [Q2: c,A6: tree_ta_c_a] :
      ( ( fmember_c @ Q2 @ ( tree_t8392456876464840804le_c_a @ A6 ) )
     => ~ ! [T2: ground_gterm_a] :
            ~ ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( ground5786051259813020158rm_a_c @ T2 ) ) ) ) ).

% ta_reachable_gtermE
thf(fact_829_ta__reachable__gtermE,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t3813786739961944781rm_a_a @ A6 ) )
     => ~ ! [T2: ground_gterm_a] :
            ~ ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( ground2947561787367041419term_a @ T2 ) ) ) ) ).

% ta_reachable_gtermE
thf(fact_830_ta__der__ctxt,axiom,
    ! [P2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,T: term_a4617543059560371146term_a,Q2: bot_bot_term_a,C: subter720051973439326747term_a] :
      ( ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
     => ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( subter8974050253859094258term_a @ C @ ( var_Bot_bot_term_a_a @ P2 ) ) ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( subter8974050253859094258term_a @ C @ T ) ) ) ) ) ).

% ta_der_ctxt
thf(fact_831_ta__der__ctxt,axiom,
    ! [P2: term_a_b,A6: tree_ta_term_a_b_a,T: term_a_term_a_b,Q2: term_a_b,C: subter2881564190832396198rm_a_b] :
      ( ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
     => ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( subter7263177522250274301rm_a_b @ C @ ( var_term_a_b_a @ P2 ) ) ) )
       => ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( subter7263177522250274301rm_a_b @ C @ T ) ) ) ) ) ).

% ta_der_ctxt
thf(fact_832_ta__der__ctxt,axiom,
    ! [P2: b,A6: tree_ta_b_a,T: term_a_b,Q2: b,C: subterm_and_ctxt_a_b] :
      ( ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A6 @ T ) )
     => ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( subter2376574525758040790rm_a_b @ C @ ( var_b_a @ P2 ) ) ) )
       => ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( subter2376574525758040790rm_a_b @ C @ T ) ) ) ) ) ).

% ta_der_ctxt
thf(fact_833_ta__der__ctxt,axiom,
    ! [P2: c,A6: tree_ta_c_a,T: term_a_c,Q2: c,C: subterm_and_ctxt_a_c] :
      ( ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A6 @ T ) )
     => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ ( var_c_a @ P2 ) ) ) )
       => ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ T ) ) ) ) ) ).

% ta_der_ctxt
thf(fact_834_ta__der__ctxt__decompose,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,C: subter720051973439326747term_a,T: term_a4617543059560371146term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( subter8974050253859094258term_a @ C @ T ) ) )
     => ? [P3: bot_bot_term_a] :
          ( ( fmembe1418995468851910914term_a @ P3 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
          & ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( subter8974050253859094258term_a @ C @ ( var_Bot_bot_term_a_a @ P3 ) ) ) ) ) ) ).

% ta_der_ctxt_decompose
thf(fact_835_ta__der__ctxt__decompose,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a,C: subter2881564190832396198rm_a_b,T: term_a_term_a_b] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( subter7263177522250274301rm_a_b @ C @ T ) ) )
     => ? [P3: term_a_b] :
          ( ( fmember_term_a_b @ P3 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
          & ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( subter7263177522250274301rm_a_b @ C @ ( var_term_a_b_a @ P3 ) ) ) ) ) ) ).

% ta_der_ctxt_decompose
thf(fact_836_ta__der__ctxt__decompose,axiom,
    ! [Q2: b,A6: tree_ta_b_a,C: subterm_and_ctxt_a_b,T: term_a_b] :
      ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( subter2376574525758040790rm_a_b @ C @ T ) ) )
     => ? [P3: b] :
          ( ( fmember_b @ P3 @ ( tree_ta_der_b_a @ A6 @ T ) )
          & ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( subter2376574525758040790rm_a_b @ C @ ( var_b_a @ P3 ) ) ) ) ) ) ).

% ta_der_ctxt_decompose
thf(fact_837_ta__der__ctxt__decompose,axiom,
    ! [Q2: c,A6: tree_ta_c_a,C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ T ) ) )
     => ? [P3: c] :
          ( ( fmember_c @ P3 @ ( tree_ta_der_c_a @ A6 @ T ) )
          & ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ ( var_c_a @ P3 ) ) ) ) ) ) ).

% ta_der_ctxt_decompose
thf(fact_838_is__ta__eps__free__def,axiom,
    ( tree_i6598266728802621794ee_b_a
    = ( ^ [A7: tree_ta_b_a] :
          ( ( tree_eps_b_a @ A7 )
          = bot_bo121928729148956601od_b_b ) ) ) ).

% is_ta_eps_free_def
thf(fact_839_is__ta__eps__free__def,axiom,
    ( tree_i3810811146829945761ee_c_a
    = ( ^ [A7: tree_ta_c_a] :
          ( ( tree_eps_c_a @ A7 )
          = bot_bo8780685805445091705od_c_c ) ) ) ).

% is_ta_eps_free_def
thf(fact_840_fthe__felem__eq,axiom,
    ! [X: bot_bot_term_a] :
      ( ( fthe_e5677522983250699724term_a @ ( finser8320498192367885249term_a @ X @ bot_bo4313458260509495885term_a ) )
      = X ) ).

% fthe_felem_eq
thf(fact_841_fthe__felem__eq,axiom,
    ! [X: term_a_b] :
      ( ( fthe_elem_term_a_b @ ( finsert_term_a_b @ X @ bot_bo1583650863486268376rm_a_b ) )
      = X ) ).

% fthe_felem_eq
thf(fact_842_fthe__felem__eq,axiom,
    ! [X: nat] :
      ( ( fthe_elem_nat @ ( finsert_nat @ X @ bot_bot_fset_nat ) )
      = X ) ).

% fthe_felem_eq
thf(fact_843_linear__term_Osimps_I1_J,axiom,
    ! [Uu: c] : ( linear_term_a_c @ ( var_c_a @ Uu ) ) ).

% linear_term.simps(1)
thf(fact_844_linear__term_Osimps_I1_J,axiom,
    ! [Uu: b] : ( linear_term_a_b @ ( var_b_a @ Uu ) ) ).

% linear_term.simps(1)
thf(fact_845__092_060L_062__epmty,axiom,
    ( ( tree_L_b_a @ ( tree_Reg_b_a @ bot_bot_fset_b @ ( tree_TA_b_a @ bot_bo6352603316032826929le_b_a @ bot_bo121928729148956601od_b_b ) ) )
    = bot_bo2731025238921882491term_a ) ).

% \<L>_epmty
thf(fact_846__092_060L_062__epmty,axiom,
    ( ( tree_L8125518851412548553rm_a_a @ ( tree_R2766333946944686633rm_a_a @ bot_bo4313458260509495885term_a @ ( tree_T877157127653060943rm_a_a @ bot_bo1498970239794229685rm_a_a @ bot_bo3304177514724984389term_a ) ) )
    = bot_bo2731025238921882491term_a ) ).

% \<L>_epmty
thf(fact_847_ground__term__of__gterm,axiom,
    ! [S: ground_gterm_a] : ( term_ground_a_b @ ( ground5786051259813020157rm_a_b @ S ) ) ).

% ground_term_of_gterm
thf(fact_848_ground__term__of__gterm,axiom,
    ! [S: ground_gterm_a] : ( term_g4884896889520838057term_a @ ( ground2947561787367041419term_a @ S ) ) ).

% ground_term_of_gterm
thf(fact_849_ground__term__of__gterm,axiom,
    ! [S: ground_gterm_a] : ( term_g2255532262831151796rm_a_b @ ( ground678623728659755670rm_a_b @ S ) ) ).

% ground_term_of_gterm
thf(fact_850_ground__term__of__gterm,axiom,
    ! [S: ground_gterm_a] : ( term_ground_a_c @ ( ground5786051259813020158rm_a_c @ S ) ) ).

% ground_term_of_gterm
thf(fact_851_ground__term__to__gtermD,axiom,
    ! [T: term_a_b] :
      ( ( term_ground_a_b @ T )
     => ? [T3: ground_gterm_a] :
          ( T
          = ( ground5786051259813020157rm_a_b @ T3 ) ) ) ).

% ground_term_to_gtermD
thf(fact_852_ground__term__to__gtermD,axiom,
    ! [T: term_a4617543059560371146term_a] :
      ( ( term_g4884896889520838057term_a @ T )
     => ? [T3: ground_gterm_a] :
          ( T
          = ( ground2947561787367041419term_a @ T3 ) ) ) ).

% ground_term_to_gtermD
thf(fact_853_ground__term__to__gtermD,axiom,
    ! [T: term_a_term_a_b] :
      ( ( term_g2255532262831151796rm_a_b @ T )
     => ? [T3: ground_gterm_a] :
          ( T
          = ( ground678623728659755670rm_a_b @ T3 ) ) ) ).

% ground_term_to_gtermD
thf(fact_854_ground__term__to__gtermD,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ? [T3: ground_gterm_a] :
          ( T
          = ( ground5786051259813020158rm_a_c @ T3 ) ) ) ).

% ground_term_to_gtermD
thf(fact_855_ta__der_H__Var,axiom,
    ! [Q2: c,A6: tree_ta_c_a,X: c] :
      ( ( fmember_term_a_c @ ( var_c_a @ Q2 ) @ ( tree_ta_der_c_a2 @ A6 @ ( var_c_a @ X ) ) )
      = ( ( X = Q2 )
        | ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A6 ) ) ) ) ) ).

% ta_der'_Var
thf(fact_856_ta__der_H__Var,axiom,
    ! [Q2: b,A6: tree_ta_b_a,X: b] :
      ( ( fmember_term_a_b @ ( var_b_a @ Q2 ) @ ( tree_ta_der_b_a2 @ A6 @ ( var_b_a @ X ) ) )
      = ( ( X = Q2 )
        | ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A6 ) ) ) ) ) ).

% ta_der'_Var
thf(fact_857_ta__der__rule__empty,axiom,
    ! [Q2: bot_bot_term_a,Delta_epsilon: fset_P6640269656128473term_a,T: term_a4617543059560371146term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ ( tree_T877157127653060943rm_a_a @ bot_bo1498970239794229685rm_a_a @ Delta_epsilon ) @ T ) )
     => ~ ! [P3: bot_bot_term_a] :
            ( ( T
              = ( var_Bot_bot_term_a_a @ P3 ) )
           => ~ ( ( P3 = Q2 )
                | ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ P3 @ Q2 ) @ ( fSet_f7983512403460980795term_a @ Delta_epsilon ) ) ) ) ) ).

% ta_der_rule_empty
thf(fact_858_ta__der__rule__empty,axiom,
    ! [Q2: c,Delta_epsilon: fset_P8014105740199022861od_c_c,T: term_a_c] :
      ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ ( tree_TA_c_a @ bot_bo5716954315427840240le_c_a @ Delta_epsilon ) @ T ) )
     => ~ ! [P3: c] :
            ( ( T
              = ( var_c_a @ P3 ) )
           => ~ ( ( P3 = Q2 )
                | ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P3 @ Q2 ) @ ( fSet_ftrancl_c @ Delta_epsilon ) ) ) ) ) ).

% ta_der_rule_empty
thf(fact_859_ta__der__rule__empty,axiom,
    ! [Q2: term_a_b,Delta_epsilon: fset_P3577670608877728751rm_a_b,T: term_a_term_a_b] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ ( tree_TA_term_a_b_a @ bot_bo3560329175948364010_a_b_a @ Delta_epsilon ) @ T ) )
     => ~ ! [P3: term_a_b] :
            ( ( T
              = ( var_term_a_b_a @ P3 ) )
           => ~ ( ( P3 = Q2 )
                | ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ P3 @ Q2 ) @ ( fSet_f6224701657781119302rm_a_b @ Delta_epsilon ) ) ) ) ) ).

% ta_der_rule_empty
thf(fact_860_ta__der__rule__empty,axiom,
    ! [Q2: b,Delta_epsilon: fset_P8578720700757663565od_b_b,T: term_a_b] :
      ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ ( tree_TA_b_a @ bot_bo6352603316032826929le_b_a @ Delta_epsilon ) @ T ) )
     => ~ ! [P3: b] :
            ( ( T
              = ( var_b_a @ P3 ) )
           => ~ ( ( P3 = Q2 )
                | ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P3 @ Q2 ) @ ( fSet_ftrancl_b @ Delta_epsilon ) ) ) ) ) ).

% ta_der_rule_empty
thf(fact_861_singelton__ftrancl,axiom,
    ! [A: product_prod_b_b] :
      ( ( fSet_ftrancl_b @ ( finser2611709832528954927od_b_b @ A @ bot_bo121928729148956601od_b_b ) )
      = ( finser2611709832528954927od_b_b @ A @ bot_bo121928729148956601od_b_b ) ) ).

% singelton_ftrancl
thf(fact_862_singelton__ftrancl,axiom,
    ! [A: produc7033602110846336755term_a] :
      ( ( fSet_f7983512403460980795term_a @ ( finser5549460190070563387term_a @ A @ bot_bo3304177514724984389term_a ) )
      = ( finser5549460190070563387term_a @ A @ bot_bo3304177514724984389term_a ) ) ).

% singelton_ftrancl
thf(fact_863_singelton__ftrancl,axiom,
    ! [A: product_prod_c_c] :
      ( ( fSet_ftrancl_c @ ( finser9047626291714283503od_c_c @ A @ bot_bo8780685805445091705od_c_c ) )
      = ( finser9047626291714283503od_c_c @ A @ bot_bo8780685805445091705od_c_c ) ) ).

% singelton_ftrancl
thf(fact_864_ftrancl__empty,axiom,
    ( ( fSet_ftrancl_b @ bot_bo121928729148956601od_b_b )
    = bot_bo121928729148956601od_b_b ) ).

% ftrancl_empty
thf(fact_865_ftrancl__empty,axiom,
    ( ( fSet_f7983512403460980795term_a @ bot_bo3304177514724984389term_a )
    = bot_bo3304177514724984389term_a ) ).

% ftrancl_empty
thf(fact_866_ftrancl__empty,axiom,
    ( ( fSet_ftrancl_c @ bot_bo8780685805445091705od_c_c )
    = bot_bo8780685805445091705od_c_c ) ).

% ftrancl_empty
thf(fact_867_fr__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
     => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f6224701657781119302rm_a_b @ R ) ) ) ).

% fr_into_trancl
thf(fact_868_fr__into__trancl,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ R )
     => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f445164888270409035term_a @ R ) ) ) ).

% fr_into_trancl
thf(fact_869_fr__into__trancl,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_ftrancl_nat @ R ) ) ) ).

% fr_into_trancl
thf(fact_870_fr__into__trancl,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
     => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_ftrancl_b @ R ) ) ) ).

% fr_into_trancl
thf(fact_871_fr__into__trancl,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
     => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_ftrancl_c @ R ) ) ) ).

% fr_into_trancl
thf(fact_872_fr__into__trancl,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
     => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f7983512403460980795term_a @ R ) ) ) ).

% fr_into_trancl
thf(fact_873_ta__der__Var,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,X: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ ( var_Bot_bot_term_a_a @ X ) ) )
      = ( ( X = Q2 )
        | ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Q2 ) @ ( fSet_f7983512403460980795term_a @ ( tree_e5393235435095066050rm_a_a @ A6 ) ) ) ) ) ).

% ta_der_Var
thf(fact_874_ta__der__Var,axiom,
    ! [Q2: c,A6: tree_ta_c_a,X: c] :
      ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( var_c_a @ X ) ) )
      = ( ( X = Q2 )
        | ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A6 ) ) ) ) ) ).

% ta_der_Var
thf(fact_875_ta__der__Var,axiom,
    ! [Q2: term_a_b,A6: tree_ta_term_a_b_a,X: term_a_b] :
      ( ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ ( var_term_a_b_a @ X ) ) )
      = ( ( X = Q2 )
        | ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Q2 ) @ ( fSet_f6224701657781119302rm_a_b @ ( tree_eps_term_a_b_a @ A6 ) ) ) ) ) ).

% ta_der_Var
thf(fact_876_ta__der__Var,axiom,
    ! [Q2: b,A6: tree_ta_b_a,X: b] :
      ( ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( var_b_a @ X ) ) )
      = ( ( X = Q2 )
        | ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A6 ) ) ) ) ) ).

% ta_der_Var
thf(fact_877_frelcomp__empty__ftrancl__simp,axiom,
    ! [R2: fset_P8578720700757663565od_b_b] :
      ( ( ( fSet_frelcomp_b_b_b @ R2 @ R2 )
        = bot_bo121928729148956601od_b_b )
     => ( ( fSet_ftrancl_b @ R2 )
        = R2 ) ) ).

% frelcomp_empty_ftrancl_simp
thf(fact_878_frelcomp__empty__ftrancl__simp,axiom,
    ! [R2: fset_P8014105740199022861od_c_c] :
      ( ( ( fSet_frelcomp_c_c_c @ R2 @ R2 )
        = bot_bo8780685805445091705od_c_c )
     => ( ( fSet_ftrancl_c @ R2 )
        = R2 ) ) ).

% frelcomp_empty_ftrancl_simp
thf(fact_879_frelcomp__empty__ftrancl__simp,axiom,
    ! [R2: fset_P6640269656128473term_a] :
      ( ( ( fSet_f5283962319602365193term_a @ R2 @ R2 )
        = bot_bo3304177514724984389term_a )
     => ( ( fSet_f7983512403460980795term_a @ R2 )
        = R2 ) ) ).

% frelcomp_empty_ftrancl_simp
thf(fact_880_ftranclD,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( fSet_f6224701657781119302rm_a_b @ R2 ) )
     => ? [Z2: term_a_b] :
          ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
          & ( ( Z2 = Y )
            | ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ ( fSet_f6224701657781119302rm_a_b @ R2 ) ) ) ) ) ).

% ftranclD
thf(fact_881_ftranclD,axiom,
    ! [X: list_Bot_bot_term_a,Y: list_Bot_bot_term_a,R2: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Y ) @ ( fSet_f445164888270409035term_a @ R2 ) )
     => ? [Z2: list_Bot_bot_term_a] :
          ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Z2 ) @ R2 )
          & ( ( Z2 = Y )
            | ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Z2 @ Y ) @ ( fSet_f445164888270409035term_a @ R2 ) ) ) ) ) ).

% ftranclD
thf(fact_882_ftranclD,axiom,
    ! [X: nat,Y: nat,R2: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R2 ) )
     => ? [Z2: nat] :
          ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 )
          & ( ( Z2 = Y )
            | ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Z2 @ Y ) @ ( fSet_ftrancl_nat @ R2 ) ) ) ) ) ).

% ftranclD
thf(fact_883_ftranclD,axiom,
    ! [X: b,Y: b,R2: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R2 ) )
     => ? [Z2: b] :
          ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Z2 ) @ R2 )
          & ( ( Z2 = Y )
            | ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Z2 @ Y ) @ ( fSet_ftrancl_b @ R2 ) ) ) ) ) ).

% ftranclD
thf(fact_884_ftranclD,axiom,
    ! [X: c,Y: c,R2: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R2 ) )
     => ? [Z2: c] :
          ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Z2 ) @ R2 )
          & ( ( Z2 = Y )
            | ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Z2 @ Y ) @ ( fSet_ftrancl_c @ R2 ) ) ) ) ) ).

% ftranclD
thf(fact_885_ftranclD,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,R2: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( fSet_f7983512403460980795term_a @ R2 ) )
     => ? [Z2: bot_bot_term_a] :
          ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Z2 ) @ R2 )
          & ( ( Z2 = Y )
            | ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Z2 @ Y ) @ ( fSet_f7983512403460980795term_a @ R2 ) ) ) ) ) ).

% ftranclD
thf(fact_886_ftranclE,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
       => ~ ! [C5: term_a_b] :
              ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C5 ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
             => ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ C5 @ B ) @ R ) ) ) ) ).

% ftranclE
thf(fact_887_ftranclE,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ R )
       => ~ ! [C5: list_Bot_bot_term_a] :
              ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ C5 ) @ ( fSet_f445164888270409035term_a @ R ) )
             => ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ C5 @ B ) @ R ) ) ) ) ).

% ftranclE
thf(fact_888_ftranclE,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ~ ! [C5: nat] :
              ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ C5 ) @ ( fSet_ftrancl_nat @ R ) )
             => ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ C5 @ B ) @ R ) ) ) ) ).

% ftranclE
thf(fact_889_ftranclE,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_ftrancl_b @ R ) )
     => ( ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
       => ~ ! [C5: b] :
              ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ C5 ) @ ( fSet_ftrancl_b @ R ) )
             => ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ C5 @ B ) @ R ) ) ) ) ).

% ftranclE
thf(fact_890_ftranclE,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_ftrancl_c @ R ) )
     => ( ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
       => ~ ! [C5: c] :
              ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ C5 ) @ ( fSet_ftrancl_c @ R ) )
             => ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ C5 @ B ) @ R ) ) ) ) ).

% ftranclE
thf(fact_891_ftranclE,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
       => ~ ! [C5: bot_bot_term_a] :
              ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ C5 ) @ ( fSet_f7983512403460980795term_a @ R ) )
             => ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ C5 @ B ) @ R ) ) ) ) ).

% ftranclE
thf(fact_892_ftranclD2,axiom,
    ! [X: term_a_b,Y: term_a_b,R2: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( fSet_f6224701657781119302rm_a_b @ R2 ) )
     => ? [Z2: term_a_b] :
          ( ( ( X = Z2 )
            | ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( fSet_f6224701657781119302rm_a_b @ R2 ) ) )
          & ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ R2 ) ) ) ).

% ftranclD2
thf(fact_893_ftranclD2,axiom,
    ! [X: list_Bot_bot_term_a,Y: list_Bot_bot_term_a,R2: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Y ) @ ( fSet_f445164888270409035term_a @ R2 ) )
     => ? [Z2: list_Bot_bot_term_a] :
          ( ( ( X = Z2 )
            | ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Z2 ) @ ( fSet_f445164888270409035term_a @ R2 ) ) )
          & ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Z2 @ Y ) @ R2 ) ) ) ).

% ftranclD2
thf(fact_894_ftranclD2,axiom,
    ! [X: nat,Y: nat,R2: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R2 ) )
     => ? [Z2: nat] :
          ( ( ( X = Z2 )
            | ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( fSet_ftrancl_nat @ R2 ) ) )
          & ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Z2 @ Y ) @ R2 ) ) ) ).

% ftranclD2
thf(fact_895_ftranclD2,axiom,
    ! [X: b,Y: b,R2: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R2 ) )
     => ? [Z2: b] :
          ( ( ( X = Z2 )
            | ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Z2 ) @ ( fSet_ftrancl_b @ R2 ) ) )
          & ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Z2 @ Y ) @ R2 ) ) ) ).

% ftranclD2
thf(fact_896_ftranclD2,axiom,
    ! [X: c,Y: c,R2: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R2 ) )
     => ? [Z2: c] :
          ( ( ( X = Z2 )
            | ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Z2 ) @ ( fSet_ftrancl_c @ R2 ) ) )
          & ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Z2 @ Y ) @ R2 ) ) ) ).

% ftranclD2
thf(fact_897_ftranclD2,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,R2: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( fSet_f7983512403460980795term_a @ R2 ) )
     => ? [Z2: bot_bot_term_a] :
          ( ( ( X = Z2 )
            | ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Z2 ) @ ( fSet_f7983512403460980795term_a @ R2 ) ) )
          & ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Z2 @ Y ) @ R2 ) ) ) ).

% ftranclD2
thf(fact_898_ftrancl__map,axiom,
    ! [R: fset_P6640269656128473term_a,F4: bot_bot_term_a > bot_bot_term_a,S: fset_P6640269656128473term_a,X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R )
         => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( fSet_f7983512403460980795term_a @ R ) )
       => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_f7983512403460980795term_a @ S ) ) ) ) ).

% ftrancl_map
thf(fact_899_ftrancl__map,axiom,
    ! [R: fset_P5670320511379867111at_nat,F4: nat > nat,S: fset_P5670320511379867111at_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R ) )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_nat @ S ) ) ) ) ).

% ftrancl_map
thf(fact_900_ftrancl__map,axiom,
    ! [R: fset_P5670320511379867111at_nat,F4: nat > b,S: fset_P8578720700757663565od_b_b,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R ) )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_b @ S ) ) ) ) ).

% ftrancl_map
thf(fact_901_ftrancl__map,axiom,
    ! [R: fset_P5670320511379867111at_nat,F4: nat > c,S: fset_P8014105740199022861od_c_c,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R ) )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_c @ S ) ) ) ) ).

% ftrancl_map
thf(fact_902_ftrancl__map,axiom,
    ! [R: fset_P8578720700757663565od_b_b,F4: b > nat,S: fset_P5670320511379867111at_nat,X: b,Y: b] :
      ( ! [X3: b,Y3: b] :
          ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
         => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R ) )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_nat @ S ) ) ) ) ).

% ftrancl_map
thf(fact_903_ftrancl__map,axiom,
    ! [R: fset_P8578720700757663565od_b_b,F4: b > b,S: fset_P8578720700757663565od_b_b,X: b,Y: b] :
      ( ! [X3: b,Y3: b] :
          ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
         => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R ) )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_b @ S ) ) ) ) ).

% ftrancl_map
thf(fact_904_ftrancl__map,axiom,
    ! [R: fset_P8578720700757663565od_b_b,F4: b > c,S: fset_P8014105740199022861od_c_c,X: b,Y: b] :
      ( ! [X3: b,Y3: b] :
          ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
         => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R ) )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_c @ S ) ) ) ) ).

% ftrancl_map
thf(fact_905_ftrancl__map,axiom,
    ! [R: fset_P8014105740199022861od_c_c,F4: c > nat,S: fset_P5670320511379867111at_nat,X: c,Y: c] :
      ( ! [X3: c,Y3: c] :
          ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X3 @ Y3 ) @ R )
         => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R ) )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_nat @ S ) ) ) ) ).

% ftrancl_map
thf(fact_906_ftrancl__map,axiom,
    ! [R: fset_P8014105740199022861od_c_c,F4: c > b,S: fset_P8578720700757663565od_b_b,X: c,Y: c] :
      ( ! [X3: c,Y3: c] :
          ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X3 @ Y3 ) @ R )
         => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R ) )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_b @ S ) ) ) ) ).

% ftrancl_map
thf(fact_907_ftrancl__map,axiom,
    ! [R: fset_P8014105740199022861od_c_c,F4: c > c,S: fset_P8014105740199022861od_c_c,X: c,Y: c] :
      ( ! [X3: c,Y3: c] :
          ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X3 @ Y3 ) @ R )
         => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R ) )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( fSet_ftrancl_c @ S ) ) ) ) ).

% ftrancl_map
thf(fact_908_ftrancl__trans,axiom,
    ! [X: term_a_b,Y: term_a_b,R: fset_P3577670608877728751rm_a_b,Z: term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
       => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( fSet_f6224701657781119302rm_a_b @ R ) ) ) ) ).

% ftrancl_trans
thf(fact_909_ftrancl__trans,axiom,
    ! [X: list_Bot_bot_term_a,Y: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a,Z: list_Bot_bot_term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Y ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Y @ Z ) @ ( fSet_f445164888270409035term_a @ R ) )
       => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Z ) @ ( fSet_f445164888270409035term_a @ R ) ) ) ) ).

% ftrancl_trans
thf(fact_910_ftrancl__trans,axiom,
    ! [X: nat,Y: nat,R: fset_P5670320511379867111at_nat,Z: nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( fSet_ftrancl_nat @ R ) )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( fSet_ftrancl_nat @ R ) ) ) ) ).

% ftrancl_trans
thf(fact_911_ftrancl__trans,axiom,
    ! [X: b,Y: b,R: fset_P8578720700757663565od_b_b,Z: b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ ( fSet_ftrancl_b @ R ) )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Z ) @ ( fSet_ftrancl_b @ R ) ) ) ) ).

% ftrancl_trans
thf(fact_912_ftrancl__trans,axiom,
    ! [X: c,Y: c,R: fset_P8014105740199022861od_c_c,Z: c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Y @ Z ) @ ( fSet_ftrancl_c @ R ) )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Z ) @ ( fSet_ftrancl_c @ R ) ) ) ) ).

% ftrancl_trans
thf(fact_913_ftrancl__trans,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,R: fset_P6640269656128473term_a,Z: bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Y @ Z ) @ ( fSet_f7983512403460980795term_a @ R ) )
       => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Z ) @ ( fSet_f7983512403460980795term_a @ R ) ) ) ) ).

% ftrancl_trans
thf(fact_914_ftrancl__induct,axiom,
    ! [X1: term_a_b,X2: term_a_b,R: fset_P3577670608877728751rm_a_b,P: term_a_b > term_a_b > $o] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X1 @ X2 ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ! [A3: term_a_b,B3: term_a_b] :
            ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: term_a_b,B3: term_a_b,C5: term_a_b] :
              ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
             => ( ( P @ A3 @ B3 )
               => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C5 ) @ R )
                 => ( P @ A3 @ C5 ) ) ) )
         => ( P @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_915_ftrancl__induct,axiom,
    ! [X1: list_Bot_bot_term_a,X2: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a,P: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X1 @ X2 ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ! [A3: list_Bot_bot_term_a,B3: list_Bot_bot_term_a] :
            ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A3 @ B3 ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: list_Bot_bot_term_a,B3: list_Bot_bot_term_a,C5: list_Bot_bot_term_a] :
              ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A3 @ B3 ) @ ( fSet_f445164888270409035term_a @ R ) )
             => ( ( P @ A3 @ B3 )
               => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ B3 @ C5 ) @ R )
                 => ( P @ A3 @ C5 ) ) ) )
         => ( P @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_916_ftrancl__induct,axiom,
    ! [X1: nat,X2: nat,R: fset_P5670320511379867111at_nat,P: nat > nat > $o] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: nat,B3: nat,C5: nat] :
              ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( fSet_ftrancl_nat @ R ) )
             => ( ( P @ A3 @ B3 )
               => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ B3 @ C5 ) @ R )
                 => ( P @ A3 @ C5 ) ) ) )
         => ( P @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_917_ftrancl__induct,axiom,
    ! [X1: b,X2: b,R: fset_P8578720700757663565od_b_b,P: b > b > $o] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X1 @ X2 ) @ ( fSet_ftrancl_b @ R ) )
     => ( ! [A3: b,B3: b] :
            ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: b,B3: b,C5: b] :
              ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ ( fSet_ftrancl_b @ R ) )
             => ( ( P @ A3 @ B3 )
               => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ B3 @ C5 ) @ R )
                 => ( P @ A3 @ C5 ) ) ) )
         => ( P @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_918_ftrancl__induct,axiom,
    ! [X1: c,X2: c,R: fset_P8014105740199022861od_c_c,P: c > c > $o] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X1 @ X2 ) @ ( fSet_ftrancl_c @ R ) )
     => ( ! [A3: c,B3: c] :
            ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A3 @ B3 ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: c,B3: c,C5: c] :
              ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A3 @ B3 ) @ ( fSet_ftrancl_c @ R ) )
             => ( ( P @ A3 @ B3 )
               => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ B3 @ C5 ) @ R )
                 => ( P @ A3 @ C5 ) ) ) )
         => ( P @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_919_ftrancl__induct,axiom,
    ! [X1: bot_bot_term_a,X2: bot_bot_term_a,R: fset_P6640269656128473term_a,P: bot_bot_term_a > bot_bot_term_a > $o] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X1 @ X2 ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a] :
            ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A3 @ B3 ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a,C5: bot_bot_term_a] :
              ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A3 @ B3 ) @ ( fSet_f7983512403460980795term_a @ R ) )
             => ( ( P @ A3 @ B3 )
               => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ B3 @ C5 ) @ R )
                 => ( P @ A3 @ C5 ) ) ) )
         => ( P @ X1 @ X2 ) ) ) ) ).

% ftrancl_induct
thf(fact_920_not__ftrancl__into,axiom,
    ! [X: term_a_b,Z: term_a_b,R: fset_P3577670608877728751rm_a_b,Y: term_a_b] :
      ( ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R )
       => ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( fSet_f6224701657781119302rm_a_b @ R ) ) ) ) ).

% not_ftrancl_into
thf(fact_921_not__ftrancl__into,axiom,
    ! [X: list_Bot_bot_term_a,Z: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a,Y: list_Bot_bot_term_a] :
      ( ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Z ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Y @ Z ) @ R )
       => ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Y ) @ ( fSet_f445164888270409035term_a @ R ) ) ) ) ).

% not_ftrancl_into
thf(fact_922_not__ftrancl__into,axiom,
    ! [X: nat,Z: nat,R: fset_P5670320511379867111at_nat,Y: nat] :
      ( ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R )
       => ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( fSet_ftrancl_nat @ R ) ) ) ) ).

% not_ftrancl_into
thf(fact_923_not__ftrancl__into,axiom,
    ! [X: b,Z: b,R: fset_P8578720700757663565od_b_b,Y: b] :
      ( ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Z ) @ ( fSet_ftrancl_b @ R ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Y @ Z ) @ R )
       => ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( fSet_ftrancl_b @ R ) ) ) ) ).

% not_ftrancl_into
thf(fact_924_not__ftrancl__into,axiom,
    ! [X: c,Z: c,R: fset_P8014105740199022861od_c_c,Y: c] :
      ( ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Z ) @ ( fSet_ftrancl_c @ R ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Y @ Z ) @ R )
       => ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y ) @ ( fSet_ftrancl_c @ R ) ) ) ) ).

% not_ftrancl_into
thf(fact_925_not__ftrancl__into,axiom,
    ! [X: bot_bot_term_a,Z: bot_bot_term_a,R: fset_P6640269656128473term_a,Y: bot_bot_term_a] :
      ( ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Z ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Y @ Z ) @ R )
       => ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( fSet_f7983512403460980795term_a @ R ) ) ) ) ).

% not_ftrancl_into
thf(fact_926_converse__ftranclE,axiom,
    ! [X: term_a_b,Z: term_a_b,R: fset_P3577670608877728751rm_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R )
       => ~ ! [Y3: term_a_b] :
              ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
             => ~ ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( fSet_f6224701657781119302rm_a_b @ R ) ) ) ) ) ).

% converse_ftranclE
thf(fact_927_converse__ftranclE,axiom,
    ! [X: list_Bot_bot_term_a,Z: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Z ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Z ) @ R )
       => ~ ! [Y3: list_Bot_bot_term_a] :
              ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ X @ Y3 ) @ R )
             => ~ ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Y3 @ Z ) @ ( fSet_f445164888270409035term_a @ R ) ) ) ) ) ).

% converse_ftranclE
thf(fact_928_converse__ftranclE,axiom,
    ! [X: nat,Z: nat,R: fset_P5670320511379867111at_nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R )
       => ~ ! [Y3: nat] :
              ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
             => ~ ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( fSet_ftrancl_nat @ R ) ) ) ) ) ).

% converse_ftranclE
thf(fact_929_converse__ftranclE,axiom,
    ! [X: b,Z: b,R: fset_P8578720700757663565od_b_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Z ) @ ( fSet_ftrancl_b @ R ) )
     => ( ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Z ) @ R )
       => ~ ! [Y3: b] :
              ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ X @ Y3 ) @ R )
             => ~ ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Y3 @ Z ) @ ( fSet_ftrancl_b @ R ) ) ) ) ) ).

% converse_ftranclE
thf(fact_930_converse__ftranclE,axiom,
    ! [X: c,Z: c,R: fset_P8014105740199022861od_c_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Z ) @ ( fSet_ftrancl_c @ R ) )
     => ( ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Z ) @ R )
       => ~ ! [Y3: c] :
              ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ X @ Y3 ) @ R )
             => ~ ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Y3 @ Z ) @ ( fSet_ftrancl_c @ R ) ) ) ) ) ).

% converse_ftranclE
thf(fact_931_converse__ftranclE,axiom,
    ! [X: bot_bot_term_a,Z: bot_bot_term_a,R: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Z ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Z ) @ R )
       => ~ ! [Y3: bot_bot_term_a] :
              ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X @ Y3 ) @ R )
             => ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Y3 @ Z ) @ ( fSet_f7983512403460980795term_a @ R ) ) ) ) ) ).

% converse_ftranclE
thf(fact_932_ftrancl__into__trancl,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b,C2: term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R )
       => ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( fSet_f6224701657781119302rm_a_b @ R ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_933_ftrancl__into__trancl,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a,C2: list_Bot_bot_term_a] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ B @ C2 ) @ R )
       => ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ C2 ) @ ( fSet_f445164888270409035term_a @ R ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_934_ftrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat,C2: nat] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R )
       => ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( fSet_ftrancl_nat @ R ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_935_ftrancl__into__trancl,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b,C2: b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_ftrancl_b @ R ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ B @ C2 ) @ R )
       => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ ( fSet_ftrancl_b @ R ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_936_ftrancl__into__trancl,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c,C2: c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_ftrancl_c @ R ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ B @ C2 ) @ R )
       => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ C2 ) @ ( fSet_ftrancl_c @ R ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_937_ftrancl__into__trancl,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a,C2: bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ B @ C2 ) @ R )
       => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ C2 ) @ ( fSet_f7983512403460980795term_a @ R ) ) ) ) ).

% ftrancl_into_trancl
thf(fact_938_converse__ftrancl__induct,axiom,
    ! [A: term_a_b,B: term_a_b,R: fset_P3577670608877728751rm_a_b,P: term_a_b > $o] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
     => ( ! [Y3: term_a_b] :
            ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: term_a_b,Z2: term_a_b] :
              ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R )
             => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ B ) @ ( fSet_f6224701657781119302rm_a_b @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_939_converse__ftrancl__induct,axiom,
    ! [A: list_Bot_bot_term_a,B: list_Bot_bot_term_a,R: fset_P7293260781336502521term_a,P: list_Bot_bot_term_a > $o] :
      ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ A @ B ) @ ( fSet_f445164888270409035term_a @ R ) )
     => ( ! [Y3: list_Bot_bot_term_a] :
            ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: list_Bot_bot_term_a,Z2: list_Bot_bot_term_a] :
              ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Y3 @ Z2 ) @ R )
             => ( ( fmembe4453961250406969178term_a @ ( produc1077782494051512011term_a @ Z2 @ B ) @ ( fSet_f445164888270409035term_a @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_940_converse__ftrancl__induct,axiom,
    ! [A: nat,B: nat,R: fset_P5670320511379867111at_nat,P: nat > $o] :
      ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( fSet_ftrancl_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: nat,Z2: nat] :
              ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R )
             => ( ( fmembe1449479052724974408at_nat @ ( product_Pair_nat_nat @ Z2 @ B ) @ ( fSet_ftrancl_nat @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_941_converse__ftrancl__induct,axiom,
    ! [A: b,B: b,R: fset_P8578720700757663565od_b_b,P: b > $o] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( fSet_ftrancl_b @ R ) )
     => ( ! [Y3: b] :
            ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: b,Z2: b] :
              ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Y3 @ Z2 ) @ R )
             => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Z2 @ B ) @ ( fSet_ftrancl_b @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_942_converse__ftrancl__induct,axiom,
    ! [A: c,B: c,R: fset_P8014105740199022861od_c_c,P: c > $o] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ A @ B ) @ ( fSet_ftrancl_c @ R ) )
     => ( ! [Y3: c] :
            ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: c,Z2: c] :
              ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Y3 @ Z2 ) @ R )
             => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Z2 @ B ) @ ( fSet_ftrancl_c @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_943_converse__ftrancl__induct,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: fset_P6640269656128473term_a,P: bot_bot_term_a > $o] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( fSet_f7983512403460980795term_a @ R ) )
     => ( ! [Y3: bot_bot_term_a] :
            ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: bot_bot_term_a,Z2: bot_bot_term_a] :
              ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Y3 @ Z2 ) @ R )
             => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Z2 @ B ) @ ( fSet_f7983512403460980795term_a @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_ftrancl_induct
thf(fact_944_ta__der__trancl__eps,axiom,
    ! [P2: bot_bot_term_a,Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,T: term_a4617543059560371146term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ P2 @ Q2 ) @ ( fSet_f7983512403460980795term_a @ ( tree_e5393235435095066050rm_a_a @ A6 ) ) )
     => ( ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) )
       => ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t7856459924191630025rm_a_a @ A6 @ T ) ) ) ) ).

% ta_der_trancl_eps
thf(fact_945_ta__der__trancl__eps,axiom,
    ! [P2: c,Q2: c,A6: tree_ta_c_a,T: term_a_c] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A6 ) ) )
     => ( ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A6 @ T ) )
       => ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ T ) ) ) ) ).

% ta_der_trancl_eps
thf(fact_946_ta__der__trancl__eps,axiom,
    ! [P2: term_a_b,Q2: term_a_b,A6: tree_ta_term_a_b_a,T: term_a_term_a_b] :
      ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ P2 @ Q2 ) @ ( fSet_f6224701657781119302rm_a_b @ ( tree_eps_term_a_b_a @ A6 ) ) )
     => ( ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) )
       => ( fmember_term_a_b @ Q2 @ ( tree_t8390983252668960894_a_b_a @ A6 @ T ) ) ) ) ).

% ta_der_trancl_eps
thf(fact_947_ta__der__trancl__eps,axiom,
    ! [P2: b,Q2: b,A6: tree_ta_b_a,T: term_a_b] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A6 ) ) )
     => ( ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A6 @ T ) )
       => ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ T ) ) ) ) ).

% ta_der_trancl_eps
thf(fact_948_ta__reachableI__eps_H,axiom,
    ! [Q2: b,A6: tree_ta_b_a,Q4: b] :
      ( ( fmember_b @ Q2 @ ( tree_t1956540421582741029le_b_a @ A6 ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Q2 @ Q4 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A6 ) ) )
       => ( fmember_b @ Q4 @ ( tree_t1956540421582741029le_b_a @ A6 ) ) ) ) ).

% ta_reachableI_eps'
thf(fact_949_ta__reachableI__eps_H,axiom,
    ! [Q2: c,A6: tree_ta_c_a,Q4: c] :
      ( ( fmember_c @ Q2 @ ( tree_t8392456876464840804le_c_a @ A6 ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Q2 @ Q4 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A6 ) ) )
       => ( fmember_c @ Q4 @ ( tree_t8392456876464840804le_c_a @ A6 ) ) ) ) ).

% ta_reachableI_eps'
thf(fact_950_ta__reachableI__eps_H,axiom,
    ! [Q2: bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a,Q4: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ Q2 @ ( tree_t3813786739961944781rm_a_a @ A6 ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Q2 @ Q4 ) @ ( fSet_f7983512403460980795term_a @ ( tree_e5393235435095066050rm_a_a @ A6 ) ) )
       => ( fmembe1418995468851910914term_a @ Q4 @ ( tree_t3813786739961944781rm_a_a @ A6 ) ) ) ) ).

% ta_reachableI_eps'
thf(fact_951_ta__der_H__eps,axiom,
    ! [P2: c,A6: tree_ta_c_a,S: term_a_c,Q2: c] :
      ( ( fmember_term_a_c @ ( var_c_a @ P2 ) @ ( tree_ta_der_c_a2 @ A6 @ S ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A6 ) ) )
       => ( fmember_term_a_c @ ( var_c_a @ Q2 ) @ ( tree_ta_der_c_a2 @ A6 @ S ) ) ) ) ).

% ta_der'_eps
thf(fact_952_ta__der_H__eps,axiom,
    ! [P2: b,A6: tree_ta_b_a,S: term_a_b,Q2: b] :
      ( ( fmember_term_a_b @ ( var_b_a @ P2 ) @ ( tree_ta_der_b_a2 @ A6 @ S ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A6 ) ) )
       => ( fmember_term_a_b @ ( var_b_a @ Q2 ) @ ( tree_ta_der_b_a2 @ A6 @ S ) ) ) ) ).

% ta_der'_eps
thf(fact_953_ta__der__eps__ctxt,axiom,
    ! [P2: bot_bot_term_a,A4: tree_t5222909361115825503rm_a_a,C: subter720051973439326747term_a,Q4: bot_bot_term_a,Q2: bot_bot_term_a] :
      ( ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A4 @ ( subter8974050253859094258term_a @ C @ ( var_Bot_bot_term_a_a @ Q4 ) ) ) )
     => ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ Q2 @ Q4 ) @ ( fSet_f7983512403460980795term_a @ ( tree_e5393235435095066050rm_a_a @ A4 ) ) )
       => ( fmembe1418995468851910914term_a @ P2 @ ( tree_t7856459924191630025rm_a_a @ A4 @ ( subter8974050253859094258term_a @ C @ ( var_Bot_bot_term_a_a @ Q2 ) ) ) ) ) ) ).

% ta_der_eps_ctxt
thf(fact_954_ta__der__eps__ctxt,axiom,
    ! [P2: term_a_b,A4: tree_ta_term_a_b_a,C: subter2881564190832396198rm_a_b,Q4: term_a_b,Q2: term_a_b] :
      ( ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A4 @ ( subter7263177522250274301rm_a_b @ C @ ( var_term_a_b_a @ Q4 ) ) ) )
     => ( ( fmembe2328057640968976720rm_a_b @ ( produc7020197800436672577rm_a_b @ Q2 @ Q4 ) @ ( fSet_f6224701657781119302rm_a_b @ ( tree_eps_term_a_b_a @ A4 ) ) )
       => ( fmember_term_a_b @ P2 @ ( tree_t8390983252668960894_a_b_a @ A4 @ ( subter7263177522250274301rm_a_b @ C @ ( var_term_a_b_a @ Q2 ) ) ) ) ) ) ).

% ta_der_eps_ctxt
thf(fact_955_ta__der__eps__ctxt,axiom,
    ! [P2: b,A4: tree_ta_b_a,C: subterm_and_ctxt_a_b,Q4: b,Q2: b] :
      ( ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A4 @ ( subter2376574525758040790rm_a_b @ C @ ( var_b_a @ Q4 ) ) ) )
     => ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ Q2 @ Q4 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A4 ) ) )
       => ( fmember_b @ P2 @ ( tree_ta_der_b_a @ A4 @ ( subter2376574525758040790rm_a_b @ C @ ( var_b_a @ Q2 ) ) ) ) ) ) ).

% ta_der_eps_ctxt
thf(fact_956_ta__der__eps__ctxt,axiom,
    ! [P2: c,A4: tree_ta_c_a,C: subterm_and_ctxt_a_c,Q4: c,Q2: c] :
      ( ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A4 @ ( subter2376574525758040791rm_a_c @ C @ ( var_c_a @ Q4 ) ) ) )
     => ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ Q2 @ Q4 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A4 ) ) )
       => ( fmember_c @ P2 @ ( tree_ta_der_c_a @ A4 @ ( subter2376574525758040791rm_a_c @ C @ ( var_c_a @ Q2 ) ) ) ) ) ) ).

% ta_der_eps_ctxt
thf(fact_957_empty__iff,axiom,
    ! [C2: produc441850963132823271term_a] :
      ~ ( member7618331267207336464term_a @ C2 @ bot_bo6119087614312989235term_a ) ).

% empty_iff
thf(fact_958_empty__iff,axiom,
    ! [C2: produc1095327201564732427rm_a_c] :
      ~ ( member6607649026611388852rm_a_c @ C2 @ bot_bo3144958465932184535rm_a_c ) ).

% empty_iff
thf(fact_959_empty__iff,axiom,
    ! [C2: produc357393685978478089rm_a_b] :
      ~ ( member5869715511025134514rm_a_b @ C2 @ bot_bo197521221353338581rm_a_b ) ).

% empty_iff
thf(fact_960_empty__iff,axiom,
    ! [C2: produc2056175833795995411term_a] :
      ~ ( member4272934500791052220term_a @ C2 @ bot_bo5671043144174269919term_a ) ).

% empty_iff
thf(fact_961_empty__iff,axiom,
    ! [C2: term_a_b] :
      ~ ( member_term_a_b @ C2 @ bot_bot_set_term_a_b ) ).

% empty_iff
thf(fact_962_empty__iff,axiom,
    ! [C2: produc7033602110846336755term_a] :
      ~ ( member2247295376459944092term_a @ C2 @ bot_bo2447593420621686463term_a ) ).

% empty_iff
thf(fact_963_empty__iff,axiom,
    ! [C2: term_a_c] :
      ~ ( member_term_a_c @ C2 @ bot_bot_set_term_a_c ) ).

% empty_iff
thf(fact_964_all__not__in__conv,axiom,
    ! [A4: set_Pr825864233230045127term_a] :
      ( ( ! [X4: produc441850963132823271term_a] :
            ~ ( member7618331267207336464term_a @ X4 @ A4 ) )
      = ( A4 = bot_bo6119087614312989235term_a ) ) ).

% all_not_in_conv
thf(fact_965_all__not__in__conv,axiom,
    ! [A4: set_Pr7334014819586186091rm_a_c] :
      ( ( ! [X4: produc1095327201564732427rm_a_c] :
            ~ ( member6607649026611388852rm_a_c @ X4 @ A4 ) )
      = ( A4 = bot_bo3144958465932184535rm_a_c ) ) ).

% all_not_in_conv
thf(fact_966_all__not__in__conv,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( ! [X4: produc357393685978478089rm_a_b] :
            ~ ( member5869715511025134514rm_a_b @ X4 @ A4 ) )
      = ( A4 = bot_bo197521221353338581rm_a_b ) ) ).

% all_not_in_conv
thf(fact_967_all__not__in__conv,axiom,
    ! [A4: set_Pr7374270731007213939term_a] :
      ( ( ! [X4: produc2056175833795995411term_a] :
            ~ ( member4272934500791052220term_a @ X4 @ A4 ) )
      = ( A4 = bot_bo5671043144174269919term_a ) ) ).

% all_not_in_conv
thf(fact_968_all__not__in__conv,axiom,
    ! [A4: set_term_a_b] :
      ( ( ! [X4: term_a_b] :
            ~ ( member_term_a_b @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_term_a_b ) ) ).

% all_not_in_conv
thf(fact_969_all__not__in__conv,axiom,
    ! [A4: set_Pr3965413743549523539term_a] :
      ( ( ! [X4: produc7033602110846336755term_a] :
            ~ ( member2247295376459944092term_a @ X4 @ A4 ) )
      = ( A4 = bot_bo2447593420621686463term_a ) ) ).

% all_not_in_conv
thf(fact_970_all__not__in__conv,axiom,
    ! [A4: set_term_a_c] :
      ( ( ! [X4: term_a_c] :
            ~ ( member_term_a_c @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_term_a_c ) ) ).

% all_not_in_conv
thf(fact_971_Collect__empty__eq,axiom,
    ! [P: produc7033602110846336755term_a > $o] :
      ( ( ( collec7536399311987840478term_a @ P )
        = bot_bo2447593420621686463term_a )
      = ( ! [X4: produc7033602110846336755term_a] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_972_Collect__empty__eq,axiom,
    ! [P: term_a_c > $o] :
      ( ( ( collect_term_a_c @ P )
        = bot_bot_set_term_a_c )
      = ( ! [X4: term_a_c] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_973_empty__Collect__eq,axiom,
    ! [P: produc7033602110846336755term_a > $o] :
      ( ( bot_bo2447593420621686463term_a
        = ( collec7536399311987840478term_a @ P ) )
      = ( ! [X4: produc7033602110846336755term_a] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_974_empty__Collect__eq,axiom,
    ! [P: term_a_c > $o] :
      ( ( bot_bot_set_term_a_c
        = ( collect_term_a_c @ P ) )
      = ( ! [X4: term_a_c] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_975_restrict__subst__domain__empty,axiom,
    ! [Sigma: b > term_a_b] :
      ( ( restri22458263168500592in_b_a @ bot_bot_set_b @ Sigma )
      = var_b_a ) ).

% restrict_subst_domain_empty
thf(fact_976_restrict__subst__domain__empty,axiom,
    ! [Sigma: c > term_a_c] :
      ( ( restri6458374718050600367in_c_a @ bot_bot_set_c @ Sigma )
      = var_c_a ) ).

% restrict_subst_domain_empty
thf(fact_977_eps__ta__union__eps_I1_J,axiom,
    ! [P2: b,Q2: b,A6: tree_ta_b_a,B7: tree_ta_b_a] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ A6 ) ) )
     => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ ( tree_ta_union_b_a @ A6 @ B7 ) ) ) ) ) ).

% eps_ta_union_eps(1)
thf(fact_978_eps__ta__union__eps_I1_J,axiom,
    ! [P2: c,Q2: c,A6: tree_ta_c_a,B7: tree_ta_c_a] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ A6 ) ) )
     => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ ( tree_ta_union_c_a @ A6 @ B7 ) ) ) ) ) ).

% eps_ta_union_eps(1)
thf(fact_979_restrict__subst__domain__Var,axiom,
    ! [V: set_b] :
      ( ( restri22458263168500592in_b_a @ V @ var_b_a )
      = var_b_a ) ).

% restrict_subst_domain_Var
thf(fact_980_restrict__subst__domain__Var,axiom,
    ! [V: set_c] :
      ( ( restri6458374718050600367in_c_a @ V @ var_c_a )
      = var_c_a ) ).

% restrict_subst_domain_Var
thf(fact_981_restrict__subst__domain__def,axiom,
    ( restri22458263168500592in_b_a
    = ( ^ [V2: set_b,Sigma3: b > term_a_b,X4: b] : ( if_term_a_b @ ( member_b @ X4 @ V2 ) @ ( Sigma3 @ X4 ) @ ( var_b_a @ X4 ) ) ) ) ).

% restrict_subst_domain_def
thf(fact_982_restrict__subst__domain__def,axiom,
    ( restri6458374718050600367in_c_a
    = ( ^ [V2: set_c,Sigma3: c > term_a_c,X4: c] : ( if_term_a_c @ ( member_c @ X4 @ V2 ) @ ( Sigma3 @ X4 ) @ ( var_c_a @ X4 ) ) ) ) ).

% restrict_subst_domain_def
thf(fact_983_bot__set__def,axiom,
    ( bot_bo2447593420621686463term_a
    = ( collec7536399311987840478term_a @ bot_bo104646228311450846rm_a_o ) ) ).

% bot_set_def
thf(fact_984_bot__set__def,axiom,
    ( bot_bot_set_term_a_c
    = ( collect_term_a_c @ bot_bot_term_a_c_o ) ) ).

% bot_set_def
thf(fact_985_ex__in__conv,axiom,
    ! [A4: set_Pr825864233230045127term_a] :
      ( ( ? [X4: produc441850963132823271term_a] : ( member7618331267207336464term_a @ X4 @ A4 ) )
      = ( A4 != bot_bo6119087614312989235term_a ) ) ).

% ex_in_conv
thf(fact_986_ex__in__conv,axiom,
    ! [A4: set_Pr7334014819586186091rm_a_c] :
      ( ( ? [X4: produc1095327201564732427rm_a_c] : ( member6607649026611388852rm_a_c @ X4 @ A4 ) )
      = ( A4 != bot_bo3144958465932184535rm_a_c ) ) ).

% ex_in_conv
thf(fact_987_ex__in__conv,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A4 ) )
      = ( A4 != bot_bo197521221353338581rm_a_b ) ) ).

% ex_in_conv
thf(fact_988_ex__in__conv,axiom,
    ! [A4: set_Pr7374270731007213939term_a] :
      ( ( ? [X4: produc2056175833795995411term_a] : ( member4272934500791052220term_a @ X4 @ A4 ) )
      = ( A4 != bot_bo5671043144174269919term_a ) ) ).

% ex_in_conv
thf(fact_989_ex__in__conv,axiom,
    ! [A4: set_term_a_b] :
      ( ( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_term_a_b ) ) ).

% ex_in_conv
thf(fact_990_ex__in__conv,axiom,
    ! [A4: set_Pr3965413743549523539term_a] :
      ( ( ? [X4: produc7033602110846336755term_a] : ( member2247295376459944092term_a @ X4 @ A4 ) )
      = ( A4 != bot_bo2447593420621686463term_a ) ) ).

% ex_in_conv
thf(fact_991_ex__in__conv,axiom,
    ! [A4: set_term_a_c] :
      ( ( ? [X4: term_a_c] : ( member_term_a_c @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_term_a_c ) ) ).

% ex_in_conv
thf(fact_992_equals0I,axiom,
    ! [A4: set_Pr825864233230045127term_a] :
      ( ! [Y3: produc441850963132823271term_a] :
          ~ ( member7618331267207336464term_a @ Y3 @ A4 )
     => ( A4 = bot_bo6119087614312989235term_a ) ) ).

% equals0I
thf(fact_993_equals0I,axiom,
    ! [A4: set_Pr7334014819586186091rm_a_c] :
      ( ! [Y3: produc1095327201564732427rm_a_c] :
          ~ ( member6607649026611388852rm_a_c @ Y3 @ A4 )
     => ( A4 = bot_bo3144958465932184535rm_a_c ) ) ).

% equals0I
thf(fact_994_equals0I,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b] :
      ( ! [Y3: produc357393685978478089rm_a_b] :
          ~ ( member5869715511025134514rm_a_b @ Y3 @ A4 )
     => ( A4 = bot_bo197521221353338581rm_a_b ) ) ).

% equals0I
thf(fact_995_equals0I,axiom,
    ! [A4: set_Pr7374270731007213939term_a] :
      ( ! [Y3: produc2056175833795995411term_a] :
          ~ ( member4272934500791052220term_a @ Y3 @ A4 )
     => ( A4 = bot_bo5671043144174269919term_a ) ) ).

% equals0I
thf(fact_996_equals0I,axiom,
    ! [A4: set_term_a_b] :
      ( ! [Y3: term_a_b] :
          ~ ( member_term_a_b @ Y3 @ A4 )
     => ( A4 = bot_bot_set_term_a_b ) ) ).

% equals0I
thf(fact_997_equals0I,axiom,
    ! [A4: set_Pr3965413743549523539term_a] :
      ( ! [Y3: produc7033602110846336755term_a] :
          ~ ( member2247295376459944092term_a @ Y3 @ A4 )
     => ( A4 = bot_bo2447593420621686463term_a ) ) ).

% equals0I
thf(fact_998_equals0I,axiom,
    ! [A4: set_term_a_c] :
      ( ! [Y3: term_a_c] :
          ~ ( member_term_a_c @ Y3 @ A4 )
     => ( A4 = bot_bot_set_term_a_c ) ) ).

% equals0I
thf(fact_999_equals0D,axiom,
    ! [A4: set_Pr825864233230045127term_a,A: produc441850963132823271term_a] :
      ( ( A4 = bot_bo6119087614312989235term_a )
     => ~ ( member7618331267207336464term_a @ A @ A4 ) ) ).

% equals0D
thf(fact_1000_equals0D,axiom,
    ! [A4: set_Pr7334014819586186091rm_a_c,A: produc1095327201564732427rm_a_c] :
      ( ( A4 = bot_bo3144958465932184535rm_a_c )
     => ~ ( member6607649026611388852rm_a_c @ A @ A4 ) ) ).

% equals0D
thf(fact_1001_equals0D,axiom,
    ! [A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
      ( ( A4 = bot_bo197521221353338581rm_a_b )
     => ~ ( member5869715511025134514rm_a_b @ A @ A4 ) ) ).

% equals0D
thf(fact_1002_equals0D,axiom,
    ! [A4: set_Pr7374270731007213939term_a,A: produc2056175833795995411term_a] :
      ( ( A4 = bot_bo5671043144174269919term_a )
     => ~ ( member4272934500791052220term_a @ A @ A4 ) ) ).

% equals0D
thf(fact_1003_equals0D,axiom,
    ! [A4: set_term_a_b,A: term_a_b] :
      ( ( A4 = bot_bot_set_term_a_b )
     => ~ ( member_term_a_b @ A @ A4 ) ) ).

% equals0D
thf(fact_1004_equals0D,axiom,
    ! [A4: set_Pr3965413743549523539term_a,A: produc7033602110846336755term_a] :
      ( ( A4 = bot_bo2447593420621686463term_a )
     => ~ ( member2247295376459944092term_a @ A @ A4 ) ) ).

% equals0D
thf(fact_1005_equals0D,axiom,
    ! [A4: set_term_a_c,A: term_a_c] :
      ( ( A4 = bot_bot_set_term_a_c )
     => ~ ( member_term_a_c @ A @ A4 ) ) ).

% equals0D
thf(fact_1006_emptyE,axiom,
    ! [A: produc441850963132823271term_a] :
      ~ ( member7618331267207336464term_a @ A @ bot_bo6119087614312989235term_a ) ).

% emptyE
thf(fact_1007_emptyE,axiom,
    ! [A: produc1095327201564732427rm_a_c] :
      ~ ( member6607649026611388852rm_a_c @ A @ bot_bo3144958465932184535rm_a_c ) ).

% emptyE
thf(fact_1008_emptyE,axiom,
    ! [A: produc357393685978478089rm_a_b] :
      ~ ( member5869715511025134514rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ).

% emptyE
thf(fact_1009_emptyE,axiom,
    ! [A: produc2056175833795995411term_a] :
      ~ ( member4272934500791052220term_a @ A @ bot_bo5671043144174269919term_a ) ).

% emptyE
thf(fact_1010_emptyE,axiom,
    ! [A: term_a_b] :
      ~ ( member_term_a_b @ A @ bot_bot_set_term_a_b ) ).

% emptyE
thf(fact_1011_emptyE,axiom,
    ! [A: produc7033602110846336755term_a] :
      ~ ( member2247295376459944092term_a @ A @ bot_bo2447593420621686463term_a ) ).

% emptyE
thf(fact_1012_emptyE,axiom,
    ! [A: term_a_c] :
      ~ ( member_term_a_c @ A @ bot_bot_set_term_a_c ) ).

% emptyE
thf(fact_1013_eps__ta__union__eps_I2_J,axiom,
    ! [P2: b,Q2: b,B7: tree_ta_b_a,A6: tree_ta_b_a] :
      ( ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ B7 ) ) )
     => ( fmembe4950632938690628014od_b_b @ ( product_Pair_b_b @ P2 @ Q2 ) @ ( fSet_ftrancl_b @ ( tree_eps_b_a @ ( tree_ta_union_b_a @ A6 @ B7 ) ) ) ) ) ).

% eps_ta_union_eps(2)
thf(fact_1014_eps__ta__union__eps_I2_J,axiom,
    ! [P2: c,Q2: c,B7: tree_ta_c_a,A6: tree_ta_c_a] :
      ( ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ B7 ) ) )
     => ( fmembe2163177361021180782od_c_c @ ( product_Pair_c_c @ P2 @ Q2 ) @ ( fSet_ftrancl_c @ ( tree_eps_c_a @ ( tree_ta_union_c_a @ A6 @ B7 ) ) ) ) ) ).

% eps_ta_union_eps(2)
thf(fact_1015_ffuns__term__Var,axiom,
    ! [X: b] :
      ( ( fSet_ffuns_term_a_b @ ( var_b_a @ X ) )
      = bot_bot_fset_a ) ).

% ffuns_term_Var
thf(fact_1016_ffuns__term__Var,axiom,
    ! [X: c] :
      ( ( fSet_ffuns_term_a_c @ ( var_c_a @ X ) )
      = bot_bot_fset_a ) ).

% ffuns_term_Var
thf(fact_1017_ta__der_H__Fun,axiom,
    ! [Q2: b,A6: tree_ta_b_a,F4: a,Ts: list_term_a_b] :
      ( ( fmember_term_a_b @ ( var_b_a @ Q2 ) @ ( tree_ta_der_b_a2 @ A6 @ ( fun_a_b @ F4 @ Ts ) ) )
      = ( fmember_b @ Q2 @ ( tree_ta_der_b_a @ A6 @ ( fun_a_b @ F4 @ Ts ) ) ) ) ).

% ta_der'_Fun
thf(fact_1018_ground__term__of__gterm_H,axiom,
    ! [S: ground_gterm_a,F4: a,Ss: list_term_a_c] :
      ( ( ( ground5786051259813020158rm_a_c @ S )
        = ( fun_a_c @ F4 @ Ss ) )
     => ( term_ground_a_c @ ( fun_a_c @ F4 @ Ss ) ) ) ).

% ground_term_of_gterm'
thf(fact_1019_term__of__gterm__in__ta__lang__conv,axiom,
    ! [T: ground_gterm_a,Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( member_term_a_c @ ( ground5786051259813020158rm_a_c @ T ) @ ( tree_t1675334710359824801_a_a_c @ Q @ A6 ) )
      = ( member5261315023172567896term_a @ T @ ( tree_g6148166148305676299rm_a_a @ Q @ A6 ) ) ) ).

% term_of_gterm_in_ta_lang_conv
thf(fact_1020_ta__productiveI_H,axiom,
    ! [Q2: c,A6: tree_ta_c_a,C: subterm_and_ctxt_a_c,P2: c,P: fset_c] :
      ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ ( var_c_a @ P2 ) ) ) )
     => ( ( fmember_c @ Q2 @ ( tree_t299928733079354318ve_c_a @ P @ A6 ) )
       => ( fmember_c @ P2 @ ( tree_t299928733079354318ve_c_a @ P @ A6 ) ) ) ) ).

% ta_productiveI'
thf(fact_1021_ta__productiveI,axiom,
    ! [Q4: c,A6: tree_ta_c_a,C: subterm_and_ctxt_a_c,Q2: c,P: fset_c] :
      ( ( fmember_c @ Q4 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ ( var_c_a @ Q2 ) ) ) )
     => ( ( fmember_c @ Q4 @ P )
       => ( fmember_c @ Q2 @ ( tree_t299928733079354318ve_c_a @ P @ A6 ) ) ) ) ).

% ta_productiveI
thf(fact_1022_ta__productiveE,axiom,
    ! [Q2: c,P: fset_c,A6: tree_ta_c_a] :
      ( ( fmember_c @ Q2 @ ( tree_t299928733079354318ve_c_a @ P @ A6 ) )
     => ~ ! [Q5: c] :
            ( ? [C3: subterm_and_ctxt_a_c] : ( fmember_c @ Q5 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C3 @ ( var_c_a @ Q2 ) ) ) )
           => ~ ( fmember_c @ Q5 @ P ) ) ) ).

% ta_productiveE
thf(fact_1023_bot__empty__eq,axiom,
    ( bot_bo104646228311450846rm_a_o
    = ( ^ [X4: produc7033602110846336755term_a] : ( member2247295376459944092term_a @ X4 @ bot_bo2447593420621686463term_a ) ) ) ).

% bot_empty_eq
thf(fact_1024_bot__empty__eq,axiom,
    ( bot_bot_term_a_c_o
    = ( ^ [X4: term_a_c] : ( member_term_a_c @ X4 @ bot_bot_set_term_a_c ) ) ) ).

% bot_empty_eq
thf(fact_1025_ground__ta__der__statesD,axiom,
    ! [T: term_a_c,C2: c,A6: tree_ta_c_a] :
      ( ( term_ground_a_c @ T )
     => ( ( fmember_c @ C2 @ ( tree_ta_der_c_a @ A6 @ T ) )
       => ( fmember_c @ C2 @ ( tree_Q_c_a @ A6 ) ) ) ) ).

% ground_ta_der_statesD
thf(fact_1026_ta__only__reach__lang,axiom,
    ! [Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( tree_t1675334710359824801_a_a_c @ Q @ ( tree_t4763908311024172656rm_a_a @ A6 @ ( tree_t3813786739961944781rm_a_a @ A6 ) ) )
      = ( tree_t1675334710359824801_a_a_c @ Q @ A6 ) ) ).

% ta_only_reach_lang
thf(fact_1027_ta__only__prod__lang,axiom,
    ! [Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( tree_t1675334710359824801_a_a_c @ Q @ ( tree_t4763908311024172656rm_a_a @ A6 @ ( tree_t7469959674368944611rm_a_a @ Q @ A6 ) ) )
      = ( tree_t1675334710359824801_a_a_c @ Q @ A6 ) ) ).

% ta_only_prod_lang
thf(fact_1028_ta__productive__aux,axiom,
    ! [A6: tree_ta_c_a,Q2: c,C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( ord_less_eq_fset_c @ ( tree_Q_c_a @ A6 ) @ ( tree_t8392456876464840804le_c_a @ A6 ) )
     => ( ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C @ T ) ) )
       => ? [C6: subterm_and_ctxt_a_c] :
            ( ( term_ground_ctxt_a_c @ C6 )
            & ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C6 @ T ) ) ) ) ) ) ).

% ta_productive_aux
thf(fact_1029_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1030_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1031_fsubsetI,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ! [X3: term_a_b] :
          ( ( fmember_term_a_b @ X3 @ A4 )
         => ( fmember_term_a_b @ X3 @ B4 ) )
     => ( ord_le8417325568038501796rm_a_b @ A4 @ B4 ) ) ).

% fsubsetI
thf(fact_1032_finter__iff,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) )
      = ( ( fmember_term_a_b @ C2 @ A4 )
        & ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% finter_iff
thf(fact_1033_finterI,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ A4 )
     => ( ( fmember_term_a_b @ C2 @ B4 )
       => ( fmember_term_a_b @ C2 @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) ) ) ) ).

% finterI
thf(fact_1034_finsert__fsubset,axiom,
    ! [X: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( ord_le8417325568038501796rm_a_b @ ( finsert_term_a_b @ X @ A4 ) @ B4 )
      = ( ( fmember_term_a_b @ X @ B4 )
        & ( ord_le8417325568038501796rm_a_b @ A4 @ B4 ) ) ) ).

% finsert_fsubset
thf(fact_1035_finter__finsert__right__ifffempty,axiom,
    ! [A: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ~ ( fmember_term_a_b @ A @ A4 )
     => ( ( inf_in7980079480601282006rm_a_b @ A4 @ ( finsert_term_a_b @ A @ B4 ) )
        = ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) ) ) ).

% finter_finsert_right_ifffempty
thf(fact_1036_finter__finsert__left__ifffempty,axiom,
    ! [A: term_a_b,C: fset_term_a_b,B4: fset_term_a_b] :
      ( ~ ( fmember_term_a_b @ A @ C )
     => ( ( inf_in7980079480601282006rm_a_b @ ( finsert_term_a_b @ A @ B4 ) @ C )
        = ( inf_in7980079480601282006rm_a_b @ B4 @ C ) ) ) ).

% finter_finsert_left_ifffempty
thf(fact_1037_finter__finsert__right__if1,axiom,
    ! [A: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ( ( inf_in7980079480601282006rm_a_b @ A4 @ ( finsert_term_a_b @ A @ B4 ) )
        = ( finsert_term_a_b @ A @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) ) ) ) ).

% finter_finsert_right_if1
thf(fact_1038_finter__finsert__left__if1,axiom,
    ! [A: term_a_b,C: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ C )
     => ( ( inf_in7980079480601282006rm_a_b @ ( finsert_term_a_b @ A @ B4 ) @ C )
        = ( finsert_term_a_b @ A @ ( inf_in7980079480601282006rm_a_b @ B4 @ C ) ) ) ) ).

% finter_finsert_left_if1
thf(fact_1039_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_1040_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1041_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1042_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1043_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1044_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1045_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_1046_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1047_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_1048_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1049_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_1050_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_1051_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1052_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1053_order__subst1,axiom,
    ! [A: nat,F4: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F4 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F4 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1054_order__subst2,axiom,
    ! [A: nat,B: nat,F4: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F4 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1055_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1056_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1057_ord__eq__le__subst,axiom,
    ! [A: nat,F4: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F4 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F4 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1058_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F4: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F4 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1059_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_1060_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1061_finterD2,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) )
     => ( fmember_term_a_b @ C2 @ B4 ) ) ).

% finterD2
thf(fact_1062_finterD1,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) )
     => ( fmember_term_a_b @ C2 @ A4 ) ) ).

% finterD1
thf(fact_1063_finterE,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) )
     => ~ ( ( fmember_term_a_b @ C2 @ A4 )
         => ~ ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% finterE
thf(fact_1064_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1065_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1066_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1067_fsubsetD,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b,C2: term_a_b] :
      ( ( ord_le8417325568038501796rm_a_b @ A4 @ B4 )
     => ( ( fmember_term_a_b @ C2 @ A4 )
       => ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% fsubsetD
thf(fact_1068_fin__mono,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b,X: term_a_b] :
      ( ( ord_le8417325568038501796rm_a_b @ A4 @ B4 )
     => ( ( fmember_term_a_b @ X @ A4 )
       => ( fmember_term_a_b @ X @ B4 ) ) ) ).

% fin_mono
thf(fact_1069_finter__finsert__right,axiom,
    ! [A: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( ( fmember_term_a_b @ A @ A4 )
       => ( ( inf_in7980079480601282006rm_a_b @ A4 @ ( finsert_term_a_b @ A @ B4 ) )
          = ( finsert_term_a_b @ A @ ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) ) ) )
      & ( ~ ( fmember_term_a_b @ A @ A4 )
       => ( ( inf_in7980079480601282006rm_a_b @ A4 @ ( finsert_term_a_b @ A @ B4 ) )
          = ( inf_in7980079480601282006rm_a_b @ A4 @ B4 ) ) ) ) ).

% finter_finsert_right
thf(fact_1070_finter__finsert__left,axiom,
    ! [A: term_a_b,C: fset_term_a_b,B4: fset_term_a_b] :
      ( ( ( fmember_term_a_b @ A @ C )
       => ( ( inf_in7980079480601282006rm_a_b @ ( finsert_term_a_b @ A @ B4 ) @ C )
          = ( finsert_term_a_b @ A @ ( inf_in7980079480601282006rm_a_b @ B4 @ C ) ) ) )
      & ( ~ ( fmember_term_a_b @ A @ C )
       => ( ( inf_in7980079480601282006rm_a_b @ ( finsert_term_a_b @ A @ B4 ) @ C )
          = ( inf_in7980079480601282006rm_a_b @ B4 @ C ) ) ) ) ).

% finter_finsert_left
thf(fact_1071_fsubset__finsert,axiom,
    ! [X: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ~ ( fmember_term_a_b @ X @ A4 )
     => ( ( ord_le8417325568038501796rm_a_b @ A4 @ ( finsert_term_a_b @ X @ B4 ) )
        = ( ord_le8417325568038501796rm_a_b @ A4 @ B4 ) ) ) ).

% fsubset_finsert
thf(fact_1072_ground__ta__der__states,axiom,
    ! [T: term_a_c,A6: tree_ta_c_a] :
      ( ( term_ground_a_c @ T )
     => ( ord_less_eq_fset_c @ ( tree_ta_der_c_a @ A6 @ T ) @ ( tree_Q_c_a @ A6 ) ) ) ).

% ground_ta_der_states
thf(fact_1073_Int__emptyI,axiom,
    ! [A4: set_Pr3965413743549523539term_a,B4: set_Pr3965413743549523539term_a] :
      ( ! [X3: produc7033602110846336755term_a] :
          ( ( member2247295376459944092term_a @ X3 @ A4 )
         => ~ ( member2247295376459944092term_a @ X3 @ B4 ) )
     => ( ( inf_in2157114169531486017term_a @ A4 @ B4 )
        = bot_bo2447593420621686463term_a ) ) ).

% Int_emptyI
thf(fact_1074_Int__emptyI,axiom,
    ! [A4: set_term_a_c,B4: set_term_a_c] :
      ( ! [X3: term_a_c] :
          ( ( member_term_a_c @ X3 @ A4 )
         => ~ ( member_term_a_c @ X3 @ B4 ) )
     => ( ( inf_inf_set_term_a_c @ A4 @ B4 )
        = bot_bot_set_term_a_c ) ) ).

% Int_emptyI
thf(fact_1075_disjoint__iff,axiom,
    ! [A4: set_Pr3965413743549523539term_a,B4: set_Pr3965413743549523539term_a] :
      ( ( ( inf_in2157114169531486017term_a @ A4 @ B4 )
        = bot_bo2447593420621686463term_a )
      = ( ! [X4: produc7033602110846336755term_a] :
            ( ( member2247295376459944092term_a @ X4 @ A4 )
           => ~ ( member2247295376459944092term_a @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1076_disjoint__iff,axiom,
    ! [A4: set_term_a_c,B4: set_term_a_c] :
      ( ( ( inf_inf_set_term_a_c @ A4 @ B4 )
        = bot_bot_set_term_a_c )
      = ( ! [X4: term_a_c] :
            ( ( member_term_a_c @ X4 @ A4 )
           => ~ ( member_term_a_c @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1077_subrelI,axiom,
    ! [R: set_Pr3965413743549523539term_a,S: set_Pr3965413743549523539term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R )
         => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ S ) )
     => ( ord_le2299786838182399475term_a @ R @ S ) ) ).

% subrelI
thf(fact_1078_subset__emptyI,axiom,
    ! [A4: set_Pr3965413743549523539term_a] :
      ( ! [X3: produc7033602110846336755term_a] :
          ~ ( member2247295376459944092term_a @ X3 @ A4 )
     => ( ord_le2299786838182399475term_a @ A4 @ bot_bo2447593420621686463term_a ) ) ).

% subset_emptyI
thf(fact_1079_subset__emptyI,axiom,
    ! [A4: set_term_a_c] :
      ( ! [X3: term_a_c] :
          ~ ( member_term_a_c @ X3 @ A4 )
     => ( ord_le2776320456296813995rm_a_c @ A4 @ bot_bot_set_term_a_c ) ) ).

% subset_emptyI
thf(fact_1080_ta__lang__mono_H,axiom,
    ! [A6: tree_t5222909361115825503rm_a_a,B7: tree_t5222909361115825503rm_a_a,Q_A: fset_Bot_bot_term_a,Q_B: fset_Bot_bot_term_a] :
      ( ( tree_t2904061461305401546rm_a_a @ A6 @ B7 )
     => ( ( ord_le7216997114146882585term_a @ Q_A @ Q_B )
       => ( ord_le2776320456296813995rm_a_c @ ( tree_t1675334710359824801_a_a_c @ Q_A @ A6 ) @ ( tree_t1675334710359824801_a_a_c @ Q_B @ B7 ) ) ) ) ).

% ta_lang_mono'
thf(fact_1081_term__fset__induct,axiom,
    ! [P: term_a_b > $o,Term: term_a_b] :
      ( ! [X3: b] : ( P @ ( var_b_a @ X3 ) )
     => ( ! [X1a: a,X22: list_term_a_b] :
            ( ! [X2a: term_a_b] :
                ( ( fmember_term_a_b @ X2a @ ( fset_o2064584849798326990rm_a_b @ X22 ) )
               => ( P @ X2a ) )
           => ( P @ ( fun_a_b @ X1a @ X22 ) ) )
       => ( P @ Term ) ) ) ).

% term_fset_induct
thf(fact_1082_funionCI,axiom,
    ! [C2: term_a_b,B4: fset_term_a_b,A4: fset_term_a_b] :
      ( ( ~ ( fmember_term_a_b @ C2 @ B4 )
       => ( fmember_term_a_b @ C2 @ A4 ) )
     => ( fmember_term_a_b @ C2 @ ( sup_su1847537981033997936rm_a_b @ A4 @ B4 ) ) ) ).

% funionCI
thf(fact_1083_funion__iff,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( sup_su1847537981033997936rm_a_b @ A4 @ B4 ) )
      = ( ( fmember_term_a_b @ C2 @ A4 )
        | ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% funion_iff
thf(fact_1084_funionE,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( sup_su1847537981033997936rm_a_b @ A4 @ B4 ) )
     => ( ~ ( fmember_term_a_b @ C2 @ A4 )
       => ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% funionE
thf(fact_1085_funionI1,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ A4 )
     => ( fmember_term_a_b @ C2 @ ( sup_su1847537981033997936rm_a_b @ A4 @ B4 ) ) ) ).

% funionI1
thf(fact_1086_funionI2,axiom,
    ! [C2: term_a_b,B4: fset_term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ B4 )
     => ( fmember_term_a_b @ C2 @ ( sup_su1847537981033997936rm_a_b @ A4 @ B4 ) ) ) ).

% funionI2
thf(fact_1087_fvars__term__ctxt__apply,axiom,
    ! [C: subterm_and_ctxt_a_c,T: term_a_c] :
      ( ( fSet_fvars_term_a_c @ ( subter2376574525758040791rm_a_c @ C @ T ) )
      = ( sup_sup_fset_c @ ( fSet_fvars_ctxt_a_c @ C ) @ ( fSet_fvars_term_a_c @ T ) ) ) ).

% fvars_term_ctxt_apply
thf(fact_1088_ta__syms__lang,axiom,
    ! [T: term_a_c,Q: fset_Bot_bot_term_a,A6: tree_t5222909361115825503rm_a_a] :
      ( ( member_term_a_c @ T @ ( tree_t1675334710359824801_a_a_c @ Q @ A6 ) )
     => ( ord_le1241528519210447629_a_nat @ ( fSet_ffunas_term_a_c @ T ) @ ( tree_t7303415228429699401rm_a_a @ A6 ) ) ) ).

% ta_syms_lang
thf(fact_1089_snd__conv,axiom,
    ! [X1: bot_bot_term_a,X2: bot_bot_term_a] :
      ( ( produc7794171008584320473term_a @ ( produc780340308816795563term_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_1090_snd__eqD,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,A: bot_bot_term_a] :
      ( ( ( produc7794171008584320473term_a @ ( produc780340308816795563term_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_1091_eq__snd__iff,axiom,
    ! [B: bot_bot_term_a,P2: produc7033602110846336755term_a] :
      ( ( B
        = ( produc7794171008584320473term_a @ P2 ) )
      = ( ? [A5: bot_bot_term_a] :
            ( P2
            = ( produc780340308816795563term_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_1092_prod_Ocollapse,axiom,
    ! [Prod: produc7033602110846336755term_a] :
      ( ( produc780340308816795563term_a @ ( produc3704134870435948439term_a @ Prod ) @ ( produc7794171008584320473term_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1093_eq__fst__iff,axiom,
    ! [A: bot_bot_term_a,P2: produc7033602110846336755term_a] :
      ( ( A
        = ( produc3704134870435948439term_a @ P2 ) )
      = ( ? [B5: bot_bot_term_a] :
            ( P2
            = ( produc780340308816795563term_a @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_1094_fst__eqD,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,A: bot_bot_term_a] :
      ( ( ( produc3704134870435948439term_a @ ( produc780340308816795563term_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_1095_fst__conv,axiom,
    ! [X1: bot_bot_term_a,X2: bot_bot_term_a] :
      ( ( produc3704134870435948439term_a @ ( produc780340308816795563term_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_1096_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7033602110846336755term_a] :
      ( Prod
      = ( produc780340308816795563term_a @ ( produc3704134870435948439term_a @ Prod ) @ ( produc7794171008584320473term_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_1097_surjective__pairing,axiom,
    ! [T: produc7033602110846336755term_a] :
      ( T
      = ( produc780340308816795563term_a @ ( produc3704134870435948439term_a @ T ) @ ( produc7794171008584320473term_a @ T ) ) ) ).

% surjective_pairing
thf(fact_1098_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc7033602110846336755term_a,A4: bot_bot_term_a > bot_bot_term_a > $o] :
      ( ( member2247295376459944092term_a @ X @ ( collec7536399311987840478term_a @ ( produc480846774855951832rm_a_o @ A4 ) ) )
     => ( A4 @ ( produc3704134870435948439term_a @ X ) @ ( produc7794171008584320473term_a @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_1099_exI__realizer,axiom,
    ! [P: bot_bot_term_a > bot_bot_term_a > $o,Y: bot_bot_term_a,X: bot_bot_term_a] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7794171008584320473term_a @ ( produc780340308816795563term_a @ X @ Y ) ) @ ( produc3704134870435948439term_a @ ( produc780340308816795563term_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_1100_conjI__realizer,axiom,
    ! [P: bot_bot_term_a > $o,P2: bot_bot_term_a,Q: bot_bot_term_a > $o,Q2: bot_bot_term_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3704134870435948439term_a @ ( produc780340308816795563term_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7794171008584320473term_a @ ( produc780340308816795563term_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1101_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: bot_bot_term_a > bot_bot_term_a > $o,X: bot_bot_term_a,Y: bot_bot_term_a,A: produc7033602110846336755term_a] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc780340308816795563term_a @ X @ Y ) )
       => ( P @ ( produc3704134870435948439term_a @ A ) @ ( produc7794171008584320473term_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1102_sndI,axiom,
    ! [X: produc7033602110846336755term_a,Y: bot_bot_term_a,Z: bot_bot_term_a] :
      ( ( X
        = ( produc780340308816795563term_a @ Y @ Z ) )
     => ( ( produc7794171008584320473term_a @ X )
        = Z ) ) ).

% sndI
thf(fact_1103_fstI,axiom,
    ! [X: produc7033602110846336755term_a,Y: bot_bot_term_a,Z: bot_bot_term_a] :
      ( ( X
        = ( produc780340308816795563term_a @ Y @ Z ) )
     => ( ( produc3704134870435948439term_a @ X )
        = Y ) ) ).

% fstI
thf(fact_1104_ffunas__gterm__gterm__of__term,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( fSet_ffunas_gterm_a @ ( ground3594116404241818628rm_a_c @ T ) )
        = ( fSet_ffunas_term_a_c @ T ) ) ) ).

% ffunas_gterm_gterm_of_term
thf(fact_1105_gterm__of__term__inv,axiom,
    ! [T: term_a_c] :
      ( ( term_ground_a_c @ T )
     => ( ( ground5786051259813020158rm_a_c @ ( ground3594116404241818628rm_a_c @ T ) )
        = T ) ) ).

% gterm_of_term_inv
thf(fact_1106_fmember_Orep__eq,axiom,
    ( fmembe4510397451897038394term_a
    = ( ^ [X4: produc7033602110846336755term_a,Xa: fset_P6640269656128473term_a] : ( member2247295376459944092term_a @ X4 @ ( fset_P6484774398675748578term_a @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_1107_fmember_Orep__eq,axiom,
    ( fmember_term_a_c
    = ( ^ [X4: term_a_c,Xa: fset_term_a_c] : ( member_term_a_c @ X4 @ ( fset_term_a_c2 @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_1108_fmember_Orep__eq,axiom,
    ( fmember_term_a_b
    = ( ^ [X4: term_a_b,Xa: fset_term_a_b] : ( member_term_a_b @ X4 @ ( fset_term_a_b2 @ Xa ) ) ) ) ).

% fmember.rep_eq
thf(fact_1109_notin__fset,axiom,
    ! [X: produc7033602110846336755term_a,S4: fset_P6640269656128473term_a] :
      ( ( ~ ( fmembe4510397451897038394term_a @ X @ S4 ) )
      = ( ~ ( member2247295376459944092term_a @ X @ ( fset_P6484774398675748578term_a @ S4 ) ) ) ) ).

% notin_fset
thf(fact_1110_notin__fset,axiom,
    ! [X: term_a_c,S4: fset_term_a_c] :
      ( ( ~ ( fmember_term_a_c @ X @ S4 ) )
      = ( ~ ( member_term_a_c @ X @ ( fset_term_a_c2 @ S4 ) ) ) ) ).

% notin_fset
thf(fact_1111_notin__fset,axiom,
    ! [X: term_a_b,S4: fset_term_a_b] :
      ( ( ~ ( fmember_term_a_b @ X @ S4 ) )
      = ( ~ ( member_term_a_b @ X @ ( fset_term_a_b2 @ S4 ) ) ) ) ).

% notin_fset
thf(fact_1112_fmember__iff__member__fset,axiom,
    ( fmembe4510397451897038394term_a
    = ( ^ [X4: produc7033602110846336755term_a,A8: fset_P6640269656128473term_a] : ( member2247295376459944092term_a @ X4 @ ( fset_P6484774398675748578term_a @ A8 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_1113_fmember__iff__member__fset,axiom,
    ( fmember_term_a_c
    = ( ^ [X4: term_a_c,A8: fset_term_a_c] : ( member_term_a_c @ X4 @ ( fset_term_a_c2 @ A8 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_1114_fmember__iff__member__fset,axiom,
    ( fmember_term_a_b
    = ( ^ [X4: term_a_b,A8: fset_term_a_b] : ( member_term_a_b @ X4 @ ( fset_term_a_b2 @ A8 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_1115_fimage__eqI,axiom,
    ! [B: term_a_b,F4: term_a_b > term_a_b,X: term_a_b,A4: fset_term_a_b] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( fmember_term_a_b @ X @ A4 )
       => ( fmember_term_a_b @ B @ ( fimage6249664427166183207rm_a_b @ F4 @ A4 ) ) ) ) ).

% fimage_eqI
thf(fact_1116_fimage__fsubsetI,axiom,
    ! [A4: fset_term_a_b,F4: term_a_b > term_a_b,B4: fset_term_a_b] :
      ( ! [X3: term_a_b] :
          ( ( fmember_term_a_b @ X3 @ A4 )
         => ( fmember_term_a_b @ ( F4 @ X3 ) @ B4 ) )
     => ( ord_le8417325568038501796rm_a_b @ ( fimage6249664427166183207rm_a_b @ F4 @ A4 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_1117_fset_Omap__ident__strong,axiom,
    ! [T: fset_P6640269656128473term_a,F4: produc7033602110846336755term_a > produc7033602110846336755term_a] :
      ( ! [Z2: produc7033602110846336755term_a] :
          ( ( member2247295376459944092term_a @ Z2 @ ( fset_P6484774398675748578term_a @ T ) )
         => ( ( F4 @ Z2 )
            = Z2 ) )
     => ( ( fimage3742054321524018237term_a @ F4 @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_1118_fset_Omap__ident__strong,axiom,
    ! [T: fset_term_a_c,F4: term_a_c > term_a_c] :
      ( ! [Z2: term_a_c] :
          ( ( member_term_a_c @ Z2 @ ( fset_term_a_c2 @ T ) )
         => ( ( F4 @ Z2 )
            = Z2 ) )
     => ( ( fimage6916563907009320361rm_a_c @ F4 @ T )
        = T ) ) ).

% fset.map_ident_strong
thf(fact_1119_rev__fimage__eqI,axiom,
    ! [X: term_a_b,A4: fset_term_a_b,B: term_a_b,F4: term_a_b > term_a_b] :
      ( ( fmember_term_a_b @ X @ A4 )
     => ( ( B
          = ( F4 @ X ) )
       => ( fmember_term_a_b @ B @ ( fimage6249664427166183207rm_a_b @ F4 @ A4 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_1120_fimageI,axiom,
    ! [X: term_a_b,A4: fset_term_a_b,F4: term_a_b > term_a_b] :
      ( ( fmember_term_a_b @ X @ A4 )
     => ( fmember_term_a_b @ ( F4 @ X ) @ ( fimage6249664427166183207rm_a_b @ F4 @ A4 ) ) ) ).

% fimageI
thf(fact_1121_fimageE,axiom,
    ! [B: term_a_b,F4: term_a_b > term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ B @ ( fimage6249664427166183207rm_a_b @ F4 @ A4 ) )
     => ~ ! [X3: term_a_b] :
            ( ( B
              = ( F4 @ X3 ) )
           => ~ ( fmember_term_a_b @ X3 @ A4 ) ) ) ).

% fimageE
thf(fact_1122_the__finv__into__into,axiom,
    ! [F4: term_a_b > term_a_b,A4: fset_term_a_b,X: term_a_b,B4: fset_term_a_b] :
      ( ( fSet_f3581483917893148177rm_a_b @ F4 @ A4 )
     => ( ( fmember_term_a_b @ X @ ( fimage6249664427166183207rm_a_b @ F4 @ A4 ) )
       => ( ( ord_le8417325568038501796rm_a_b @ A4 @ B4 )
         => ( fmember_term_a_b @ ( fSet_t5505035234354303037rm_a_b @ A4 @ F4 @ X ) @ B4 ) ) ) ) ).

% the_finv_into_into
thf(fact_1123_map__prod__simp,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a,G: bot_bot_term_a > bot_bot_term_a,A: bot_bot_term_a,B: bot_bot_term_a] :
      ( ( produc5208308128219516446term_a @ F4 @ G @ ( produc780340308816795563term_a @ A @ B ) )
      = ( produc780340308816795563term_a @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_1124_fminusI,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ A4 )
     => ( ~ ( fmember_term_a_b @ C2 @ B4 )
       => ( fmember_term_a_b @ C2 @ ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) ) ) ) ).

% fminusI
thf(fact_1125_fminus__iff,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) )
      = ( ( fmember_term_a_b @ C2 @ A4 )
        & ~ ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% fminus_iff
thf(fact_1126_ffUnionI,axiom,
    ! [X5: fset_term_a_b,C: fset_fset_term_a_b,A4: term_a_b] :
      ( ( fmembe3754333024224722621rm_a_b @ X5 @ C )
     => ( ( fmember_term_a_b @ A4 @ X5 )
       => ( fmember_term_a_b @ A4 @ ( ffUnion_term_a_b @ C ) ) ) ) ).

% ffUnionI
thf(fact_1127_finsert__fminus1,axiom,
    ! [X: term_a_b,B4: fset_term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ X @ B4 )
     => ( ( minus_5522361424680520477rm_a_b @ ( finsert_term_a_b @ X @ A4 ) @ B4 )
        = ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) ) ) ).

% finsert_fminus1
thf(fact_1128_fminusE,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) )
     => ~ ( ( fmember_term_a_b @ C2 @ A4 )
         => ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% fminusE
thf(fact_1129_fminusD1,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) )
     => ( fmember_term_a_b @ C2 @ A4 ) ) ).

% fminusD1
thf(fact_1130_fminusD2,axiom,
    ! [C2: term_a_b,A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( fmember_term_a_b @ C2 @ ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) )
     => ~ ( fmember_term_a_b @ C2 @ B4 ) ) ).

% fminusD2
thf(fact_1131_fmap__prod__fimageI,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R2: fset_P6640269656128473term_a,F4: bot_bot_term_a > bot_bot_term_a,G: bot_bot_term_a > bot_bot_term_a] :
      ( ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ A @ B ) @ R2 )
     => ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ ( F4 @ A ) @ ( G @ B ) ) @ ( fimage3742054321524018237term_a @ ( produc5208308128219516446term_a @ F4 @ G ) @ R2 ) ) ) ).

% fmap_prod_fimageI
thf(fact_1132_prod__fun__fimageE,axiom,
    ! [C2: produc7033602110846336755term_a,F4: bot_bot_term_a > bot_bot_term_a,G: bot_bot_term_a > bot_bot_term_a,R2: fset_P6640269656128473term_a] :
      ( ( fmembe4510397451897038394term_a @ C2 @ ( fimage3742054321524018237term_a @ ( produc5208308128219516446term_a @ F4 @ G ) @ R2 ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( C2
              = ( produc780340308816795563term_a @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( fmembe4510397451897038394term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_fimageE
thf(fact_1133_finsert__fminus__if,axiom,
    ! [X: term_a_b,B4: fset_term_a_b,A4: fset_term_a_b] :
      ( ( ( fmember_term_a_b @ X @ B4 )
       => ( ( minus_5522361424680520477rm_a_b @ ( finsert_term_a_b @ X @ A4 ) @ B4 )
          = ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) ) )
      & ( ~ ( fmember_term_a_b @ X @ B4 )
       => ( ( minus_5522361424680520477rm_a_b @ ( finsert_term_a_b @ X @ A4 ) @ B4 )
          = ( finsert_term_a_b @ X @ ( minus_5522361424680520477rm_a_b @ A4 @ B4 ) ) ) ) ) ).

% finsert_fminus_if
thf(fact_1134_finsert__fminus,axiom,
    ! [A: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ A @ A4 )
     => ( ( finsert_term_a_b @ A @ ( minus_5522361424680520477rm_a_b @ A4 @ ( finsert_term_a_b @ A @ bot_bo1583650863486268376rm_a_b ) ) )
        = A4 ) ) ).

% finsert_fminus
thf(fact_1135_fminus__finsert__absorb,axiom,
    ! [X: term_a_b,A4: fset_term_a_b] :
      ( ~ ( fmember_term_a_b @ X @ A4 )
     => ( ( minus_5522361424680520477rm_a_b @ ( finsert_term_a_b @ X @ A4 ) @ ( finsert_term_a_b @ X @ bot_bo1583650863486268376rm_a_b ) )
        = A4 ) ) ).

% fminus_finsert_absorb
thf(fact_1136_fsubset__finsert__iff,axiom,
    ! [A4: fset_term_a_b,X: term_a_b,B4: fset_term_a_b] :
      ( ( ord_le8417325568038501796rm_a_b @ A4 @ ( finsert_term_a_b @ X @ B4 ) )
      = ( ( ( fmember_term_a_b @ X @ A4 )
         => ( ord_le8417325568038501796rm_a_b @ ( minus_5522361424680520477rm_a_b @ A4 @ ( finsert_term_a_b @ X @ bot_bo1583650863486268376rm_a_b ) ) @ B4 ) )
        & ( ~ ( fmember_term_a_b @ X @ A4 )
         => ( ord_le8417325568038501796rm_a_b @ A4 @ B4 ) ) ) ) ).

% fsubset_finsert_iff
thf(fact_1137_pfsubset__finsert__iff,axiom,
    ! [A4: fset_term_a_b,X: term_a_b,B4: fset_term_a_b] :
      ( ( ord_le3921303545160784792rm_a_b @ A4 @ ( finsert_term_a_b @ X @ B4 ) )
      = ( ( ( fmember_term_a_b @ X @ B4 )
         => ( ord_le3921303545160784792rm_a_b @ A4 @ B4 ) )
        & ( ~ ( fmember_term_a_b @ X @ B4 )
         => ( ( ( fmember_term_a_b @ X @ A4 )
             => ( ord_le3921303545160784792rm_a_b @ ( minus_5522361424680520477rm_a_b @ A4 @ ( finsert_term_a_b @ X @ bot_bo1583650863486268376rm_a_b ) ) @ B4 ) )
            & ( ~ ( fmember_term_a_b @ X @ A4 )
             => ( ord_le8417325568038501796rm_a_b @ A4 @ B4 ) ) ) ) ) ) ).

% pfsubset_finsert_iff
thf(fact_1138_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_1139_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_1140_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1141_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1142_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F4: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F4 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ ( F4 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1143_order__less__le__subst1,axiom,
    ! [A: nat,F4: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F4 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F4 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1144_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F4: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F4 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ ( F4 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1145_order__le__less__subst1,axiom,
    ! [A: nat,F4: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F4 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F4 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1146_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1147_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1148_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_1149_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_1150_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1151_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1152_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1153_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1154_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1155_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_1156_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_1157_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1158_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1159_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1160_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1161_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_nat @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1162_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1163_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1164_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1165_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1166_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1167_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1168_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1169_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1170_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1171_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1172_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_1173_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_1174_prod__less__def,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y5 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y5 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ) ).

% prod_less_def
thf(fact_1175_less__prod__def_H,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y5 ) )
          | ( ( ( product_fst_nat_nat @ X4 )
              = ( product_fst_nat_nat @ Y5 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_1176_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1177_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1178_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_1179_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1180_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1181_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_1182_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1183_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_1184_order__less__subst2,axiom,
    ! [A: nat,B: nat,F4: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F4 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ ( F4 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1185_order__less__subst1,axiom,
    ! [A: nat,F4: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F4 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F4 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1186_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_1187_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F4: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F4 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ ( F4 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1188_ord__eq__less__subst,axiom,
    ! [A: nat,F4: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F4 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F4 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1189_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1190_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1191_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1192_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1193_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1194_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_1195_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1196_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1197_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1198_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1199_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_1200_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X6: nat] : ( P4 @ X6 ) )
    = ( ^ [P5: nat > $o] :
        ? [N: nat] :
          ( ( P5 @ N )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N )
             => ~ ( P5 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_1201_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1202_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1203_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1204_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1205_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X3 )
             => ( P @ Y6 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1206_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1207_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1208_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1209_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1210_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_1211_pfsubsetD,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b,C2: term_a_b] :
      ( ( ord_le3921303545160784792rm_a_b @ A4 @ B4 )
     => ( ( fmember_term_a_b @ C2 @ A4 )
       => ( fmember_term_a_b @ C2 @ B4 ) ) ) ).

% pfsubsetD
thf(fact_1212_pfsubset__imp__ex__fmem,axiom,
    ! [A4: fset_term_a_b,B4: fset_term_a_b] :
      ( ( ord_le3921303545160784792rm_a_b @ A4 @ B4 )
     => ? [B3: term_a_b] : ( fmember_term_a_b @ B3 @ ( minus_5522361424680520477rm_a_b @ B4 @ A4 ) ) ) ).

% pfsubset_imp_ex_fmem
thf(fact_1213_less__eq__prod__def,axiom,
    ( ord_le8460144461188290721at_nat
    = ( ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y5 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y5 ) )
            & ( ord_less_eq_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_1214_fset__linorder__min__induct,axiom,
    ! [P: fset_nat > $o,S4: fset_nat] :
      ( ( P @ bot_bot_fset_nat )
     => ( ! [X3: nat,S6: fset_nat] :
            ( ! [Y6: nat] :
                ( ( fmember_nat @ Y6 @ S6 )
               => ( ord_less_nat @ X3 @ Y6 ) )
           => ( ( P @ S6 )
             => ( P @ ( finsert_nat @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_linorder_min_induct
thf(fact_1215_fset__linorder__max__induct,axiom,
    ! [P: fset_nat > $o,S4: fset_nat] :
      ( ( P @ bot_bot_fset_nat )
     => ( ! [X3: nat,S6: fset_nat] :
            ( ! [Y6: nat] :
                ( ( fmember_nat @ Y6 @ S6 )
               => ( ord_less_nat @ Y6 @ X3 ) )
           => ( ( P @ S6 )
             => ( P @ ( finsert_nat @ X3 @ S6 ) ) ) )
       => ( P @ S4 ) ) ) ).

% fset_linorder_max_induct
thf(fact_1216_acyclicI__order,axiom,
    ! [R: set_Pr3965413743549523539term_a,F4: bot_bot_term_a > nat] :
      ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a] :
          ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A3 @ B3 ) @ R )
         => ( ord_less_nat @ ( F4 @ B3 ) @ ( F4 @ A3 ) ) )
     => ( transi4548540958286804730term_a @ R ) ) ).

% acyclicI_order
thf(fact_1217_rule__reachable__ctxt__exist,axiom,
    ! [F4: a,Qs: list_c,Q2: c,A6: tree_ta_c_a,I: nat] :
      ( ( fmembe5561272363050924325le_c_a @ ( tree_ta_TA_rule_a_c @ F4 @ Qs @ Q2 ) @ ( tree_rules_c_a @ A6 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Qs ) )
       => ? [C3: subterm_and_ctxt_a_c] : ( fmember_c @ Q2 @ ( tree_ta_der_c_a @ A6 @ ( subter2376574525758040791rm_a_c @ C3 @ ( var_c_a @ ( nth_c @ Qs @ I ) ) ) ) ) ) ) ).

% rule_reachable_ctxt_exist
thf(fact_1218_apsnd__conv,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a,X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ( produc6734519164561896987term_a @ F4 @ ( produc780340308816795563term_a @ X @ Y ) )
      = ( produc780340308816795563term_a @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_1219_ta__der_H__Fun2,axiom,
    ! [F4: a,Ps: list_term_a_b,A6: tree_ta_b_a,G: a,Ts: list_term_a_b] :
      ( ( fmember_term_a_b @ ( fun_a_b @ F4 @ Ps ) @ ( tree_ta_der_b_a2 @ A6 @ ( fun_a_b @ G @ Ts ) ) )
      = ( ( F4 = G )
        & ( ( size_s8906293707977694520rm_a_b @ Ps )
          = ( size_s8906293707977694520rm_a_b @ Ts ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
           => ( fmember_term_a_b @ ( nth_term_a_b @ Ps @ I2 ) @ ( tree_ta_der_b_a2 @ A6 @ ( nth_term_a_b @ Ts @ I2 ) ) ) ) ) ) ).

% ta_der'_Fun2
thf(fact_1220_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N2 @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N2 )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1221_in__fset__idx,axiom,
    ! [X: term_a_b,Xs: list_term_a_b] :
      ( ( fmember_term_a_b @ X @ ( fset_o2064584849798326990rm_a_b @ Xs ) )
     => ? [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
          & ( ( nth_term_a_b @ Xs @ I4 )
            = X ) ) ) ).

% in_fset_idx
thf(fact_1222_fnth__mem,axiom,
    ! [N2: nat,Xs: list_term_a_b] :
      ( ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
     => ( fmember_term_a_b @ ( nth_term_a_b @ Xs @ N2 ) @ ( fset_o2064584849798326990rm_a_b @ Xs ) ) ) ).

% fnth_mem
thf(fact_1223_in__fset__conv__nth,axiom,
    ! [X: term_a_b,Xs: list_term_a_b] :
      ( ( fmember_term_a_b @ X @ ( fset_o2064584849798326990rm_a_b @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
            & ( ( nth_term_a_b @ Xs @ I2 )
              = X ) ) ) ) ).

% in_fset_conv_nth
thf(fact_1224_linear__term__comb__subst,axiom,
    ! [F4: a,Ss: list_term_a_b,Ts: list_term_a_c,Sigma: nat > b > term_a_c] :
      ( ( linear_term_a_b @ ( fun_a_b @ F4 @ Ss ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ss )
          = ( size_s8977327748024040505rm_a_c @ Ts ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s8977327748024040505rm_a_c @ Ts ) )
             => ( ( subst_7999470309526761005_a_b_c @ ( nth_term_a_b @ Ss @ I4 ) @ ( Sigma @ I4 ) )
                = ( nth_term_a_c @ Ts @ I4 ) ) )
         => ? [Sigma2: b > term_a_c] :
              ( ( subst_7999470309526761005_a_b_c @ ( fun_a_b @ F4 @ Ss ) @ Sigma2 )
              = ( fun_a_c @ F4 @ Ts ) ) ) ) ) ).

% linear_term_comb_subst
thf(fact_1225_linear__term__comb__subst,axiom,
    ! [F4: a,Ss: list_term_a_c,Ts: list_term_a_c,Sigma: nat > c > term_a_c] :
      ( ( linear_term_a_c @ ( fun_a_c @ F4 @ Ss ) )
     => ( ( ( size_s8977327748024040505rm_a_c @ Ss )
          = ( size_s8977327748024040505rm_a_c @ Ts ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s8977327748024040505rm_a_c @ Ts ) )
             => ( ( subst_5212014727554084972_a_c_c @ ( nth_term_a_c @ Ss @ I4 ) @ ( Sigma @ I4 ) )
                = ( nth_term_a_c @ Ts @ I4 ) ) )
         => ? [Sigma2: c > term_a_c] :
              ( ( subst_5212014727554084972_a_c_c @ ( fun_a_c @ F4 @ Ss ) @ Sigma2 )
              = ( fun_a_c @ F4 @ Ts ) ) ) ) ) ).

% linear_term_comb_subst
thf(fact_1226_linear__term__comb__subst,axiom,
    ! [F4: a,Ss: list_term_a_c,Ts: list_term_a_b,Sigma: nat > c > term_a_b] :
      ( ( linear_term_a_c @ ( fun_a_c @ F4 @ Ss ) )
     => ( ( ( size_s8977327748024040505rm_a_c @ Ss )
          = ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
             => ( ( subst_5212014727554084971_a_c_b @ ( nth_term_a_c @ Ss @ I4 ) @ ( Sigma @ I4 ) )
                = ( nth_term_a_b @ Ts @ I4 ) ) )
         => ? [Sigma2: c > term_a_b] :
              ( ( subst_5212014727554084971_a_c_b @ ( fun_a_c @ F4 @ Ss ) @ Sigma2 )
              = ( fun_a_b @ F4 @ Ts ) ) ) ) ) ).

% linear_term_comb_subst
thf(fact_1227_linear__term__comb__subst,axiom,
    ! [F4: a,Ss: list_term_a_b,Ts: list_term_a_b,Sigma: nat > b > term_a_b] :
      ( ( linear_term_a_b @ ( fun_a_b @ F4 @ Ss ) )
     => ( ( ( size_s8906293707977694520rm_a_b @ Ss )
          = ( size_s8906293707977694520rm_a_b @ Ts ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
             => ( ( subst_7999470309526761004_a_b_b @ ( nth_term_a_b @ Ss @ I4 ) @ ( Sigma @ I4 ) )
                = ( nth_term_a_b @ Ts @ I4 ) ) )
         => ? [Sigma2: b > term_a_b] :
              ( ( subst_7999470309526761004_a_b_b @ ( fun_a_b @ F4 @ Ss ) @ Sigma2 )
              = ( fun_a_b @ F4 @ Ts ) ) ) ) ) ).

% linear_term_comb_subst
thf(fact_1228_fsubseteq__fset__conv__nth,axiom,
    ! [Ss: list_term_a_b,T5: fset_term_a_b] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ss ) )
           => ( fmember_term_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ T5 ) ) )
      = ( ord_le8417325568038501796rm_a_b @ ( fset_o2064584849798326990rm_a_b @ Ss ) @ T5 ) ) ).

% fsubseteq_fset_conv_nth
thf(fact_1229_fset__list__fsubset__eq__nth__conv,axiom,
    ! [Xs: list_term_a_b,A4: fset_term_a_b] :
      ( ( ord_le8417325568038501796rm_a_b @ ( fset_o2064584849798326990rm_a_b @ Xs ) @ A4 )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
           => ( fmember_term_a_b @ ( nth_term_a_b @ Xs @ I2 ) @ A4 ) ) ) ) ).

% fset_list_fsubset_eq_nth_conv
thf(fact_1230_ta__reachableI__rule_I2_J,axiom,
    ! [Qs: list_c,A6: tree_ta_c_a,F4: a,Q2: c] :
      ( ( ord_less_eq_fset_c @ ( fset_of_list_c @ Qs ) @ ( tree_t8392456876464840804le_c_a @ A6 ) )
     => ( ( fmembe5561272363050924325le_c_a @ ( tree_ta_TA_rule_a_c @ F4 @ Qs @ Q2 ) @ ( tree_rules_c_a @ A6 ) )
       => ? [Ts2: list_term_a_c] :
            ( ( ( size_size_list_c @ Qs )
              = ( size_s8977327748024040505rm_a_c @ Ts2 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s8977327748024040505rm_a_c @ Ts2 ) )
               => ( term_ground_a_c @ ( nth_term_a_c @ Ts2 @ I3 ) ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s8977327748024040505rm_a_c @ Ts2 ) )
               => ( fmember_c @ ( nth_c @ Qs @ I3 ) @ ( tree_ta_der_c_a @ A6 @ ( nth_term_a_c @ Ts2 @ I3 ) ) ) ) ) ) ) ).

% ta_reachableI_rule(2)
thf(fact_1231_apfst__conv,axiom,
    ! [F4: bot_bot_term_a > bot_bot_term_a,X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ( produc285492028240800985term_a @ F4 @ ( produc780340308816795563term_a @ X @ Y ) )
      = ( produc780340308816795563term_a @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_1232_apfst__convE,axiom,
    ! [Q2: produc7033602110846336755term_a,F4: bot_bot_term_a > bot_bot_term_a,P2: produc7033602110846336755term_a] :
      ( ( Q2
        = ( produc285492028240800985term_a @ F4 @ P2 ) )
     => ~ ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( P2
              = ( produc780340308816795563term_a @ X3 @ Y3 ) )
           => ( Q2
             != ( produc780340308816795563term_a @ ( F4 @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1233_case__prod__Pair,axiom,
    ( ( produc1985598078976903475term_a @ produc780340308816795563term_a )
    = id_Pro7590044230637570276term_a ) ).

% case_prod_Pair
thf(fact_1234_trancl__list_Obase,axiom,
    ! [Xs: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a] :
      ( ( ( size_s1103687553077312429term_a @ Xs )
        = ( size_s1103687553077312429term_a @ Ys2 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ys2 ) )
           => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Xs @ I4 ) @ ( nth_Bot_bot_term_a @ Ys2 @ I4 ) ) @ R4 ) )
       => ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs @ Ys2 ) @ ( trancl6444742763886629761term_a @ R4 ) ) ) ) ).

% trancl_list.base
thf(fact_1235_swap__simp,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ( produc5746675070383261003term_a @ ( produc780340308816795563term_a @ X @ Y ) )
      = ( produc780340308816795563term_a @ Y @ X ) ) ).

% swap_simp
thf(fact_1236_prod_Oswap__def,axiom,
    ( produc5746675070383261003term_a
    = ( ^ [P6: produc7033602110846336755term_a] : ( produc780340308816795563term_a @ ( produc7794171008584320473term_a @ P6 ) @ ( produc3704134870435948439term_a @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1237_trancl__list_Olist__trancl,axiom,
    ! [Xs: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a,I: nat,Z: bot_bot_term_a] :
      ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs @ Ys2 ) @ ( trancl6444742763886629761term_a @ R4 ) )
     => ( ( ord_less_nat @ I @ ( size_s1103687553077312429term_a @ Ys2 ) )
       => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Ys2 @ I ) @ Z ) @ R4 )
         => ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs @ ( list_u5503441972953478683term_a @ Ys2 @ I @ Z ) ) @ ( trancl6444742763886629761term_a @ R4 ) ) ) ) ) ).

% trancl_list.list_trancl
thf(fact_1238_trancl__list_Ocases,axiom,
    ! [A1: list_Bot_bot_term_a,A22: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a] :
      ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ A1 @ A22 ) @ ( trancl6444742763886629761term_a @ R4 ) )
     => ( ( ( ( size_s1103687553077312429term_a @ A1 )
            = ( size_s1103687553077312429term_a @ A22 ) )
         => ~ ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1103687553077312429term_a @ A22 ) )
               => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ A1 @ I3 ) @ ( nth_Bot_bot_term_a @ A22 @ I3 ) ) @ R4 ) ) )
       => ~ ! [Ys: list_Bot_bot_term_a,I4: nat,Z2: bot_bot_term_a] :
              ( ( A22
                = ( list_u5503441972953478683term_a @ Ys @ I4 @ Z2 ) )
             => ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ A1 @ Ys ) @ ( trancl6444742763886629761term_a @ R4 ) )
               => ( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ys ) )
                 => ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Ys @ I4 ) @ Z2 ) @ R4 ) ) ) ) ) ) ).

% trancl_list.cases
thf(fact_1239_trancl__list_Osimps,axiom,
    ! [A1: list_Bot_bot_term_a,A22: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a] :
      ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ A1 @ A22 ) @ ( trancl6444742763886629761term_a @ R4 ) )
      = ( ? [Xs3: list_Bot_bot_term_a,Ys3: list_Bot_bot_term_a] :
            ( ( A1 = Xs3 )
            & ( A22 = Ys3 )
            & ( ( size_s1103687553077312429term_a @ Xs3 )
              = ( size_s1103687553077312429term_a @ Ys3 ) )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Ys3 ) )
               => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Xs3 @ I2 ) @ ( nth_Bot_bot_term_a @ Ys3 @ I2 ) ) @ R4 ) ) )
        | ? [Xs3: list_Bot_bot_term_a,Ys3: list_Bot_bot_term_a,I2: nat,Z4: bot_bot_term_a] :
            ( ( A1 = Xs3 )
            & ( A22
              = ( list_u5503441972953478683term_a @ Ys3 @ I2 @ Z4 ) )
            & ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs3 @ Ys3 ) @ ( trancl6444742763886629761term_a @ R4 ) )
            & ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Ys3 ) )
            & ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Ys3 @ I2 ) @ Z4 ) @ R4 ) ) ) ) ).

% trancl_list.simps
thf(fact_1240_trancl__list__conv,axiom,
    ! [Xs: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a] :
      ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs @ Ys2 ) @ ( trancl6444742763886629761term_a @ R4 ) )
      = ( ( ( size_s1103687553077312429term_a @ Xs )
          = ( size_s1103687553077312429term_a @ Ys2 ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1103687553077312429term_a @ Ys2 ) )
           => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Xs @ I2 ) @ ( nth_Bot_bot_term_a @ Ys2 @ I2 ) ) @ ( transi2774326411154761896term_a @ R4 ) ) ) ) ) ).

% trancl_list_conv
thf(fact_1241_trancl__induct2,axiom,
    ! [Ax: bot_bot_term_a,Ay: bot_bot_term_a,Bx: bot_bot_term_a,By: bot_bot_term_a,R: set_Pr825864233230045127term_a,P: bot_bot_term_a > bot_bot_term_a > $o] :
      ( ( member7618331267207336464term_a @ ( produc1145950395756198999term_a @ ( produc780340308816795563term_a @ Ax @ Ay ) @ ( produc780340308816795563term_a @ Bx @ By ) ) @ ( transi306844429749061140term_a @ R ) )
     => ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a] :
            ( ( member7618331267207336464term_a @ ( produc1145950395756198999term_a @ ( produc780340308816795563term_a @ Ax @ Ay ) @ ( produc780340308816795563term_a @ A3 @ B3 ) ) @ R )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: bot_bot_term_a,B3: bot_bot_term_a,Aa: bot_bot_term_a,Ba: bot_bot_term_a] :
              ( ( member7618331267207336464term_a @ ( produc1145950395756198999term_a @ ( produc780340308816795563term_a @ Ax @ Ay ) @ ( produc780340308816795563term_a @ A3 @ B3 ) ) @ ( transi306844429749061140term_a @ R ) )
             => ( ( member7618331267207336464term_a @ ( produc1145950395756198999term_a @ ( produc780340308816795563term_a @ A3 @ B3 ) @ ( produc780340308816795563term_a @ Aa @ Ba ) ) @ R )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1242_trancl_Ocases,axiom,
    ! [A1: bot_bot_term_a,A22: bot_bot_term_a,R: set_Pr3965413743549523539term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A1 @ A22 ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A1 @ A22 ) @ R )
       => ~ ! [B3: bot_bot_term_a] :
              ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A1 @ B3 ) @ ( transi2774326411154761896term_a @ R ) )
             => ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ B3 @ A22 ) @ R ) ) ) ) ).

% trancl.cases
thf(fact_1243_trancl_Osimps,axiom,
    ! [A1: bot_bot_term_a,A22: bot_bot_term_a,R: set_Pr3965413743549523539term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A1 @ A22 ) @ ( transi2774326411154761896term_a @ R ) )
      = ( ? [A5: bot_bot_term_a,B5: bot_bot_term_a] :
            ( ( A1 = A5 )
            & ( A22 = B5 )
            & ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A5 @ B5 ) @ R ) )
        | ? [A5: bot_bot_term_a,B5: bot_bot_term_a,C7: bot_bot_term_a] :
            ( ( A1 = A5 )
            & ( A22 = C7 )
            & ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A5 @ B5 ) @ ( transi2774326411154761896term_a @ R ) )
            & ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ B5 @ C7 ) @ R ) ) ) ) ).

% trancl.simps
thf(fact_1244_trancl_Or__into__trancl,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: set_Pr3965413743549523539term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
     => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( transi2774326411154761896term_a @ R ) ) ) ).

% trancl.r_into_trancl
thf(fact_1245_tranclE,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: set_Pr3965413743549523539term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
       => ~ ! [C5: bot_bot_term_a] :
              ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ C5 ) @ ( transi2774326411154761896term_a @ R ) )
             => ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ C5 @ B ) @ R ) ) ) ) ).

% tranclE
thf(fact_1246_trancl__trans,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,R: set_Pr3965413743549523539term_a,Z: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y @ Z ) @ ( transi2774326411154761896term_a @ R ) )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Z ) @ ( transi2774326411154761896term_a @ R ) ) ) ) ).

% trancl_trans
thf(fact_1247_trancl__induct,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: set_Pr3965413743549523539term_a,P: bot_bot_term_a > $o] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ! [Y3: bot_bot_term_a] :
            ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ Y3 ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: bot_bot_term_a,Z2: bot_bot_term_a] :
              ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ Y3 ) @ ( transi2774326411154761896term_a @ R ) )
             => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y3 @ Z2 ) @ R )
               => ( ( P @ Y3 )
                 => ( P @ Z2 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_1248_r__r__into__trancl,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R2: set_Pr3965413743549523539term_a,C2: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ R2 )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ B @ C2 ) @ R2 )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ C2 ) @ ( transi2774326411154761896term_a @ R2 ) ) ) ) ).

% r_r_into_trancl
thf(fact_1249_converse__tranclE,axiom,
    ! [X: bot_bot_term_a,Z: bot_bot_term_a,R: set_Pr3965413743549523539term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Z ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Z ) @ R )
       => ~ ! [Y3: bot_bot_term_a] :
              ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y3 ) @ R )
             => ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y3 @ Z ) @ ( transi2774326411154761896term_a @ R ) ) ) ) ) ).

% converse_tranclE
thf(fact_1250_irrefl__trancl__rD,axiom,
    ! [R: set_Pr3965413743549523539term_a,X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ! [X3: bot_bot_term_a] :
          ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ X3 ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ R )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_1251_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: set_Pr3965413743549523539term_a,C2: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ B @ C2 ) @ R )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ C2 ) @ ( transi2774326411154761896term_a @ R ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_1252_trancl__into__trancl2,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: set_Pr3965413743549523539term_a,C2: bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ R )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ B @ C2 ) @ ( transi2774326411154761896term_a @ R ) )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ C2 ) @ ( transi2774326411154761896term_a @ R ) ) ) ) ).

% trancl_into_trancl2
thf(fact_1253_trancl__trans__induct,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,R: set_Pr3965413743549523539term_a,P: bot_bot_term_a > bot_bot_term_a > $o] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
            ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R )
           => ( P @ X3 @ Y3 ) )
       => ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a,Z2: bot_bot_term_a] :
              ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ ( transi2774326411154761896term_a @ R ) )
             => ( ( P @ X3 @ Y3 )
               => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y3 @ Z2 ) @ ( transi2774326411154761896term_a @ R ) )
                 => ( ( P @ Y3 @ Z2 )
                   => ( P @ X3 @ Z2 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_1254_converse__trancl__induct,axiom,
    ! [A: bot_bot_term_a,B: bot_bot_term_a,R: set_Pr3965413743549523539term_a,P: bot_bot_term_a > $o] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ A @ B ) @ ( transi2774326411154761896term_a @ R ) )
     => ( ! [Y3: bot_bot_term_a] :
            ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y3 @ B ) @ R )
           => ( P @ Y3 ) )
       => ( ! [Y3: bot_bot_term_a,Z2: bot_bot_term_a] :
              ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Y3 @ Z2 ) @ R )
             => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ Z2 @ B ) @ ( transi2774326411154761896term_a @ R ) )
               => ( ( P @ Z2 )
                 => ( P @ Y3 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_1255_acyclic__def,axiom,
    ( transi4548540958286804730term_a
    = ( ^ [R5: set_Pr3965413743549523539term_a] :
        ! [X4: bot_bot_term_a] :
          ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X4 @ X4 ) @ ( transi2774326411154761896term_a @ R5 ) ) ) ) ).

% acyclic_def
thf(fact_1256_acyclicI,axiom,
    ! [R: set_Pr3965413743549523539term_a] :
      ( ! [X3: bot_bot_term_a] :
          ~ ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ X3 ) @ ( transi2774326411154761896term_a @ R ) )
     => ( transi4548540958286804730term_a @ R ) ) ).

% acyclicI
thf(fact_1257_trancl__map,axiom,
    ! [R: set_Pr3965413743549523539term_a,F4: bot_bot_term_a > bot_bot_term_a,S: set_Pr3965413743549523539term_a,X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ! [X3: bot_bot_term_a,Y3: bot_bot_term_a] :
          ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X3 @ Y3 ) @ R )
         => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ S ) )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( transi2774326411154761896term_a @ R ) )
       => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( F4 @ X ) @ ( F4 @ Y ) ) @ ( transi2774326411154761896term_a @ S ) ) ) ) ).

% trancl_map
thf(fact_1258_trancl__list__induct,axiom,
    ! [Ss: list_Bot_bot_term_a,Ts: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a,P: list_Bot_bot_term_a > list_Bot_bot_term_a > $o] :
      ( ( ( size_s1103687553077312429term_a @ Ss )
        = ( size_s1103687553077312429term_a @ Ts ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ts ) )
           => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Ss @ I4 ) @ ( nth_Bot_bot_term_a @ Ts @ I4 ) ) @ ( transi2774326411154761896term_a @ R4 ) ) )
       => ( ! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a] :
              ( ( ( size_s1103687553077312429term_a @ Xs2 )
                = ( size_s1103687553077312429term_a @ Ys ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( size_s1103687553077312429term_a @ Ys ) )
                   => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Xs2 @ I3 ) @ ( nth_Bot_bot_term_a @ Ys @ I3 ) ) @ R4 ) )
               => ( P @ Xs2 @ Ys ) ) )
         => ( ! [Xs2: list_Bot_bot_term_a,Ys: list_Bot_bot_term_a,I4: nat,Z2: bot_bot_term_a] :
                ( ( ( size_s1103687553077312429term_a @ Xs2 )
                  = ( size_s1103687553077312429term_a @ Ys ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s1103687553077312429term_a @ Ys ) )
                     => ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Xs2 @ Ia ) @ ( nth_Bot_bot_term_a @ Ys @ Ia ) ) @ ( transi2774326411154761896term_a @ R4 ) ) )
                 => ( ( P @ Xs2 @ Ys )
                   => ( ( ord_less_nat @ I4 @ ( size_s1103687553077312429term_a @ Ys ) )
                     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( nth_Bot_bot_term_a @ Ys @ I4 ) @ Z2 ) @ R4 )
                       => ( P @ Xs2 @ ( list_u5503441972953478683term_a @ Ys @ I4 @ Z2 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_induct
thf(fact_1259_fcard__fminus2__less,axiom,
    ! [X: term_a_b,A4: fset_term_a_b,Y: term_a_b] :
      ( ( fmember_term_a_b @ X @ A4 )
     => ( ( fmember_term_a_b @ Y @ A4 )
       => ( ord_less_nat @ ( fcard_term_a_b @ ( minus_5522361424680520477rm_a_b @ ( minus_5522361424680520477rm_a_b @ A4 @ ( finsert_term_a_b @ X @ bot_bo1583650863486268376rm_a_b ) ) @ ( finsert_term_a_b @ Y @ bot_bo1583650863486268376rm_a_b ) ) ) @ ( fcard_term_a_b @ A4 ) ) ) ) ).

% fcard_fminus2_less
thf(fact_1260_trancl__list__appendI,axiom,
    ! [Xs: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a,R4: set_Pr3965413743549523539term_a,X: bot_bot_term_a,Y: bot_bot_term_a] :
      ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs @ Ys2 ) @ ( trancl6444742763886629761term_a @ R4 ) )
     => ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ R4 )
       => ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ ( cons_Bot_bot_term_a @ X @ Xs ) @ ( cons_Bot_bot_term_a @ Y @ Ys2 ) ) @ ( trancl6444742763886629761term_a @ R4 ) ) ) ) ).

% trancl_list_appendI
thf(fact_1261_trancl__list__append__tranclI,axiom,
    ! [X: bot_bot_term_a,Y: bot_bot_term_a,R4: set_Pr3965413743549523539term_a,Xs: list_Bot_bot_term_a,Ys2: list_Bot_bot_term_a] :
      ( ( member2247295376459944092term_a @ ( produc780340308816795563term_a @ X @ Y ) @ ( transi2774326411154761896term_a @ R4 ) )
     => ( ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ Xs @ Ys2 ) @ ( trancl6444742763886629761term_a @ R4 ) )
       => ( member4272934500791052220term_a @ ( produc1077782494051512011term_a @ ( cons_Bot_bot_term_a @ X @ Xs ) @ ( cons_Bot_bot_term_a @ Y @ Ys2 ) ) @ ( trancl6444742763886629761term_a @ R4 ) ) ) ) ).

% trancl_list_append_tranclI
thf(fact_1262_fmember__def,axiom,
    ( fmembe4510397451897038394term_a
    = ( map_fu4212813072579612070rm_a_o @ id_Pro7590044230637570276term_a @ ( map_fu7453585163258293968_a_o_o @ fset_P6484774398675748578term_a @ id_o ) @ member2247295376459944092term_a ) ) ).

% fmember_def
thf(fact_1263_fmember__def,axiom,
    ( fmember_term_a_c
    = ( map_fu2492594514265942088_a_c_o @ id_term_a_c @ ( map_fu5563206862702799912_c_o_o @ fset_term_a_c2 @ id_o ) @ member_term_a_c ) ) ).

% fmember_def
thf(fact_1264_fmember__def,axiom,
    ( fmember_term_a_b
    = ( map_fu2757774360759362054_a_b_o @ id_term_a_b @ ( map_fu435557360475869606_b_o_o @ fset_term_a_b2 @ id_o ) @ member_term_a_b ) ) ).

% fmember_def
thf(fact_1265_fcard__fminus1__less,axiom,
    ! [X: term_a_b,A4: fset_term_a_b] :
      ( ( fmember_term_a_b @ X @ A4 )
     => ( ord_less_nat @ ( fcard_term_a_b @ ( minus_5522361424680520477rm_a_b @ A4 @ ( finsert_term_a_b @ X @ bot_bo1583650863486268376rm_a_b ) ) ) @ ( fcard_term_a_b @ A4 ) ) ) ).

% fcard_fminus1_less
thf(fact_1266_singletonI,axiom,
    ! [A: produc7033602110846336755term_a] : ( member2247295376459944092term_a @ A @ ( insert7072317625840424131term_a @ A @ bot_bo2447593420621686463term_a ) ) ).

% singletonI
thf(fact_1267_singletonI,axiom,
    ! [A: term_a_c] : ( member_term_a_c @ A @ ( insert_term_a_c @ A @ bot_bot_set_term_a_c ) ) ).

% singletonI
thf(fact_1268_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr3965413743549523539term_a,B: produc7033602110846336755term_a,B4: set_Pr3965413743549523539term_a] :
      ( ( bot_bo2447593420621686463term_a
        = ( inf_in2157114169531486017term_a @ A4 @ ( insert7072317625840424131term_a @ B @ B4 ) ) )
      = ( ~ ( member2247295376459944092term_a @ B @ A4 )
        & ( bot_bo2447593420621686463term_a
          = ( inf_in2157114169531486017term_a @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1269_disjoint__insert_I2_J,axiom,
    ! [A4: set_term_a_c,B: term_a_c,B4: set_term_a_c] :
      ( ( bot_bot_set_term_a_c
        = ( inf_inf_set_term_a_c @ A4 @ ( insert_term_a_c @ B @ B4 ) ) )
      = ( ~ ( member_term_a_c @ B @ A4 )
        & ( bot_bot_set_term_a_c
          = ( inf_inf_set_term_a_c @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1270_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr3965413743549523539term_a,A: produc7033602110846336755term_a,A4: set_Pr3965413743549523539term_a] :
      ( ( ( inf_in2157114169531486017term_a @ B4 @ ( insert7072317625840424131term_a @ A @ A4 ) )
        = bot_bo2447593420621686463term_a )
      = ( ~ ( member2247295376459944092term_a @ A @ B4 )
        & ( ( inf_in2157114169531486017term_a @ B4 @ A4 )
          = bot_bo2447593420621686463term_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1271_disjoint__insert_I1_J,axiom,
    ! [B4: set_term_a_c,A: term_a_c,A4: set_term_a_c] :
      ( ( ( inf_inf_set_term_a_c @ B4 @ ( insert_term_a_c @ A @ A4 ) )
        = bot_bot_set_term_a_c )
      = ( ~ ( member_term_a_c @ A @ B4 )
        & ( ( inf_inf_set_term_a_c @ B4 @ A4 )
          = bot_bot_set_term_a_c ) ) ) ).

% disjoint_insert(1)
thf(fact_1272_insert__disjoint_I2_J,axiom,
    ! [A: produc7033602110846336755term_a,A4: set_Pr3965413743549523539term_a,B4: set_Pr3965413743549523539term_a] :
      ( ( bot_bo2447593420621686463term_a
        = ( inf_in2157114169531486017term_a @ ( insert7072317625840424131term_a @ A @ A4 ) @ B4 ) )
      = ( ~ ( member2247295376459944092term_a @ A @ B4 )
        & ( bot_bo2447593420621686463term_a
          = ( inf_in2157114169531486017term_a @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1273_insert__disjoint_I2_J,axiom,
    ! [A: term_a_c,A4: set_term_a_c,B4: set_term_a_c] :
      ( ( bot_bot_set_term_a_c
        = ( inf_inf_set_term_a_c @ ( insert_term_a_c @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_term_a_c @ A @ B4 )
        & ( bot_bot_set_term_a_c
          = ( inf_inf_set_term_a_c @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1274_insert__disjoint_I1_J,axiom,
    ! [A: produc7033602110846336755term_a,A4: set_Pr3965413743549523539term_a,B4: set_Pr3965413743549523539term_a] :
      ( ( ( inf_in2157114169531486017term_a @ ( insert7072317625840424131term_a @ A @ A4 ) @ B4 )
        = bot_bo2447593420621686463term_a )
      = ( ~ ( member2247295376459944092term_a @ A @ B4 )
        & ( ( inf_in2157114169531486017term_a @ A4 @ B4 )
          = bot_bo2447593420621686463term_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1275_insert__disjoint_I1_J,axiom,
    ! [A: term_a_c,A4: set_term_a_c,B4: set_term_a_c] :
      ( ( ( inf_inf_set_term_a_c @ ( insert_term_a_c @ A @ A4 ) @ B4 )
        = bot_bot_set_term_a_c )
      = ( ~ ( member_term_a_c @ A @ B4 )
        & ( ( inf_inf_set_term_a_c @ A4 @ B4 )
          = bot_bot_set_term_a_c ) ) ) ).

% insert_disjoint(1)
thf(fact_1276_insert__Diff,axiom,
    ! [A: produc7033602110846336755term_a,A4: set_Pr3965413743549523539term_a] :
      ( ( member2247295376459944092term_a @ A @ A4 )
     => ( ( insert7072317625840424131term_a @ A @ ( minus_4931416064860695866term_a @ A4 @ ( insert7072317625840424131term_a @ A @ bot_bo2447593420621686463term_a ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1277_insert__Diff,axiom,
    ! [A: term_a_c,A4: set_term_a_c] :
      ( ( member_term_a_c @ A @ A4 )
     => ( ( insert_term_a_c @ A @ ( minus_4684887306374467492rm_a_c @ A4 @ ( insert_term_a_c @ A @ bot_bot_set_term_a_c ) ) )
        = A4 ) ) ).

% insert_Diff

% Helper facts (23)
thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001tf__c_T,axiom,
    ! [X: c,Y: c] :
      ( ( if_c @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__c_T,axiom,
    ! [X: c,Y: c] :
      ( ( if_c @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_Itf__b_J_T,axiom,
    ! [X: fset_b,Y: fset_b] :
      ( ( if_fset_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_Itf__b_J_T,axiom,
    ! [X: fset_b,Y: fset_b] :
      ( ( if_fset_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_Itf__c_J_T,axiom,
    ! [X: fset_c,Y: fset_c] :
      ( ( if_fset_c @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_Itf__c_J_T,axiom,
    ! [X: fset_c,Y: fset_c] :
      ( ( if_fset_c @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Nat__Onat_J_T,axiom,
    ! [X: fset_nat,Y: fset_nat] :
      ( ( if_fset_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Nat__Onat_J_T,axiom,
    ! [X: fset_nat,Y: fset_nat] :
      ( ( if_fset_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( if_term_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
    ! [X: term_a_b,Y: term_a_b] :
      ( ( if_term_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Term__Oterm_Itf__a_Mtf__c_J_T,axiom,
    ! [X: term_a_c,Y: term_a_c] :
      ( ( if_term_a_c @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Term__Oterm_Itf__a_Mtf__c_J_T,axiom,
    ! [X: term_a_c,Y: term_a_c] :
      ( ( if_term_a_c @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: fset_term_a_b,Y: fset_term_a_b] :
      ( ( if_fset_term_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: fset_term_a_b,Y: fset_term_a_b] :
      ( ( if_fset_term_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J_T,axiom,
    ! [X: fset_term_a_c,Y: fset_term_a_c] :
      ( ( if_fset_term_a_c @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__c_J_J_T,axiom,
    ! [X: fset_term_a_c,Y: fset_term_a_c] :
      ( ( if_fset_term_a_c @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [X: fset_fset_term_a_b,Y: fset_fset_term_a_b] :
      ( ( if_fse3313186383936193710rm_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__FSet__Ofset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [X: fset_fset_term_a_b,Y: fset_fset_term_a_b] :
      ( ( if_fse3313186383936193710rm_a_b @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member2247295376459944092term_a @ ( produc780340308816795563term_a @ ( bot_te3394087401593457953rm_a_b @ l ) @ ( bot_te3394087401593457954rm_a_c @ ( subst_7999470309526761005_a_b_c @ l @ sigma ) ) ) @ bot_bless_eq_a ).

%------------------------------------------------------------------------------