%------------------------------------------------------------------------------
% 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    : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_01415_052945__16338110_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1419 ( 572 unt; 147 typ;   0 def)
%            Number of atoms       : 3419 ( 996 equ;   0 cnn)
%            Maximal formula atoms :   23 (   2 avg)
%            Number of connectives : 10069 ( 402   ~;  50   |; 162   &;7968   @)
%                                         (   0 <=>;1487  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  :  720 ( 720   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  137 ( 136 usr;  23 con; 0-5 aty)
%            Number of variables   : 3315 ( 301   ^;2948   !;  66   ?;3315   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:50:18.054
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    set_set_set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_set_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
    set_Mo2626137824023173004mula_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Monotone____Formula__Omformula_Itf__a_J,type,
    monotone_mformula_a: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

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

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

% Explicit typings (136)
thf(sy_c_Assumptions__and__Approximations_OL0,type,
    assumptions_and_L0: nat ).

thf(sy_c_Assumptions__and__Approximations_OL0_H,type,
    assumptions_and_L02: nat ).

thf(sy_c_Assumptions__and__Approximations_OM0,type,
    assumptions_and_M0: nat ).

thf(sy_c_Assumptions__and__Approximations_OM0_H,type,
    assumptions_and_M02: nat ).

thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
    assump5453534214990993103ptions: nat > nat > nat > $o ).

thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_OL,type,
    assump1710595444109740301irst_L: nat > nat > nat ).

thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
    assump1710595444109740334irst_m: nat > nat ).

thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions,type,
    assump2881078719466019805ptions: nat > nat > nat > $o ).

thf(sy_c_Assumptions__and__Approximations_Othird__assumptions,type,
    assump2119784843035796504ptions: nat > nat > nat > $o ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC,type,
    clique3210737319928189260st_ACC: nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC__cf,type,
    clique951075384711337423ACC_cf: nat > set_set_set_nat > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OC,type,
    clique5033774636164728462irst_C: nat > ( nat > nat ) > set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
    clique363107459185959606CLIQUE: nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_ONEG,type,
    clique3210737375870294875st_NEG: nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060F_062,type,
    clique2971579238625216137irst_F: nat > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060G_062l,type,
    clique7840962075309931874st_G_l: nat > nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
    clique3326749438856946062irst_K: nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060P_062L_092_060G_062l,type,
    clique2294137941332549862_L_G_l: nat > nat > nat > set_set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oaccepts,type,
    clique3686358387679108662ccepts: set_set_set_nat > set_set_nat > $o ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodot,type,
    clique5469973757772500719t_odot: set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodotl,type,
    clique7966186356931407165_odotl: nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oplucking__step,type,
    clique4095374090462327202g_step: nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov__gs,type,
    clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001tf__a,type,
    clique8563529963003110213ions_a: nat > nat > nat > set_a > ( a > set_nat ) > $o ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__cf__mf_001tf__a,type,
    clique8961599393750669800f_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__mf_001tf__a,type,
    clique4708818501384062891C_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR_001tf__a,type,
    clique3873310923663319714_APR_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET_001tf__a,type,
    clique6509092761774629891_SET_a: ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001tf__a,type,
    clique5987991184601036204th_A_a: set_a > set_Mo2626137824023173004mula_a ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__neg_001tf__a,type,
    clique6623365555141101007_neg_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__neg__rel_001tf__a,type,
    clique6353239774569474354_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__pos_001tf__a,type,
    clique8538548958085942603_pos_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__pos__rel_001tf__a,type,
    clique4465983624924118198_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Odeviate__neg_001tf__a,type,
    clique2019076642914533763_neg_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Odeviate__pos_001tf__a,type,
    clique3934260045859375359_pos_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU,type,
    clique2699557479641037314nd_PLU: nat > nat > nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__rel,type,
    clique8954521387433384062in_rel: nat > nat > nat > set_set_set_nat > set_set_set_nat > $o ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cap,type,
    clique1591571987438064265eg_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cup,type,
    clique1591571987439376245eg_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cap,type,
    clique3314026705535538693os_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cup,type,
    clique3314026705536850673os_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcap,type,
    clique2586627118206219037_sqcap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcup,type,
    clique2586627118207531017_sqcup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    finite1149291290879098388et_nat: set_set_set_nat > nat ).

thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite2115694454571419734at_nat: set_nat_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite3586981331298542604at_nat: set_set_nat_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    finite6739761609112101331et_nat: set_set_set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    finite5926941155766903689et_nat: set_set_set_set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
    inj_on_a_set_nat: ( a > set_nat ) > set_a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
    minus_3028096444314564325mula_a: set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    minus_3113942175840221057et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    sup_sup_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
    sup_su7438456061012554424mula_a: set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    sup_su3906748206781935060et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_Monotone__Formula_Omformula_OConj_001tf__a,type,
    monotone_Conj_a: monotone_mformula_a > monotone_mformula_a > monotone_mformula_a ).

thf(sy_c_Monotone__Formula_Omformula_ODisj_001tf__a,type,
    monotone_Disj_a: monotone_mformula_a > monotone_mformula_a > monotone_mformula_a ).

thf(sy_c_Monotone__Formula_Omformula_OFALSE_001tf__a,type,
    monotone_FALSE_a: monotone_mformula_a ).

thf(sy_c_Monotone__Formula_Omformula_OTRUE_001tf__a,type,
    monotone_TRUE_a: monotone_mformula_a ).

thf(sy_c_Monotone__Formula_Omformula_OVar_001tf__a,type,
    monotone_Var_a: a > monotone_mformula_a ).

thf(sy_c_Monotone__Formula_Otf__mformula_001tf__a,type,
    monoto4877036962378694605mula_a: set_Mo2626137824023173004mula_a ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bot_bot_nat_nat_o: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    bot_bo6227097192321305471_nat_o: set_set_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bot_set_nat_nat: set_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
    bot_bo3042613601904376864mula_a: set_Mo2626137824023173004mula_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bot_bo7376149671870096959at_nat: set_set_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    bot_bo7198184520161983622et_nat: set_set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    bot_bo193956671110832956et_nat: set_set_set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
    bot_bot_set_a: set_a ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
    ord_le3562860513790772192mula_a: set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    ord_le52856854838348540et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
    ord_less_set_a: set_a > set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
    ord_le5054881893329012716mula_a: set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
    ord_less_eq_set_a: set_a > set_a > $o ).

thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).

thf(sy_c_Set_OCollect_001t__Monotone____Formula__Omformula_Itf__a_J,type,
    collec4794253742848188331mula_a: ( monotone_mformula_a > $o ) > set_Mo2626137824023173004mula_a ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).

thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_Itf__a_J,type,
    accp_M6162913489380515981mula_a: ( monotone_mformula_a > monotone_mformula_a > $o ) > monotone_mformula_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    accp_set_set_set_nat: ( set_set_set_nat > set_set_set_nat > $o ) > set_set_set_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).

thf(sy_c_member_001t__Monotone____Formula__Omformula_Itf__a_J,type,
    member535913909593306477mula_a: monotone_mformula_a > set_Mo2626137824023173004mula_a > $o ).

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

thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).

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

thf(sy_v__092_060V_062,type,
    v: set_a ).

thf(sy_v__092_060phi_062,type,
    phi: monotone_mformula_a ).

thf(sy_v__092_060phi_062_H____,type,
    phi2: monotone_mformula_a ).

thf(sy_v__092_060pi_062,type,
    pi: a > set_nat ).

thf(sy_v__092_060psi_062____,type,
    psi: monotone_mformula_a ).

thf(sy_v_k,type,
    k: nat ).

thf(sy_v_l,type,
    l: nat ).

thf(sy_v_p,type,
    p: nat ).

% Relevant facts (1271)
thf(fact_0_tf__Conj_Ohyps_I3_J,axiom,
    member535913909593306477mula_a @ psi @ monoto4877036962378694605mula_a ).

% tf_Conj.hyps(3)
thf(fact_1_tf__Conj_Ohyps_I1_J,axiom,
    member535913909593306477mula_a @ phi2 @ monoto4877036962378694605mula_a ).

% tf_Conj.hyps(1)
thf(fact_2_deviate__subset__Conj_I1_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Psi ) ) ) ).

% deviate_subset_Conj(1)
thf(fact_3_forth__assumptions_OAPR_Ocong,axiom,
    clique3873310923663319714_APR_a = clique3873310923663319714_APR_a ).

% forth_assumptions.APR.cong
thf(fact_4_forth__assumptions_Odeviate__pos_Ocong,axiom,
    clique3934260045859375359_pos_a = clique3934260045859375359_pos_a ).

% forth_assumptions.deviate_pos.cong
thf(fact_5_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
    clique3314026705535538693os_cap = clique3314026705535538693os_cap ).

% second_assumptions.deviate_pos_cap.cong
thf(fact_6_APR_Osimps_I4_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) )
      = ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) ) ).

% APR.simps(4)
thf(fact_7_approx__pos_Osimps_I1_J,axiom,
    ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
      ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi2 @ Psi2 ) )
      = ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) ).

% approx_pos.simps(1)
thf(fact_8_third__assumptions__axioms,axiom,
    assump2119784843035796504ptions @ l @ p @ k ).

% third_assumptions_axioms
thf(fact_9_deviate__finite_I1_J,axiom,
    ! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) ).

% deviate_finite(1)
thf(fact_10_Un__subset__iff,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
      = ( ( ord_le9131159989063066194et_nat @ A @ C )
        & ( ord_le9131159989063066194et_nat @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_11_Un__subset__iff,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
      = ( ( ord_le9059583361652607317at_nat @ A @ C )
        & ( ord_le9059583361652607317at_nat @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_12_Un__subset__iff,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
      = ( ( ord_le6893508408891458716et_nat @ A @ C )
        & ( ord_le6893508408891458716et_nat @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_13_deviate__finite_I5_J,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] : ( finite6739761609112101331et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ A @ B ) ) ).

% deviate_finite(5)
thf(fact_14_le__sup__iff,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ Z )
      = ( ( ord_le9131159989063066194et_nat @ X @ Z )
        & ( ord_le9131159989063066194et_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_15_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_16_le__sup__iff,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ Z )
      = ( ( ord_le9059583361652607317at_nat @ X @ Z )
        & ( ord_le9059583361652607317at_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_17_le__sup__iff,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ Z )
      = ( ( ord_le6893508408891458716et_nat @ X @ Z )
        & ( ord_le6893508408891458716et_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_18_le__sup__iff,axiom,
    ! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
      ( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat_nat @ X @ Z )
        & ( ord_less_eq_nat_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_19_sup_Obounded__iff,axiom,
    ! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
        & ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_20_sup_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_21_sup_Obounded__iff,axiom,
    ! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
        & ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_22_sup_Obounded__iff,axiom,
    ! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
        & ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_23_sup_Obounded__iff,axiom,
    ! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_nat_nat @ B2 @ A2 )
        & ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_24_second__assumptions__axioms,axiom,
    assump2881078719466019805ptions @ l @ p @ k ).

% second_assumptions_axioms
thf(fact_25_phi_I1_J,axiom,
    member535913909593306477mula_a @ phi @ monoto4877036962378694605mula_a ).

% phi(1)
thf(fact_26_first__assumptions__axioms,axiom,
    assump5453534214990993103ptions @ l @ p @ k ).

% first_assumptions_axioms
thf(fact_27_approx__neg_Osimps_I1_J,axiom,
    ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
      ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi2 @ Psi2 ) )
      = ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) ).

% approx_neg.simps(1)
thf(fact_28_subset__antisym,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( ord_le9131159989063066194et_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_29_subset__antisym,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( ord_le9059583361652607317at_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_30_subset__antisym,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( ord_le6893508408891458716et_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_31_subsetI,axiom,
    ! [A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ! [X2: monotone_mformula_a] :
          ( ( member535913909593306477mula_a @ X2 @ A )
         => ( member535913909593306477mula_a @ X2 @ B ) )
     => ( ord_le5054881893329012716mula_a @ A @ B ) ) ).

% subsetI
thf(fact_32_subsetI,axiom,
    ! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ! [X2: set_set_set_nat] :
          ( ( member2946998982187404937et_nat @ X2 @ A )
         => ( member2946998982187404937et_nat @ X2 @ B ) )
     => ( ord_le572741076514265352et_nat @ A @ B ) ) ).

% subsetI
thf(fact_33_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ( member_a @ X2 @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_34_subsetI,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ! [X2: set_set_nat] :
          ( ( member_set_set_nat @ X2 @ A )
         => ( member_set_set_nat @ X2 @ B ) )
     => ( ord_le9131159989063066194et_nat @ A @ B ) ) ).

% subsetI
thf(fact_35_subsetI,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ! [X2: nat > nat] :
          ( ( member_nat_nat @ X2 @ A )
         => ( member_nat_nat @ X2 @ B ) )
     => ( ord_le9059583361652607317at_nat @ A @ B ) ) ).

% subsetI
thf(fact_36_subsetI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X2: set_nat] :
          ( ( member_set_nat @ X2 @ A )
         => ( member_set_nat @ X2 @ B ) )
     => ( ord_le6893508408891458716et_nat @ A @ B ) ) ).

% subsetI
thf(fact_37_sup_Oright__idem,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ B2 )
      = ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_38_sup_Oright__idem,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_39_sup_Oright__idem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_40_sup__left__idem,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
      = ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_41_sup__left__idem,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) )
      = ( sup_sup_set_nat_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_42_sup__left__idem,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) )
      = ( sup_sup_set_set_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_43_sup_Oleft__idem,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
      = ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_44_sup_Oleft__idem,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_45_sup_Oleft__idem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_46_sup__idem,axiom,
    ! [X: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_47_sup__idem,axiom,
    ! [X: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_48_sup__idem,axiom,
    ! [X: set_set_nat] :
      ( ( sup_sup_set_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_49_sup_Oidem,axiom,
    ! [A2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_50_sup_Oidem,axiom,
    ! [A2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_51_sup_Oidem,axiom,
    ! [A2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_52_Un__iff,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ ( sup_su7438456061012554424mula_a @ A @ B ) )
      = ( ( member535913909593306477mula_a @ C2 @ A )
        | ( member535913909593306477mula_a @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_53_Un__iff,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) )
      = ( ( member2946998982187404937et_nat @ C2 @ A )
        | ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_54_Un__iff,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
      = ( ( member_a @ C2 @ A )
        | ( member_a @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_55_Un__iff,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
      = ( ( member_set_set_nat @ C2 @ A )
        | ( member_set_set_nat @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_56_Un__iff,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
      = ( ( member_nat_nat @ C2 @ A )
        | ( member_nat_nat @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_57_Un__iff,axiom,
    ! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
      = ( ( member_set_nat @ C2 @ A )
        | ( member_set_nat @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_58_UnCI,axiom,
    ! [C2: monotone_mformula_a,B: set_Mo2626137824023173004mula_a,A: set_Mo2626137824023173004mula_a] :
      ( ( ~ ( member535913909593306477mula_a @ C2 @ B )
       => ( member535913909593306477mula_a @ C2 @ A ) )
     => ( member535913909593306477mula_a @ C2 @ ( sup_su7438456061012554424mula_a @ A @ B ) ) ) ).

% UnCI
thf(fact_59_UnCI,axiom,
    ! [C2: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
      ( ( ~ ( member2946998982187404937et_nat @ C2 @ B )
       => ( member2946998982187404937et_nat @ C2 @ A ) )
     => ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_60_UnCI,axiom,
    ! [C2: a,B: set_a,A: set_a] :
      ( ( ~ ( member_a @ C2 @ B )
       => ( member_a @ C2 @ A ) )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnCI
thf(fact_61_UnCI,axiom,
    ! [C2: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
      ( ( ~ ( member_set_set_nat @ C2 @ B )
       => ( member_set_set_nat @ C2 @ A ) )
     => ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_62_UnCI,axiom,
    ! [C2: nat > nat,B: set_nat_nat,A: set_nat_nat] :
      ( ( ~ ( member_nat_nat @ C2 @ B )
       => ( member_nat_nat @ C2 @ A ) )
     => ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_63_UnCI,axiom,
    ! [C2: set_nat,B: set_set_nat,A: set_set_nat] :
      ( ( ~ ( member_set_nat @ C2 @ B )
       => ( member_set_nat @ C2 @ A ) )
     => ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_64_L0_H,axiom,
    ord_less_eq_nat @ assumptions_and_L02 @ l ).

% L0'
thf(fact_65_L0,axiom,
    ord_less_eq_nat @ assumptions_and_L0 @ l ).

% L0
thf(fact_66_finite__approx__pos,axiom,
    ! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ Phi ) ) ).

% finite_approx_pos
thf(fact_67_deviate__finite_I3_J,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] : ( finite6739761609112101331et_nat @ ( clique3314026705536850673os_cup @ l @ p @ k @ A @ B ) ) ).

% deviate_finite(3)
thf(fact_68_second__assumptions_Osqcap_Ocong,axiom,
    clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).

% second_assumptions.sqcap.cong
thf(fact_69_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
    clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).

% second_assumptions.deviate_neg_cap.cong
thf(fact_70_forth__assumptions_Oapprox__pos_Ocong,axiom,
    clique8538548958085942603_pos_a = clique8538548958085942603_pos_a ).

% forth_assumptions.approx_pos.cong
thf(fact_71_forth__assumptions_Oapprox__neg_Ocong,axiom,
    clique6623365555141101007_neg_a = clique6623365555141101007_neg_a ).

% forth_assumptions.approx_neg.cong
thf(fact_72_Collect__mono__iff,axiom,
    ! [P: set_set_nat > $o,Q: set_set_nat > $o] :
      ( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
      = ( ! [X3: set_set_nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_73_Collect__mono__iff,axiom,
    ! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
      ( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
      = ( ! [X3: nat > nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_74_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X3: set_nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_75_set__eq__subset,axiom,
    ( ( ^ [Y2: set_set_set_nat,Z2: set_set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
          & ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_76_set__eq__subset,axiom,
    ( ( ^ [Y2: set_nat_nat,Z2: set_nat_nat] : ( Y2 = Z2 ) )
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
          & ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_77_set__eq__subset,axiom,
    ( ( ^ [Y2: set_set_nat,Z2: set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
          & ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_78_subset__trans,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( ord_le9131159989063066194et_nat @ B @ C )
       => ( ord_le9131159989063066194et_nat @ A @ C ) ) ) ).

% subset_trans
thf(fact_79_subset__trans,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( ord_le9059583361652607317at_nat @ B @ C )
       => ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).

% subset_trans
thf(fact_80_subset__trans,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( ord_le6893508408891458716et_nat @ B @ C )
       => ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).

% subset_trans
thf(fact_81_Collect__mono,axiom,
    ! [P: set_set_nat > $o,Q: set_set_nat > $o] :
      ( ! [X2: set_set_nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_82_Collect__mono,axiom,
    ! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
      ( ! [X2: nat > nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_83_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X2: set_nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_84_mem__Collect__eq,axiom,
    ! [A2: monotone_mformula_a,P: monotone_mformula_a > $o] :
      ( ( member535913909593306477mula_a @ A2 @ ( collec4794253742848188331mula_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
    ! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
      ( ( member2946998982187404937et_nat @ A2 @ ( collec7201453139178570183et_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
    ! [A2: set_set_nat,P: set_set_nat > $o] :
      ( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
    ! [A2: nat > nat,P: ( nat > nat ) > $o] :
      ( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_88_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_89_Collect__mem__eq,axiom,
    ! [A: set_Mo2626137824023173004mula_a] :
      ( ( collec4794253742848188331mula_a
        @ ^ [X3: monotone_mformula_a] : ( member535913909593306477mula_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
    ! [A: set_set_set_set_nat] :
      ( ( collec7201453139178570183et_nat
        @ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
    ! [A: set_set_set_nat] :
      ( ( collect_set_set_nat
        @ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
    ! [A: set_nat_nat] :
      ( ( collect_nat_nat
        @ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_94_subset__refl,axiom,
    ! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).

% subset_refl
thf(fact_95_subset__refl,axiom,
    ! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).

% subset_refl
thf(fact_96_subset__refl,axiom,
    ! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).

% subset_refl
thf(fact_97_subset__iff,axiom,
    ( ord_le5054881893329012716mula_a
    = ( ^ [A3: set_Mo2626137824023173004mula_a,B3: set_Mo2626137824023173004mula_a] :
        ! [T: monotone_mformula_a] :
          ( ( member535913909593306477mula_a @ T @ A3 )
         => ( member535913909593306477mula_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_98_subset__iff,axiom,
    ( ord_le572741076514265352et_nat
    = ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
        ! [T: set_set_set_nat] :
          ( ( member2946998982187404937et_nat @ T @ A3 )
         => ( member2946998982187404937et_nat @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_99_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A3 )
         => ( member_a @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_100_subset__iff,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
        ! [T: set_set_nat] :
          ( ( member_set_set_nat @ T @ A3 )
         => ( member_set_set_nat @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_101_subset__iff,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
        ! [T: nat > nat] :
          ( ( member_nat_nat @ T @ A3 )
         => ( member_nat_nat @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_102_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
        ! [T: set_nat] :
          ( ( member_set_nat @ T @ A3 )
         => ( member_set_nat @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_103_equalityD2,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( A = B )
     => ( ord_le9131159989063066194et_nat @ B @ A ) ) ).

% equalityD2
thf(fact_104_equalityD2,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( A = B )
     => ( ord_le9059583361652607317at_nat @ B @ A ) ) ).

% equalityD2
thf(fact_105_equalityD2,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( A = B )
     => ( ord_le6893508408891458716et_nat @ B @ A ) ) ).

% equalityD2
thf(fact_106_equalityD1,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( A = B )
     => ( ord_le9131159989063066194et_nat @ A @ B ) ) ).

% equalityD1
thf(fact_107_equalityD1,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( A = B )
     => ( ord_le9059583361652607317at_nat @ A @ B ) ) ).

% equalityD1
thf(fact_108_equalityD1,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( A = B )
     => ( ord_le6893508408891458716et_nat @ A @ B ) ) ).

% equalityD1
thf(fact_109_subset__eq,axiom,
    ( ord_le5054881893329012716mula_a
    = ( ^ [A3: set_Mo2626137824023173004mula_a,B3: set_Mo2626137824023173004mula_a] :
        ! [X3: monotone_mformula_a] :
          ( ( member535913909593306477mula_a @ X3 @ A3 )
         => ( member535913909593306477mula_a @ X3 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_110_subset__eq,axiom,
    ( ord_le572741076514265352et_nat
    = ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
        ! [X3: set_set_set_nat] :
          ( ( member2946998982187404937et_nat @ X3 @ A3 )
         => ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_111_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [X3: a] :
          ( ( member_a @ X3 @ A3 )
         => ( member_a @ X3 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_112_subset__eq,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
        ! [X3: set_set_nat] :
          ( ( member_set_set_nat @ X3 @ A3 )
         => ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_113_subset__eq,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
        ! [X3: nat > nat] :
          ( ( member_nat_nat @ X3 @ A3 )
         => ( member_nat_nat @ X3 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_114_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
        ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( member_set_nat @ X3 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_115_equalityE,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( A = B )
     => ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
         => ~ ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).

% equalityE
thf(fact_116_equalityE,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( A = B )
     => ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
         => ~ ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).

% equalityE
thf(fact_117_equalityE,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( A = B )
     => ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
         => ~ ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).

% equalityE
thf(fact_118_subsetD,axiom,
    ! [A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,C2: monotone_mformula_a] :
      ( ( ord_le5054881893329012716mula_a @ A @ B )
     => ( ( member535913909593306477mula_a @ C2 @ A )
       => ( member535913909593306477mula_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_119_subsetD,axiom,
    ! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le572741076514265352et_nat @ A @ B )
     => ( ( member2946998982187404937et_nat @ C2 @ A )
       => ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).

% subsetD
thf(fact_120_subsetD,axiom,
    ! [A: set_a,B: set_a,C2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_121_subsetD,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( member_set_set_nat @ C2 @ A )
       => ( member_set_set_nat @ C2 @ B ) ) ) ).

% subsetD
thf(fact_122_subsetD,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C2: nat > nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( member_nat_nat @ C2 @ A )
       => ( member_nat_nat @ C2 @ B ) ) ) ).

% subsetD
thf(fact_123_subsetD,axiom,
    ! [A: set_set_nat,B: set_set_nat,C2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( member_set_nat @ C2 @ A )
       => ( member_set_nat @ C2 @ B ) ) ) ).

% subsetD
thf(fact_124_in__mono,axiom,
    ! [A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,X: monotone_mformula_a] :
      ( ( ord_le5054881893329012716mula_a @ A @ B )
     => ( ( member535913909593306477mula_a @ X @ A )
       => ( member535913909593306477mula_a @ X @ B ) ) ) ).

% in_mono
thf(fact_125_in__mono,axiom,
    ! [A: set_set_set_set_nat,B: set_set_set_set_nat,X: set_set_set_nat] :
      ( ( ord_le572741076514265352et_nat @ A @ B )
     => ( ( member2946998982187404937et_nat @ X @ A )
       => ( member2946998982187404937et_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_126_in__mono,axiom,
    ! [A: set_a,B: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X @ A )
       => ( member_a @ X @ B ) ) ) ).

% in_mono
thf(fact_127_in__mono,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,X: set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( member_set_set_nat @ X @ A )
       => ( member_set_set_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_128_in__mono,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,X: nat > nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( member_nat_nat @ X @ A )
       => ( member_nat_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_129_in__mono,axiom,
    ! [A: set_set_nat,B: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( member_set_nat @ X @ A )
       => ( member_set_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_130_sup__left__commute,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
      = ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_131_sup__left__commute,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) )
      = ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_132_sup__left__commute,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) )
      = ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_133_sup_Oleft__commute,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) )
      = ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_134_sup_Oleft__commute,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ C2 ) )
      = ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_135_sup_Oleft__commute,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C2 ) )
      = ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_136_sup__commute,axiom,
    ( sup_su4213647025997063966et_nat
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y3 @ X3 ) ) ) ).

% sup_commute
thf(fact_137_sup__commute,axiom,
    ( sup_sup_set_nat_nat
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] : ( sup_sup_set_nat_nat @ Y3 @ X3 ) ) ) ).

% sup_commute
thf(fact_138_sup__commute,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] : ( sup_sup_set_set_nat @ Y3 @ X3 ) ) ) ).

% sup_commute
thf(fact_139_sup_Ocommute,axiom,
    ( sup_su4213647025997063966et_nat
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_140_sup_Ocommute,axiom,
    ( sup_sup_set_nat_nat
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_141_sup_Ocommute,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_142_sup__assoc,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ Z )
      = ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_143_sup__assoc,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_144_sup__assoc,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_145_sup_Oassoc,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ C2 )
      = ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_146_sup_Oassoc,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ C2 )
      = ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_147_sup_Oassoc,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C2 )
      = ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_148_inf__sup__aci_I5_J,axiom,
    ( sup_su4213647025997063966et_nat
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_149_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat_nat
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] : ( sup_sup_set_nat_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_150_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] : ( sup_sup_set_set_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_151_inf__sup__aci_I6_J,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y ) @ Z )
      = ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_152_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_153_inf__sup__aci_I6_J,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_154_inf__sup__aci_I7_J,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
      = ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_155_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y @ Z ) )
      = ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_156_inf__sup__aci_I7_J,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y @ Z ) )
      = ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_157_inf__sup__aci_I8_J,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
      = ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_158_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) )
      = ( sup_sup_set_nat_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_159_inf__sup__aci_I8_J,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) )
      = ( sup_sup_set_set_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_160_Un__left__commute,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) )
      = ( sup_su4213647025997063966et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ C ) ) ) ).

% Un_left_commute
thf(fact_161_Un__left__commute,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) )
      = ( sup_sup_set_nat_nat @ B @ ( sup_sup_set_nat_nat @ A @ C ) ) ) ).

% Un_left_commute
thf(fact_162_Un__left__commute,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) )
      = ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C ) ) ) ).

% Un_left_commute
thf(fact_163_Un__left__absorb,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) )
      = ( sup_su4213647025997063966et_nat @ A @ B ) ) ).

% Un_left_absorb
thf(fact_164_Un__left__absorb,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) )
      = ( sup_sup_set_nat_nat @ A @ B ) ) ).

% Un_left_absorb
thf(fact_165_Un__left__absorb,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) )
      = ( sup_sup_set_set_nat @ A @ B ) ) ).

% Un_left_absorb
thf(fact_166_Un__commute,axiom,
    ( sup_su4213647025997063966et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_167_Un__commute,axiom,
    ( sup_sup_set_nat_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_168_Un__commute,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_169_Un__absorb,axiom,
    ! [A: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_170_Un__absorb,axiom,
    ! [A: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_171_Un__absorb,axiom,
    ! [A: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_172_Un__assoc,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
      = ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) ) ) ).

% Un_assoc
thf(fact_173_Un__assoc,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
      = ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) ) ) ).

% Un_assoc
thf(fact_174_Un__assoc,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
      = ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).

% Un_assoc
thf(fact_175_ball__Un,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
      ( ( ! [X3: set_set_nat] :
            ( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_set_nat] :
            ( ( member_set_set_nat @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: set_set_nat] :
            ( ( member_set_set_nat @ X3 @ B )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_176_ball__Un,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
      ( ( ! [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A @ B ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ B )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_177_ball__Un,axiom,
    ! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
      ( ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_178_bex__Un,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
      ( ( ? [X3: set_set_nat] :
            ( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_set_nat] :
            ( ( member_set_set_nat @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: set_set_nat] :
            ( ( member_set_set_nat @ X3 @ B )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_179_bex__Un,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
      ( ( ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A @ B ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ B )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_180_bex__Un,axiom,
    ! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
      ( ( ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_181_UnI2,axiom,
    ! [C2: monotone_mformula_a,B: set_Mo2626137824023173004mula_a,A: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ B )
     => ( member535913909593306477mula_a @ C2 @ ( sup_su7438456061012554424mula_a @ A @ B ) ) ) ).

% UnI2
thf(fact_182_UnI2,axiom,
    ! [C2: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ B )
     => ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_183_UnI2,axiom,
    ! [C2: a,B: set_a,A: set_a] :
      ( ( member_a @ C2 @ B )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI2
thf(fact_184_UnI2,axiom,
    ! [C2: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ B )
     => ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_185_UnI2,axiom,
    ! [C2: nat > nat,B: set_nat_nat,A: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ B )
     => ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_186_UnI2,axiom,
    ! [C2: set_nat,B: set_set_nat,A: set_set_nat] :
      ( ( member_set_nat @ C2 @ B )
     => ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_187_UnI1,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ A )
     => ( member535913909593306477mula_a @ C2 @ ( sup_su7438456061012554424mula_a @ A @ B ) ) ) ).

% UnI1
thf(fact_188_UnI1,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ A )
     => ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_189_UnI1,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ A )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI1
thf(fact_190_UnI1,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ A )
     => ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_191_UnI1,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ A )
     => ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_192_UnI1,axiom,
    ! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C2 @ A )
     => ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_193_UnE,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ ( sup_su7438456061012554424mula_a @ A @ B ) )
     => ( ~ ( member535913909593306477mula_a @ C2 @ A )
       => ( member535913909593306477mula_a @ C2 @ B ) ) ) ).

% UnE
thf(fact_194_UnE,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) )
     => ( ~ ( member2946998982187404937et_nat @ C2 @ A )
       => ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).

% UnE
thf(fact_195_UnE,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
     => ( ~ ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% UnE
thf(fact_196_UnE,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
     => ( ~ ( member_set_set_nat @ C2 @ A )
       => ( member_set_set_nat @ C2 @ B ) ) ) ).

% UnE
thf(fact_197_UnE,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
     => ( ~ ( member_nat_nat @ C2 @ A )
       => ( member_nat_nat @ C2 @ B ) ) ) ).

% UnE
thf(fact_198_UnE,axiom,
    ! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
     => ( ~ ( member_set_nat @ C2 @ A )
       => ( member_set_nat @ C2 @ B ) ) ) ).

% UnE
thf(fact_199_sup_OcoboundedI2,axiom,
    ! [C2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ C2 @ B2 )
     => ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_200_sup_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C2 @ B2 )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_201_sup_OcoboundedI2,axiom,
    ! [C2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
     => ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_202_sup_OcoboundedI2,axiom,
    ! [C2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ C2 @ B2 )
     => ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_203_sup_OcoboundedI2,axiom,
    ! [C2: nat > nat,B2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ C2 @ B2 )
     => ( ord_less_eq_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_204_sup_OcoboundedI1,axiom,
    ! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
     => ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_205_sup_OcoboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A2 )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_206_sup_OcoboundedI1,axiom,
    ! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
     => ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_207_sup_OcoboundedI1,axiom,
    ! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
     => ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_208_sup_OcoboundedI1,axiom,
    ! [C2: nat > nat,A2: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ C2 @ A2 )
     => ( ord_less_eq_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_209_sup_Oabsorb__iff2,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
          ( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_210_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_211_sup_Oabsorb__iff2,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
          ( ( sup_sup_set_nat_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_212_sup_Oabsorb__iff2,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( sup_sup_set_set_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_213_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [A4: nat > nat,B4: nat > nat] :
          ( ( sup_sup_nat_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_214_sup_Oabsorb__iff1,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
          ( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_215_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_216_sup_Oabsorb__iff1,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
          ( ( sup_sup_set_nat_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_217_sup_Oabsorb__iff1,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [B4: set_set_nat,A4: set_set_nat] :
          ( ( sup_sup_set_set_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_218_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [B4: nat > nat,A4: nat > nat] :
          ( ( sup_sup_nat_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_219_sup_Ocobounded2,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_220_sup_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_221_sup_Ocobounded2,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_222_sup_Ocobounded2,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_223_sup_Ocobounded2,axiom,
    ! [B2: nat > nat,A2: nat > nat] : ( ord_less_eq_nat_nat @ B2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_224_sup_Ocobounded1,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_225_sup_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_226_sup_Ocobounded1,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_227_sup_Ocobounded1,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_228_sup_Ocobounded1,axiom,
    ! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_229_sup_Oorder__iff,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
          ( A4
          = ( sup_su4213647025997063966et_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_230_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_231_sup_Oorder__iff,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
          ( A4
          = ( sup_sup_set_nat_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_232_sup_Oorder__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [B4: set_set_nat,A4: set_set_nat] :
          ( A4
          = ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_233_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [B4: nat > nat,A4: nat > nat] :
          ( A4
          = ( sup_sup_nat_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_234_sup_OboundedI,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
     => ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_235_sup_OboundedI,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_236_sup_OboundedI,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
     => ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_237_sup_OboundedI,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_238_sup_OboundedI,axiom,
    ! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat_nat @ C2 @ A2 )
       => ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_239_sup_OboundedE,axiom,
    ! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
         => ~ ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_240_sup_OboundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_241_sup_OboundedE,axiom,
    ! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
         => ~ ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_242_sup_OboundedE,axiom,
    ! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
         => ~ ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_243_sup_OboundedE,axiom,
    ! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_nat_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_244_sup__absorb2,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ Y )
     => ( ( sup_su4213647025997063966et_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_245_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_246_sup__absorb2,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ Y )
     => ( ( sup_sup_set_nat_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_247_sup__absorb2,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( sup_sup_set_set_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_248_sup__absorb2,axiom,
    ! [X: nat > nat,Y: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ Y )
     => ( ( sup_sup_nat_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_249_sup__absorb1,axiom,
    ! [Y: set_set_set_nat,X: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ Y @ X )
     => ( ( sup_su4213647025997063966et_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_250_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_251_sup__absorb1,axiom,
    ! [Y: set_nat_nat,X: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ Y @ X )
     => ( ( sup_sup_set_nat_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_252_sup__absorb1,axiom,
    ! [Y: set_set_nat,X: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ Y @ X )
     => ( ( sup_sup_set_set_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_253_sup__absorb1,axiom,
    ! [Y: nat > nat,X: nat > nat] :
      ( ( ord_less_eq_nat_nat @ Y @ X )
     => ( ( sup_sup_nat_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_254_sup_Oabsorb2,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_255_sup_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_256_sup_Oabsorb2,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_257_sup_Oabsorb2,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( sup_sup_set_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_258_sup_Oabsorb2,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( sup_sup_nat_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_259_sup_Oabsorb1,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
     => ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_260_sup_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_261_sup_Oabsorb1,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_262_sup_Oabsorb1,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ( sup_sup_set_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_263_sup_Oabsorb1,axiom,
    ! [B2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ B2 @ A2 )
     => ( ( sup_sup_nat_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_264_sup__unique,axiom,
    ! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X: set_set_set_nat,Y: set_set_set_nat] :
      ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat,Z3: set_set_set_nat] :
              ( ( ord_le9131159989063066194et_nat @ Y4 @ X2 )
             => ( ( ord_le9131159989063066194et_nat @ Z3 @ X2 )
               => ( ord_le9131159989063066194et_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_su4213647025997063966et_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_265_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X2 )
             => ( ( ord_less_eq_nat @ Z3 @ X2 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_266_sup__unique,axiom,
    ! [F: set_nat_nat > set_nat_nat > set_nat_nat,X: set_nat_nat,Y: set_nat_nat] :
      ( ! [X2: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: set_nat_nat,Y4: set_nat_nat,Z3: set_nat_nat] :
              ( ( ord_le9059583361652607317at_nat @ Y4 @ X2 )
             => ( ( ord_le9059583361652607317at_nat @ Z3 @ X2 )
               => ( ord_le9059583361652607317at_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_set_nat_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_267_sup__unique,axiom,
    ! [F: set_set_nat > set_set_nat > set_set_nat,X: set_set_nat,Y: set_set_nat] :
      ( ! [X2: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: set_set_nat,Y4: set_set_nat,Z3: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ Y4 @ X2 )
             => ( ( ord_le6893508408891458716et_nat @ Z3 @ X2 )
               => ( ord_le6893508408891458716et_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_set_set_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_268_sup__unique,axiom,
    ! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X: nat > nat,Y: nat > nat] :
      ( ! [X2: nat > nat,Y4: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: nat > nat,Y4: nat > nat] : ( ord_less_eq_nat_nat @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: nat > nat,Y4: nat > nat,Z3: nat > nat] :
              ( ( ord_less_eq_nat_nat @ Y4 @ X2 )
             => ( ( ord_less_eq_nat_nat @ Z3 @ X2 )
               => ( ord_less_eq_nat_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_nat_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_269_sup_OorderI,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( A2
        = ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
     => ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_270_sup_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_271_sup_OorderI,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( A2
        = ( sup_sup_set_nat_nat @ A2 @ B2 ) )
     => ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_272_sup_OorderI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( A2
        = ( sup_sup_set_set_nat @ A2 @ B2 ) )
     => ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_273_sup_OorderI,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( A2
        = ( sup_sup_nat_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_274_sup_OorderE,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
     => ( A2
        = ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_275_sup_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_276_sup_OorderE,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_277_sup_OorderE,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_278_sup_OorderE,axiom,
    ! [B2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_279_le__iff__sup,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
          ( ( sup_su4213647025997063966et_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_280_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( sup_sup_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_281_le__iff__sup,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
          ( ( sup_sup_set_nat_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_282_le__iff__sup,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] :
          ( ( sup_sup_set_set_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_283_le__iff__sup,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [X3: nat > nat,Y3: nat > nat] :
          ( ( sup_sup_nat_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_284_sup__least,axiom,
    ! [Y: set_set_set_nat,X: set_set_set_nat,Z: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ Y @ X )
     => ( ( ord_le9131159989063066194et_nat @ Z @ X )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_285_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_286_sup__least,axiom,
    ! [Y: set_nat_nat,X: set_nat_nat,Z: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ Y @ X )
     => ( ( ord_le9059583361652607317at_nat @ Z @ X )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_287_sup__least,axiom,
    ! [Y: set_set_nat,X: set_set_nat,Z: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ Y @ X )
     => ( ( ord_le6893508408891458716et_nat @ Z @ X )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_288_sup__least,axiom,
    ! [Y: nat > nat,X: nat > nat,Z: nat > nat] :
      ( ( ord_less_eq_nat_nat @ Y @ X )
     => ( ( ord_less_eq_nat_nat @ Z @ X )
       => ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_289_sup__mono,axiom,
    ! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ D )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_290_sup__mono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_291_sup__mono,axiom,
    ! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ D )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_292_sup__mono,axiom,
    ! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ D )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_293_sup__mono,axiom,
    ! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat_nat @ B2 @ D )
       => ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ ( sup_sup_nat_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_294_sup_Omono,axiom,
    ! [C2: set_set_set_nat,A2: set_set_set_nat,D: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
     => ( ( ord_le9131159989063066194et_nat @ D @ B2 )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C2 @ D ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_295_sup_Omono,axiom,
    ! [C2: nat,A2: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A2 )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_296_sup_Omono,axiom,
    ! [C2: set_nat_nat,A2: set_nat_nat,D: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
     => ( ( ord_le9059583361652607317at_nat @ D @ B2 )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C2 @ D ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_297_sup_Omono,axiom,
    ! [C2: set_set_nat,A2: set_set_nat,D: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ D @ B2 )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C2 @ D ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_298_sup_Omono,axiom,
    ! [C2: nat > nat,A2: nat > nat,D: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ C2 @ A2 )
     => ( ( ord_less_eq_nat_nat @ D @ B2 )
       => ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ C2 @ D ) @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_299_le__supI2,axiom,
    ! [X: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ B2 )
     => ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_300_le__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X @ B2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_301_le__supI2,axiom,
    ! [X: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ B2 )
     => ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_302_le__supI2,axiom,
    ! [X: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ B2 )
     => ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_303_le__supI2,axiom,
    ! [X: nat > nat,B2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ B2 )
     => ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_304_le__supI1,axiom,
    ! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ A2 )
     => ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_305_le__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_306_le__supI1,axiom,
    ! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ A2 )
     => ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_307_le__supI1,axiom,
    ! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ A2 )
     => ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_308_le__supI1,axiom,
    ! [X: nat > nat,A2: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ A2 )
     => ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_309_sup__ge2,axiom,
    ! [Y: set_set_set_nat,X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_310_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_311_sup__ge2,axiom,
    ! [Y: set_nat_nat,X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_312_sup__ge2,axiom,
    ! [Y: set_set_nat,X: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_313_sup__ge2,axiom,
    ! [Y: nat > nat,X: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_314_sup__ge1,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_315_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_316_sup__ge1,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_317_sup__ge1,axiom,
    ! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_318_sup__ge1,axiom,
    ! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_319_le__supI,axiom,
    ! [A2: set_set_set_nat,X: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ X )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ X )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_320_le__supI,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ( ord_less_eq_nat @ B2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_321_le__supI,axiom,
    ! [A2: set_nat_nat,X: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ X )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ X )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_322_le__supI,axiom,
    ! [A2: set_set_nat,X: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ X )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ X )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_323_le__supI,axiom,
    ! [A2: nat > nat,X: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ X )
     => ( ( ord_less_eq_nat_nat @ B2 @ X )
       => ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_324_le__supE,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,X: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le9131159989063066194et_nat @ A2 @ X )
         => ~ ( ord_le9131159989063066194et_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_325_le__supE,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_326_le__supE,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,X: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X )
         => ~ ( ord_le9059583361652607317at_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_327_le__supE,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X )
         => ~ ( ord_le6893508408891458716et_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_328_le__supE,axiom,
    ! [A2: nat > nat,B2: nat > nat,X: nat > nat] :
      ( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_329_inf__sup__ord_I3_J,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_330_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_331_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_332_inf__sup__ord_I3_J,axiom,
    ! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_333_inf__sup__ord_I3_J,axiom,
    ! [X: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X @ ( sup_sup_nat_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_334_inf__sup__ord_I4_J,axiom,
    ! [Y: set_set_set_nat,X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_335_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_336_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat_nat,X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_337_inf__sup__ord_I4_J,axiom,
    ! [Y: set_set_nat,X: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_338_inf__sup__ord_I4_J,axiom,
    ! [Y: nat > nat,X: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_339_subset__Un__eq,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
          ( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_340_subset__Un__eq,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
          ( ( sup_sup_set_nat_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_341_subset__Un__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
          ( ( sup_sup_set_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_342_subset__UnE,axiom,
    ! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
     => ~ ! [A5: set_set_set_nat] :
            ( ( ord_le9131159989063066194et_nat @ A5 @ A )
           => ! [B5: set_set_set_nat] :
                ( ( ord_le9131159989063066194et_nat @ B5 @ B )
               => ( C
                 != ( sup_su4213647025997063966et_nat @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_343_subset__UnE,axiom,
    ! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
     => ~ ! [A5: set_nat_nat] :
            ( ( ord_le9059583361652607317at_nat @ A5 @ A )
           => ! [B5: set_nat_nat] :
                ( ( ord_le9059583361652607317at_nat @ B5 @ B )
               => ( C
                 != ( sup_sup_set_nat_nat @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_344_subset__UnE,axiom,
    ! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
     => ~ ! [A5: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ A5 @ A )
           => ! [B5: set_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ B5 @ B )
               => ( C
                 != ( sup_sup_set_set_nat @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_345_Un__absorb2,axiom,
    ! [B: set_set_set_nat,A: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B @ A )
     => ( ( sup_su4213647025997063966et_nat @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_346_Un__absorb2,axiom,
    ! [B: set_nat_nat,A: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B @ A )
     => ( ( sup_sup_set_nat_nat @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_347_Un__absorb2,axiom,
    ! [B: set_set_nat,A: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B @ A )
     => ( ( sup_sup_set_set_nat @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_348_Un__absorb1,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( sup_su4213647025997063966et_nat @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_349_Un__absorb1,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( sup_sup_set_nat_nat @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_350_Un__absorb1,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( sup_sup_set_set_nat @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_351_Un__upper2,axiom,
    ! [B: set_set_set_nat,A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).

% Un_upper2
thf(fact_352_Un__upper2,axiom,
    ! [B: set_nat_nat,A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A @ B ) ) ).

% Un_upper2
thf(fact_353_Un__upper2,axiom,
    ! [B: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A @ B ) ) ).

% Un_upper2
thf(fact_354_Un__upper1,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).

% Un_upper1
thf(fact_355_Un__upper1,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) ) ).

% Un_upper1
thf(fact_356_Un__upper1,axiom,
    ! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) ) ).

% Un_upper1
thf(fact_357_Un__least,axiom,
    ! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ C )
     => ( ( ord_le9131159989063066194et_nat @ B @ C )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C ) ) ) ).

% Un_least
thf(fact_358_Un__least,axiom,
    ! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ C )
     => ( ( ord_le9059583361652607317at_nat @ B @ C )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C ) ) ) ).

% Un_least
thf(fact_359_Un__least,axiom,
    ! [A: set_set_nat,C: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ C )
     => ( ( ord_le6893508408891458716et_nat @ B @ C )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C ) ) ) ).

% Un_least
thf(fact_360_Un__mono,axiom,
    ! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ C )
     => ( ( ord_le9131159989063066194et_nat @ B @ D2 )
       => ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C @ D2 ) ) ) ) ).

% Un_mono
thf(fact_361_Un__mono,axiom,
    ! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ C )
     => ( ( ord_le9059583361652607317at_nat @ B @ D2 )
       => ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C @ D2 ) ) ) ) ).

% Un_mono
thf(fact_362_Un__mono,axiom,
    ! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ C )
     => ( ( ord_le6893508408891458716et_nat @ B @ D2 )
       => ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C @ D2 ) ) ) ) ).

% Un_mono
thf(fact_363_deviate__subset__Conj_I2_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Psi ) ) ) ).

% deviate_subset_Conj(2)
thf(fact_364_accepts__def,axiom,
    ( clique3686358387679108662ccepts
    = ( ^ [X4: set_set_set_nat,G: set_set_nat] :
        ? [X3: set_set_nat] :
          ( ( member_set_set_nat @ X3 @ X4 )
          & ( ord_le6893508408891458716et_nat @ X3 @ G ) ) ) ) ).

% accepts_def
thf(fact_365_finite__approx__neg,axiom,
    ! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ Phi ) ) ).

% finite_approx_neg
thf(fact_366_finite__Un,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_367_finite__Un,axiom,
    ! [F2: set_set_set_nat,G2: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F2 @ G2 ) )
      = ( ( finite6739761609112101331et_nat @ F2 )
        & ( finite6739761609112101331et_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_368_finite__Un,axiom,
    ! [F2: set_nat_nat,G2: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F2 @ G2 ) )
      = ( ( finite2115694454571419734at_nat @ F2 )
        & ( finite2115694454571419734at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_369_finite__Un,axiom,
    ! [F2: set_set_nat,G2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G2 ) )
      = ( ( finite1152437895449049373et_nat @ F2 )
        & ( finite1152437895449049373et_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_370_sqcap,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
     => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
       => ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X5 @ Y5 ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).

% sqcap
thf(fact_371_deviate__finite_I6_J,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k @ A @ B ) ) ).

% deviate_finite(6)
thf(fact_372_sqcap__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique2586627118206219037_sqcap @ l @ p @ k @ X5 @ Y5 )
      = ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y5 ) ) ) ).

% sqcap_def
thf(fact_373_deviate__subset__Disj_I1_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705536850673os_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Psi ) ) ) ).

% deviate_subset_Disj(1)
thf(fact_374_pl,axiom,
    ord_less_nat @ l @ p ).

% pl
thf(fact_375_kp,axiom,
    ord_less_nat @ p @ k ).

% kp
thf(fact_376_k,axiom,
    ord_less_nat @ l @ k ).

% k
thf(fact_377_PLU__union,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
     => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
       => ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).

% PLU_union
thf(fact_378_PLU__joinl,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
     => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
       => ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y5 ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).

% PLU_joinl
thf(fact_379_deviate__finite_I2_J,axiom,
    ! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) ).

% deviate_finite(2)
thf(fact_380_acceptsI,axiom,
    ! [D2: set_set_nat,G2: set_set_nat,X5: set_set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
     => ( ( member_set_set_nat @ D2 @ X5 )
       => ( clique3686358387679108662ccepts @ X5 @ G2 ) ) ) ).

% acceptsI
thf(fact_381_forth__assumptions_Odeviate__neg_Ocong,axiom,
    clique2019076642914533763_neg_a = clique2019076642914533763_neg_a ).

% forth_assumptions.deviate_neg.cong
thf(fact_382_first__assumptions_O_092_060P_062L_092_060G_062l_Ocong,axiom,
    clique2294137941332549862_L_G_l = clique2294137941332549862_L_G_l ).

% first_assumptions.\<P>L\<G>l.cong
thf(fact_383_second__assumptions_OPLU_Ocong,axiom,
    clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).

% second_assumptions.PLU.cong
thf(fact_384_first__assumptions_Oodotl_Ocong,axiom,
    clique7966186356931407165_odotl = clique7966186356931407165_odotl ).

% first_assumptions.odotl.cong
thf(fact_385_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
    clique3314026705536850673os_cup = clique3314026705536850673os_cup ).

% second_assumptions.deviate_pos_cup.cong
thf(fact_386_second__assumptions_OPLU__joinl,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
       => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
         => ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y5 ) ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).

% second_assumptions.PLU_joinl
thf(fact_387_less__supI1,axiom,
    ! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ X @ A2 )
     => ( ord_less_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_388_less__supI1,axiom,
    ! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ X @ A2 )
     => ( ord_less_set_set_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_389_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_390_less__supI1,axiom,
    ! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ A2 )
     => ( ord_le152980574450754630et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_391_less__supI2,axiom,
    ! [X: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ X @ B2 )
     => ( ord_less_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_392_less__supI2,axiom,
    ! [X: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_less_set_set_nat @ X @ B2 )
     => ( ord_less_set_set_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_393_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_394_less__supI2,axiom,
    ! [X: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ B2 )
     => ( ord_le152980574450754630et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_395_sup_Oabsorb3,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_396_sup_Oabsorb3,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_less_set_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_397_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_398_sup_Oabsorb3,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
     => ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_399_sup_Oabsorb4,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_400_sup_Oabsorb4,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_401_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_402_sup_Oabsorb4,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_403_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_set_nat_nat @ B2 @ A2 )
         => ~ ( ord_less_set_nat_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_404_sup_Ostrict__boundedE,axiom,
    ! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
      ( ( ord_less_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_set_set_nat @ B2 @ A2 )
         => ~ ( ord_less_set_set_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_405_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_406_sup_Ostrict__boundedE,axiom,
    ! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
         => ~ ( ord_le152980574450754630et_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_407_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
          ( ( A4
            = ( sup_sup_set_nat_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_408_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_set_nat
    = ( ^ [B4: set_set_nat,A4: set_set_nat] :
          ( ( A4
            = ( sup_sup_set_set_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_409_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_410_sup_Ostrict__order__iff,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
          ( ( A4
            = ( sup_su4213647025997063966et_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_411_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ C2 @ A2 )
     => ( ord_less_set_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_412_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ C2 @ A2 )
     => ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_413_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C2 @ A2 )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_414_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ C2 @ A2 )
     => ( ord_le152980574450754630et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_415_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ C2 @ B2 )
     => ( ord_less_set_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_416_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_less_set_set_nat @ C2 @ B2 )
     => ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_417_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C2 @ B2 )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_418_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ C2 @ B2 )
     => ( ord_le152980574450754630et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_419_second__assumptions_OPLU__union,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
       => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
         => ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).

% second_assumptions.PLU_union
thf(fact_420_second__assumptions_Osqcap__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X5 @ Y5 )
        = ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y5 ) ) ) ) ).

% second_assumptions.sqcap_def
thf(fact_421_first__assumptions_Oaccepts__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,G2: set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique3686358387679108662ccepts @ X5 @ G2 )
        = ( ? [X3: set_set_nat] :
              ( ( member_set_set_nat @ X3 @ X5 )
              & ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ) ).

% first_assumptions.accepts_def
thf(fact_422_first__assumptions_OacceptsI,axiom,
    ! [L: nat,P2: nat,K: nat,D2: set_set_nat,G2: set_set_nat,X5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
       => ( ( member_set_set_nat @ D2 @ X5 )
         => ( clique3686358387679108662ccepts @ X5 @ G2 ) ) ) ) ).

% first_assumptions.acceptsI
thf(fact_423_second__assumptions_Osqcap,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
       => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
         => ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X5 @ Y5 ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).

% second_assumptions.sqcap
thf(fact_424_finite__has__maximal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ( ord_less_eq_set_nat @ A2 @ X2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_425_finite__has__maximal2,axiom,
    ! [A: set_set_set_set_nat,A2: set_set_set_nat] :
      ( ( finite5926941155766903689et_nat @ A )
     => ( ( member2946998982187404937et_nat @ A2 @ A )
       => ? [X2: set_set_set_nat] :
            ( ( member2946998982187404937et_nat @ X2 @ A )
            & ( ord_le9131159989063066194et_nat @ A2 @ X2 )
            & ! [Xa: set_set_set_nat] :
                ( ( member2946998982187404937et_nat @ Xa @ A )
               => ( ( ord_le9131159989063066194et_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_426_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ A2 @ X2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_427_finite__has__maximal2,axiom,
    ! [A: set_set_nat_nat,A2: set_nat_nat] :
      ( ( finite3586981331298542604at_nat @ A )
     => ( ( member_set_nat_nat @ A2 @ A )
       => ? [X2: set_nat_nat] :
            ( ( member_set_nat_nat @ X2 @ A )
            & ( ord_le9059583361652607317at_nat @ A2 @ X2 )
            & ! [Xa: set_nat_nat] :
                ( ( member_set_nat_nat @ Xa @ A )
               => ( ( ord_le9059583361652607317at_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_428_finite__has__maximal2,axiom,
    ! [A: set_set_set_nat,A2: set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A )
     => ( ( member_set_set_nat @ A2 @ A )
       => ? [X2: set_set_nat] :
            ( ( member_set_set_nat @ X2 @ A )
            & ( ord_le6893508408891458716et_nat @ A2 @ X2 )
            & ! [Xa: set_set_nat] :
                ( ( member_set_set_nat @ Xa @ A )
               => ( ( ord_le6893508408891458716et_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_429_finite__has__maximal2,axiom,
    ! [A: set_nat_nat,A2: nat > nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( member_nat_nat @ A2 @ A )
       => ? [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ A )
            & ( ord_less_eq_nat_nat @ A2 @ X2 )
            & ! [Xa: nat > nat] :
                ( ( member_nat_nat @ Xa @ A )
               => ( ( ord_less_eq_nat_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_430_finite__has__minimal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ( ord_less_eq_set_nat @ X2 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_431_finite__has__minimal2,axiom,
    ! [A: set_set_set_set_nat,A2: set_set_set_nat] :
      ( ( finite5926941155766903689et_nat @ A )
     => ( ( member2946998982187404937et_nat @ A2 @ A )
       => ? [X2: set_set_set_nat] :
            ( ( member2946998982187404937et_nat @ X2 @ A )
            & ( ord_le9131159989063066194et_nat @ X2 @ A2 )
            & ! [Xa: set_set_set_nat] :
                ( ( member2946998982187404937et_nat @ Xa @ A )
               => ( ( ord_le9131159989063066194et_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_432_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ X2 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_433_finite__has__minimal2,axiom,
    ! [A: set_set_nat_nat,A2: set_nat_nat] :
      ( ( finite3586981331298542604at_nat @ A )
     => ( ( member_set_nat_nat @ A2 @ A )
       => ? [X2: set_nat_nat] :
            ( ( member_set_nat_nat @ X2 @ A )
            & ( ord_le9059583361652607317at_nat @ X2 @ A2 )
            & ! [Xa: set_nat_nat] :
                ( ( member_set_nat_nat @ Xa @ A )
               => ( ( ord_le9059583361652607317at_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_434_finite__has__minimal2,axiom,
    ! [A: set_set_set_nat,A2: set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A )
     => ( ( member_set_set_nat @ A2 @ A )
       => ? [X2: set_set_nat] :
            ( ( member_set_set_nat @ X2 @ A )
            & ( ord_le6893508408891458716et_nat @ X2 @ A2 )
            & ! [Xa: set_set_nat] :
                ( ( member_set_set_nat @ Xa @ A )
               => ( ( ord_le6893508408891458716et_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_435_finite__has__minimal2,axiom,
    ! [A: set_nat_nat,A2: nat > nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( member_nat_nat @ A2 @ A )
       => ? [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ A )
            & ( ord_less_eq_nat_nat @ X2 @ A2 )
            & ! [Xa: nat > nat] :
                ( ( member_nat_nat @ Xa @ A )
               => ( ( ord_less_eq_nat_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_436_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_437_finite__subset,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( finite6739761609112101331et_nat @ B )
       => ( finite6739761609112101331et_nat @ A ) ) ) ).

% finite_subset
thf(fact_438_finite__subset,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( finite2115694454571419734at_nat @ B )
       => ( finite2115694454571419734at_nat @ A ) ) ) ).

% finite_subset
thf(fact_439_finite__subset,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( finite1152437895449049373et_nat @ B )
       => ( finite1152437895449049373et_nat @ A ) ) ) ).

% finite_subset
thf(fact_440_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_441_infinite__super,axiom,
    ! [S: set_set_set_nat,T2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ S @ T2 )
     => ( ~ ( finite6739761609112101331et_nat @ S )
       => ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_442_infinite__super,axiom,
    ! [S: set_nat_nat,T2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ S @ T2 )
     => ( ~ ( finite2115694454571419734at_nat @ S )
       => ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_443_infinite__super,axiom,
    ! [S: set_set_nat,T2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ S @ T2 )
     => ( ~ ( finite1152437895449049373et_nat @ S )
       => ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_444_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_445_rev__finite__subset,axiom,
    ! [B: set_set_set_nat,A: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ B )
     => ( ( ord_le9131159989063066194et_nat @ A @ B )
       => ( finite6739761609112101331et_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_446_rev__finite__subset,axiom,
    ! [B: set_nat_nat,A: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ B )
     => ( ( ord_le9059583361652607317at_nat @ A @ B )
       => ( finite2115694454571419734at_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_447_rev__finite__subset,axiom,
    ! [B: set_set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B )
     => ( ( ord_le6893508408891458716et_nat @ A @ B )
       => ( finite1152437895449049373et_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_448_finite__UnI,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( finite_finite_nat @ G2 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_449_finite__UnI,axiom,
    ! [F2: set_set_set_nat,G2: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ F2 )
     => ( ( finite6739761609112101331et_nat @ G2 )
       => ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_450_finite__UnI,axiom,
    ! [F2: set_nat_nat,G2: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ F2 )
     => ( ( finite2115694454571419734at_nat @ G2 )
       => ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_451_finite__UnI,axiom,
    ! [F2: set_set_nat,G2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( finite1152437895449049373et_nat @ G2 )
       => ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_452_Un__infinite,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_453_Un__infinite,axiom,
    ! [S: set_set_set_nat,T2: set_set_set_nat] :
      ( ~ ( finite6739761609112101331et_nat @ S )
     => ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_454_Un__infinite,axiom,
    ! [S: set_nat_nat,T2: set_nat_nat] :
      ( ~ ( finite2115694454571419734at_nat @ S )
     => ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_455_Un__infinite,axiom,
    ! [S: set_set_nat,T2: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ S )
     => ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_456_infinite__Un,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_457_infinite__Un,axiom,
    ! [S: set_set_set_nat,T2: set_set_set_nat] :
      ( ( ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) ) )
      = ( ~ ( finite6739761609112101331et_nat @ S )
        | ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_458_infinite__Un,axiom,
    ! [S: set_nat_nat,T2: set_nat_nat] :
      ( ( ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) ) )
      = ( ~ ( finite2115694454571419734at_nat @ S )
        | ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_459_infinite__Un,axiom,
    ! [S: set_set_nat,T2: set_set_nat] :
      ( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) )
      = ( ~ ( finite1152437895449049373et_nat @ S )
        | ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_460_deviate__subset__Disj_I2_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Psi ) ) ) ).

% deviate_subset_Disj(2)
thf(fact_461_approx__neg_Osimps_I2_J,axiom,
    ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
      ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi2 @ Psi2 ) )
      = ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) ).

% approx_neg.simps(2)
thf(fact_462_APR_Osimps_I3_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) )
      = ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) ) ).

% APR.simps(3)
thf(fact_463_deviate__finite_I4_J,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k @ A @ B ) ) ).

% deviate_finite(4)
thf(fact_464_sqcup__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique2586627118207531017_sqcup @ l @ p @ k @ X5 @ Y5 )
      = ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) ).

% sqcup_def
thf(fact_465_sqcup,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
     => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
       => ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X5 @ Y5 ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).

% sqcup
thf(fact_466_finite___092_060F_062,axiom,
    finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).

% finite_\<F>
thf(fact_467_joinl__join,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y5 ) @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) ) ).

% joinl_join
thf(fact_468_approx__pos_Osimps_I5_J,axiom,
    ! [V: monotone_mformula_a,Va: monotone_mformula_a] :
      ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ V @ Va ) )
      = bot_bo7198184520161983622et_nat ) ).

% approx_pos.simps(5)
thf(fact_469_psubsetI,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( A != B )
       => ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).

% psubsetI
thf(fact_470_psubsetI,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat_nat @ A @ B ) ) ) ).

% psubsetI
thf(fact_471_psubsetI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_set_nat @ A @ B ) ) ) ).

% psubsetI
thf(fact_472_empty__Collect__eq,axiom,
    ! [P: set_set_nat > $o] :
      ( ( bot_bo7198184520161983622et_nat
        = ( collect_set_set_nat @ P ) )
      = ( ! [X3: set_set_nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_473_empty__Collect__eq,axiom,
    ! [P: ( nat > nat ) > $o] :
      ( ( bot_bot_set_nat_nat
        = ( collect_nat_nat @ P ) )
      = ( ! [X3: nat > nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_474_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X3: set_nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_475_Collect__empty__eq,axiom,
    ! [P: set_set_nat > $o] :
      ( ( ( collect_set_set_nat @ P )
        = bot_bo7198184520161983622et_nat )
      = ( ! [X3: set_set_nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_476_Collect__empty__eq,axiom,
    ! [P: ( nat > nat ) > $o] :
      ( ( ( collect_nat_nat @ P )
        = bot_bot_set_nat_nat )
      = ( ! [X3: nat > nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_477_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X3: set_nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_478_all__not__in__conv,axiom,
    ! [A: set_Mo2626137824023173004mula_a] :
      ( ( ! [X3: monotone_mformula_a] :
            ~ ( member535913909593306477mula_a @ X3 @ A ) )
      = ( A = bot_bo3042613601904376864mula_a ) ) ).

% all_not_in_conv
thf(fact_479_all__not__in__conv,axiom,
    ! [A: set_set_set_set_nat] :
      ( ( ! [X3: set_set_set_nat] :
            ~ ( member2946998982187404937et_nat @ X3 @ A ) )
      = ( A = bot_bo193956671110832956et_nat ) ) ).

% all_not_in_conv
thf(fact_480_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X3: a] :
            ~ ( member_a @ X3 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_481_all__not__in__conv,axiom,
    ! [A: set_set_set_nat] :
      ( ( ! [X3: set_set_nat] :
            ~ ( member_set_set_nat @ X3 @ A ) )
      = ( A = bot_bo7198184520161983622et_nat ) ) ).

% all_not_in_conv
thf(fact_482_all__not__in__conv,axiom,
    ! [A: set_nat_nat] :
      ( ( ! [X3: nat > nat] :
            ~ ( member_nat_nat @ X3 @ A ) )
      = ( A = bot_bot_set_nat_nat ) ) ).

% all_not_in_conv
thf(fact_483_all__not__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ! [X3: set_nat] :
            ~ ( member_set_nat @ X3 @ A ) )
      = ( A = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_484_empty__iff,axiom,
    ! [C2: monotone_mformula_a] :
      ~ ( member535913909593306477mula_a @ C2 @ bot_bo3042613601904376864mula_a ) ).

% empty_iff
thf(fact_485_empty__iff,axiom,
    ! [C2: set_set_set_nat] :
      ~ ( member2946998982187404937et_nat @ C2 @ bot_bo193956671110832956et_nat ) ).

% empty_iff
thf(fact_486_empty__iff,axiom,
    ! [C2: a] :
      ~ ( member_a @ C2 @ bot_bot_set_a ) ).

% empty_iff
thf(fact_487_empty__iff,axiom,
    ! [C2: set_set_nat] :
      ~ ( member_set_set_nat @ C2 @ bot_bo7198184520161983622et_nat ) ).

% empty_iff
thf(fact_488_empty__iff,axiom,
    ! [C2: nat > nat] :
      ~ ( member_nat_nat @ C2 @ bot_bot_set_nat_nat ) ).

% empty_iff
thf(fact_489_empty__iff,axiom,
    ! [C2: set_nat] :
      ~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_490_empty__subsetI,axiom,
    ! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).

% empty_subsetI
thf(fact_491_empty__subsetI,axiom,
    ! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).

% empty_subsetI
thf(fact_492_empty__subsetI,axiom,
    ! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).

% empty_subsetI
thf(fact_493_subset__empty,axiom,
    ! [A: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
      = ( A = bot_bo7198184520161983622et_nat ) ) ).

% subset_empty
thf(fact_494_subset__empty,axiom,
    ! [A: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
      = ( A = bot_bot_set_nat_nat ) ) ).

% subset_empty
thf(fact_495_subset__empty,axiom,
    ! [A: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
      = ( A = bot_bot_set_set_nat ) ) ).

% subset_empty
thf(fact_496_sup__bot_Oright__neutral,axiom,
    ! [A2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_497_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_498_sup__bot_Oright__neutral,axiom,
    ! [A2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_499_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( bot_bo7198184520161983622et_nat
        = ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bo7198184520161983622et_nat )
        & ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_500_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( bot_bot_set_nat_nat
        = ( sup_sup_set_nat_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_nat_nat )
        & ( B2 = bot_bot_set_nat_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_501_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( sup_sup_set_set_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_set_nat )
        & ( B2 = bot_bot_set_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_502_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_503_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_504_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_set_nat] :
      ( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_505_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
        = bot_bo7198184520161983622et_nat )
      = ( ( A2 = bot_bo7198184520161983622et_nat )
        & ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_506_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ( sup_sup_set_nat_nat @ A2 @ B2 )
        = bot_bot_set_nat_nat )
      = ( ( A2 = bot_bot_set_nat_nat )
        & ( B2 = bot_bot_set_nat_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_507_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ( sup_sup_set_set_nat @ A2 @ B2 )
        = bot_bot_set_set_nat )
      = ( ( A2 = bot_bot_set_set_nat )
        & ( B2 = bot_bot_set_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_508_sup__eq__bot__iff,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ( sup_su4213647025997063966et_nat @ X @ Y )
        = bot_bo7198184520161983622et_nat )
      = ( ( X = bot_bo7198184520161983622et_nat )
        & ( Y = bot_bo7198184520161983622et_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_509_sup__eq__bot__iff,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( ( sup_sup_set_nat_nat @ X @ Y )
        = bot_bot_set_nat_nat )
      = ( ( X = bot_bot_set_nat_nat )
        & ( Y = bot_bot_set_nat_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_510_sup__eq__bot__iff,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( ( sup_sup_set_set_nat @ X @ Y )
        = bot_bot_set_set_nat )
      = ( ( X = bot_bot_set_set_nat )
        & ( Y = bot_bot_set_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_511_bot__eq__sup__iff,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( bot_bo7198184520161983622et_nat
        = ( sup_su4213647025997063966et_nat @ X @ Y ) )
      = ( ( X = bot_bo7198184520161983622et_nat )
        & ( Y = bot_bo7198184520161983622et_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_512_bot__eq__sup__iff,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( bot_bot_set_nat_nat
        = ( sup_sup_set_nat_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat_nat )
        & ( Y = bot_bot_set_nat_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_513_bot__eq__sup__iff,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( sup_sup_set_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_set_nat )
        & ( Y = bot_bot_set_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_514_sup__bot__right,axiom,
    ! [X: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ X @ bot_bo7198184520161983622et_nat )
      = X ) ).

% sup_bot_right
thf(fact_515_sup__bot__right,axiom,
    ! [X: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ X @ bot_bot_set_nat_nat )
      = X ) ).

% sup_bot_right
thf(fact_516_sup__bot__right,axiom,
    ! [X: set_set_nat] :
      ( ( sup_sup_set_set_nat @ X @ bot_bot_set_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_517_sup__bot__left,axiom,
    ! [X: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_518_sup__bot__left,axiom,
    ! [X: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_519_sup__bot__left,axiom,
    ! [X: set_set_nat] :
      ( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_520_Un__empty,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ( sup_su4213647025997063966et_nat @ A @ B )
        = bot_bo7198184520161983622et_nat )
      = ( ( A = bot_bo7198184520161983622et_nat )
        & ( B = bot_bo7198184520161983622et_nat ) ) ) ).

% Un_empty
thf(fact_521_Un__empty,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ( sup_sup_set_nat_nat @ A @ B )
        = bot_bot_set_nat_nat )
      = ( ( A = bot_bot_set_nat_nat )
        & ( B = bot_bot_set_nat_nat ) ) ) ).

% Un_empty
thf(fact_522_Un__empty,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ( sup_sup_set_set_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ( A = bot_bot_set_set_nat )
        & ( B = bot_bot_set_set_nat ) ) ) ).

% Un_empty
thf(fact_523_deviate__pos__cup,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
     => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
       => ( ( clique3314026705536850673os_cup @ l @ p @ k @ X5 @ Y5 )
          = bot_bo7198184520161983622et_nat ) ) ) ).

% deviate_pos_cup
thf(fact_524_empty___092_060P_062L_092_060G_062l,axiom,
    member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ).

% empty_\<P>L\<G>l
thf(fact_525_not__psubset__empty,axiom,
    ! [A: set_nat_nat] :
      ~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).

% not_psubset_empty
thf(fact_526_not__psubset__empty,axiom,
    ! [A: set_set_nat] :
      ~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).

% not_psubset_empty
thf(fact_527_not__psubset__empty,axiom,
    ! [A: set_set_set_nat] :
      ~ ( ord_le152980574450754630et_nat @ A @ bot_bo7198184520161983622et_nat ) ).

% not_psubset_empty
thf(fact_528_first__assumptions_O_092_060F_062_Ocong,axiom,
    clique2971579238625216137irst_F = clique2971579238625216137irst_F ).

% first_assumptions.\<F>.cong
thf(fact_529_second__assumptions_Osqcup_Ocong,axiom,
    clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).

% second_assumptions.sqcup.cong
thf(fact_530_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
    clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).

% second_assumptions.deviate_neg_cup.cong
thf(fact_531_ex__in__conv,axiom,
    ! [A: set_Mo2626137824023173004mula_a] :
      ( ( ? [X3: monotone_mformula_a] : ( member535913909593306477mula_a @ X3 @ A ) )
      = ( A != bot_bo3042613601904376864mula_a ) ) ).

% ex_in_conv
thf(fact_532_ex__in__conv,axiom,
    ! [A: set_set_set_set_nat] :
      ( ( ? [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A ) )
      = ( A != bot_bo193956671110832956et_nat ) ) ).

% ex_in_conv
thf(fact_533_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X3: a] : ( member_a @ X3 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_534_ex__in__conv,axiom,
    ! [A: set_set_set_nat] :
      ( ( ? [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
      = ( A != bot_bo7198184520161983622et_nat ) ) ).

% ex_in_conv
thf(fact_535_ex__in__conv,axiom,
    ! [A: set_nat_nat] :
      ( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
      = ( A != bot_bot_set_nat_nat ) ) ).

% ex_in_conv
thf(fact_536_ex__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
      = ( A != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_537_equals0I,axiom,
    ! [A: set_Mo2626137824023173004mula_a] :
      ( ! [Y4: monotone_mformula_a] :
          ~ ( member535913909593306477mula_a @ Y4 @ A )
     => ( A = bot_bo3042613601904376864mula_a ) ) ).

% equals0I
thf(fact_538_equals0I,axiom,
    ! [A: set_set_set_set_nat] :
      ( ! [Y4: set_set_set_nat] :
          ~ ( member2946998982187404937et_nat @ Y4 @ A )
     => ( A = bot_bo193956671110832956et_nat ) ) ).

% equals0I
thf(fact_539_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y4: a] :
          ~ ( member_a @ Y4 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_540_equals0I,axiom,
    ! [A: set_set_set_nat] :
      ( ! [Y4: set_set_nat] :
          ~ ( member_set_set_nat @ Y4 @ A )
     => ( A = bot_bo7198184520161983622et_nat ) ) ).

% equals0I
thf(fact_541_equals0I,axiom,
    ! [A: set_nat_nat] :
      ( ! [Y4: nat > nat] :
          ~ ( member_nat_nat @ Y4 @ A )
     => ( A = bot_bot_set_nat_nat ) ) ).

% equals0I
thf(fact_542_equals0I,axiom,
    ! [A: set_set_nat] :
      ( ! [Y4: set_nat] :
          ~ ( member_set_nat @ Y4 @ A )
     => ( A = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_543_equals0D,axiom,
    ! [A: set_Mo2626137824023173004mula_a,A2: monotone_mformula_a] :
      ( ( A = bot_bo3042613601904376864mula_a )
     => ~ ( member535913909593306477mula_a @ A2 @ A ) ) ).

% equals0D
thf(fact_544_equals0D,axiom,
    ! [A: set_set_set_set_nat,A2: set_set_set_nat] :
      ( ( A = bot_bo193956671110832956et_nat )
     => ~ ( member2946998982187404937et_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_545_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_546_equals0D,axiom,
    ! [A: set_set_set_nat,A2: set_set_nat] :
      ( ( A = bot_bo7198184520161983622et_nat )
     => ~ ( member_set_set_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_547_equals0D,axiom,
    ! [A: set_nat_nat,A2: nat > nat] :
      ( ( A = bot_bot_set_nat_nat )
     => ~ ( member_nat_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_548_equals0D,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( A = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_549_emptyE,axiom,
    ! [A2: monotone_mformula_a] :
      ~ ( member535913909593306477mula_a @ A2 @ bot_bo3042613601904376864mula_a ) ).

% emptyE
thf(fact_550_emptyE,axiom,
    ! [A2: set_set_set_nat] :
      ~ ( member2946998982187404937et_nat @ A2 @ bot_bo193956671110832956et_nat ) ).

% emptyE
thf(fact_551_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_552_emptyE,axiom,
    ! [A2: set_set_nat] :
      ~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).

% emptyE
thf(fact_553_emptyE,axiom,
    ! [A2: nat > nat] :
      ~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).

% emptyE
thf(fact_554_emptyE,axiom,
    ! [A2: set_nat] :
      ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_555_subset__iff__psubset__eq,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
          ( ( ord_le152980574450754630et_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_556_subset__iff__psubset__eq,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
          ( ( ord_less_set_nat_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_557_subset__iff__psubset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
          ( ( ord_less_set_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_558_subset__psubset__trans,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( ord_le152980574450754630et_nat @ B @ C )
       => ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).

% subset_psubset_trans
thf(fact_559_subset__psubset__trans,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( ord_less_set_nat_nat @ B @ C )
       => ( ord_less_set_nat_nat @ A @ C ) ) ) ).

% subset_psubset_trans
thf(fact_560_subset__psubset__trans,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( ord_less_set_set_nat @ B @ C )
       => ( ord_less_set_set_nat @ A @ C ) ) ) ).

% subset_psubset_trans
thf(fact_561_subset__not__subset__eq,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
          & ~ ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_562_subset__not__subset__eq,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
          & ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_563_subset__not__subset__eq,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
          & ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_564_psubset__subset__trans,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A @ B )
     => ( ( ord_le9131159989063066194et_nat @ B @ C )
       => ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).

% psubset_subset_trans
thf(fact_565_psubset__subset__trans,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A @ B )
     => ( ( ord_le9059583361652607317at_nat @ B @ C )
       => ( ord_less_set_nat_nat @ A @ C ) ) ) ).

% psubset_subset_trans
thf(fact_566_psubset__subset__trans,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( ord_less_set_set_nat @ A @ B )
     => ( ( ord_le6893508408891458716et_nat @ B @ C )
       => ( ord_less_set_set_nat @ A @ C ) ) ) ).

% psubset_subset_trans
thf(fact_567_psubset__imp__subset,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A @ B )
     => ( ord_le9131159989063066194et_nat @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_568_psubset__imp__subset,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A @ B )
     => ( ord_le9059583361652607317at_nat @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_569_psubset__imp__subset,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_less_set_set_nat @ A @ B )
     => ( ord_le6893508408891458716et_nat @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_570_psubset__eq,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% psubset_eq
thf(fact_571_psubset__eq,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% psubset_eq
thf(fact_572_psubset__eq,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A3: set_set_nat,B3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% psubset_eq
thf(fact_573_psubsetE,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A @ B )
     => ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
         => ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).

% psubsetE
thf(fact_574_psubsetE,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A @ B )
     => ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
         => ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).

% psubsetE
thf(fact_575_psubsetE,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_less_set_set_nat @ A @ B )
     => ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
         => ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).

% psubsetE
thf(fact_576_finite__psubset__induct,axiom,
    ! [A: set_nat_nat,P: set_nat_nat > $o] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ! [A6: set_nat_nat] :
            ( ( finite2115694454571419734at_nat @ A6 )
           => ( ! [B6: set_nat_nat] :
                  ( ( ord_less_set_nat_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_577_finite__psubset__induct,axiom,
    ! [A: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ! [A6: set_set_nat] :
            ( ( finite1152437895449049373et_nat @ A6 )
           => ( ! [B6: set_set_nat] :
                  ( ( ord_less_set_set_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_578_finite__psubset__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [A6: set_nat] :
            ( ( finite_finite_nat @ A6 )
           => ( ! [B6: set_nat] :
                  ( ( ord_less_set_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_579_finite__psubset__induct,axiom,
    ! [A: set_set_set_nat,P: set_set_set_nat > $o] :
      ( ( finite6739761609112101331et_nat @ A )
     => ( ! [A6: set_set_set_nat] :
            ( ( finite6739761609112101331et_nat @ A6 )
           => ( ! [B6: set_set_set_nat] :
                  ( ( ord_le152980574450754630et_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_580_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_581_finite_OemptyI,axiom,
    finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).

% finite.emptyI
thf(fact_582_finite_OemptyI,axiom,
    finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).

% finite.emptyI
thf(fact_583_finite_OemptyI,axiom,
    finite1152437895449049373et_nat @ bot_bot_set_set_nat ).

% finite.emptyI
thf(fact_584_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_585_infinite__imp__nonempty,axiom,
    ! [S: set_set_set_nat] :
      ( ~ ( finite6739761609112101331et_nat @ S )
     => ( S != bot_bo7198184520161983622et_nat ) ) ).

% infinite_imp_nonempty
thf(fact_586_infinite__imp__nonempty,axiom,
    ! [S: set_nat_nat] :
      ( ~ ( finite2115694454571419734at_nat @ S )
     => ( S != bot_bot_set_nat_nat ) ) ).

% infinite_imp_nonempty
thf(fact_587_infinite__imp__nonempty,axiom,
    ! [S: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ S )
     => ( S != bot_bot_set_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_588_Un__empty__right,axiom,
    ! [A: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
      = A ) ).

% Un_empty_right
thf(fact_589_Un__empty__right,axiom,
    ! [A: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A @ bot_bot_set_nat_nat )
      = A ) ).

% Un_empty_right
thf(fact_590_Un__empty__right,axiom,
    ! [A: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
      = A ) ).

% Un_empty_right
thf(fact_591_Un__empty__left,axiom,
    ! [B: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_592_Un__empty__left,axiom,
    ! [B: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_593_Un__empty__left,axiom,
    ! [B: set_set_nat] :
      ( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_594_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_595_finite__has__maximal,axiom,
    ! [A: set_set_set_set_nat] :
      ( ( finite5926941155766903689et_nat @ A )
     => ( ( A != bot_bo193956671110832956et_nat )
       => ? [X2: set_set_set_nat] :
            ( ( member2946998982187404937et_nat @ X2 @ A )
            & ! [Xa: set_set_set_nat] :
                ( ( member2946998982187404937et_nat @ Xa @ A )
               => ( ( ord_le9131159989063066194et_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_596_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_597_finite__has__maximal,axiom,
    ! [A: set_set_nat_nat] :
      ( ( finite3586981331298542604at_nat @ A )
     => ( ( A != bot_bo7376149671870096959at_nat )
       => ? [X2: set_nat_nat] :
            ( ( member_set_nat_nat @ X2 @ A )
            & ! [Xa: set_nat_nat] :
                ( ( member_set_nat_nat @ Xa @ A )
               => ( ( ord_le9059583361652607317at_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_598_finite__has__maximal,axiom,
    ! [A: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A )
     => ( ( A != bot_bo7198184520161983622et_nat )
       => ? [X2: set_set_nat] :
            ( ( member_set_set_nat @ X2 @ A )
            & ! [Xa: set_set_nat] :
                ( ( member_set_set_nat @ Xa @ A )
               => ( ( ord_le6893508408891458716et_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_599_finite__has__maximal,axiom,
    ! [A: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( A != bot_bot_set_nat_nat )
       => ? [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ A )
            & ! [Xa: nat > nat] :
                ( ( member_nat_nat @ Xa @ A )
               => ( ( ord_less_eq_nat_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_600_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_601_finite__has__minimal,axiom,
    ! [A: set_set_set_set_nat] :
      ( ( finite5926941155766903689et_nat @ A )
     => ( ( A != bot_bo193956671110832956et_nat )
       => ? [X2: set_set_set_nat] :
            ( ( member2946998982187404937et_nat @ X2 @ A )
            & ! [Xa: set_set_set_nat] :
                ( ( member2946998982187404937et_nat @ Xa @ A )
               => ( ( ord_le9131159989063066194et_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_602_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_603_finite__has__minimal,axiom,
    ! [A: set_set_nat_nat] :
      ( ( finite3586981331298542604at_nat @ A )
     => ( ( A != bot_bo7376149671870096959at_nat )
       => ? [X2: set_nat_nat] :
            ( ( member_set_nat_nat @ X2 @ A )
            & ! [Xa: set_nat_nat] :
                ( ( member_set_nat_nat @ Xa @ A )
               => ( ( ord_le9059583361652607317at_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_604_finite__has__minimal,axiom,
    ! [A: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A )
     => ( ( A != bot_bo7198184520161983622et_nat )
       => ? [X2: set_set_nat] :
            ( ( member_set_set_nat @ X2 @ A )
            & ! [Xa: set_set_nat] :
                ( ( member_set_set_nat @ Xa @ A )
               => ( ( ord_le6893508408891458716et_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_605_finite__has__minimal,axiom,
    ! [A: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( A != bot_bot_set_nat_nat )
       => ? [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ A )
            & ! [Xa: nat > nat] :
                ( ( member_nat_nat @ Xa @ A )
               => ( ( ord_less_eq_nat_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_606_second__assumptions_Osqcup,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
       => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
         => ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X5 @ Y5 ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).

% second_assumptions.sqcup
thf(fact_607_first__assumptions_Ofinite___092_060F_062,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).

% first_assumptions.finite_\<F>
thf(fact_608_second__assumptions_Odeviate__pos__cup,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
       => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
         => ( ( clique3314026705536850673os_cup @ L @ P2 @ K @ X5 @ Y5 )
            = bot_bo7198184520161983622et_nat ) ) ) ) ).

% second_assumptions.deviate_pos_cup
thf(fact_609_first__assumptions_Ojoinl__join,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y5 ) @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) ) ) ).

% first_assumptions.joinl_join
thf(fact_610_second__assumptions_Osqcup__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X5 @ Y5 )
        = ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) ) ).

% second_assumptions.sqcup_def
thf(fact_611_approx__pos_Osimps_I3_J,axiom,
    ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
    = bot_bo7198184520161983622et_nat ) ).

% approx_pos.simps(3)
thf(fact_612_approx__pos_Osimps_I2_J,axiom,
    ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ monotone_TRUE_a )
    = bot_bo7198184520161983622et_nat ) ).

% approx_pos.simps(2)
thf(fact_613_APR_Osimps_I1_J,axiom,
    ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ monotone_FALSE_a )
    = bot_bo7198184520161983622et_nat ) ).

% APR.simps(1)
thf(fact_614_approx__pos_Osimps_I4_J,axiom,
    ! [V: a] :
      ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ V ) )
      = bot_bo7198184520161983622et_nat ) ).

% approx_pos.simps(4)
thf(fact_615_third__assumptions_OL0,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P2 @ K )
     => ( ord_less_eq_nat @ assumptions_and_L0 @ L ) ) ).

% third_assumptions.L0
thf(fact_616_third__assumptions_OL0_H,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P2 @ K )
     => ( ord_less_eq_nat @ assumptions_and_L02 @ L ) ) ).

% third_assumptions.L0'
thf(fact_617_Lp,axiom,
    ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).

% Lp
thf(fact_618_ACC__cf___092_060F_062,axiom,
    ! [X5: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique2971579238625216137irst_F @ k ) ) ).

% ACC_cf_\<F>
thf(fact_619_ACC__cf__empty,axiom,
    ( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
    = bot_bot_set_nat_nat ) ).

% ACC_cf_empty
thf(fact_620_ACC__cf__union,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) )
      = ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y5 ) ) ) ).

% ACC_cf_union
thf(fact_621_ACC__cf__mono,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
     => ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y5 ) ) ) ).

% ACC_cf_mono
thf(fact_622_approx__pos_Ocases,axiom,
    ! [X: monotone_mformula_a] :
      ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
          ( X
         != ( monotone_Conj_a @ Phi3 @ Psi3 ) )
     => ( ( X != monotone_TRUE_a )
       => ( ( X != monotone_FALSE_a )
         => ( ! [V2: a] :
                ( X
               != ( monotone_Var_a @ V2 ) )
           => ~ ! [V2: monotone_mformula_a,Va2: monotone_mformula_a] :
                  ( X
                 != ( monotone_Disj_a @ V2 @ Va2 ) ) ) ) ) ) ).

% approx_pos.cases
thf(fact_623_approx__neg_Ocases,axiom,
    ! [X: monotone_mformula_a] :
      ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
          ( X
         != ( monotone_Conj_a @ Phi3 @ Psi3 ) )
     => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
            ( X
           != ( monotone_Disj_a @ Phi3 @ Psi3 ) )
       => ( ( X != monotone_TRUE_a )
         => ( ( X != monotone_FALSE_a )
           => ~ ! [V2: a] :
                  ( X
                 != ( monotone_Var_a @ V2 ) ) ) ) ) ) ).

% approx_neg.cases
thf(fact_624_SET_Ocases,axiom,
    ! [X: monotone_mformula_a] :
      ( ( X != monotone_FALSE_a )
     => ( ! [X2: a] :
            ( X
           != ( monotone_Var_a @ X2 ) )
       => ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
              ( X
             != ( monotone_Disj_a @ Phi4 @ Psi4 ) )
         => ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
                ( X
               != ( monotone_Conj_a @ Phi4 @ Psi4 ) )
           => ( X = monotone_TRUE_a ) ) ) ) ) ).

% SET.cases
thf(fact_625_approx__neg_Osimps_I5_J,axiom,
    ! [V: a] :
      ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ V ) )
      = bot_bot_set_nat_nat ) ).

% approx_neg.simps(5)
thf(fact_626_approx__neg_Osimps_I4_J,axiom,
    ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ monotone_FALSE_a )
    = bot_bot_set_nat_nat ) ).

% approx_neg.simps(4)
thf(fact_627_approx__neg_Osimps_I3_J,axiom,
    ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ monotone_TRUE_a )
    = bot_bot_set_nat_nat ) ).

% approx_neg.simps(3)
thf(fact_628_finite__ACC,axiom,
    ! [X5: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) ) ).

% finite_ACC
thf(fact_629_approx__pos_Oelims,axiom,
    ! [X: monotone_mformula_a,Y: set_set_set_nat] :
      ( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X )
        = Y )
     => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
            ( ( X
              = ( monotone_Conj_a @ Phi3 @ Psi3 ) )
           => ( Y
             != ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
       => ( ( ( X = monotone_TRUE_a )
           => ( Y != bot_bo7198184520161983622et_nat ) )
         => ( ( ( X = monotone_FALSE_a )
             => ( Y != bot_bo7198184520161983622et_nat ) )
           => ( ( ? [V2: a] :
                    ( X
                    = ( monotone_Var_a @ V2 ) )
               => ( Y != bot_bo7198184520161983622et_nat ) )
             => ~ ( ? [V2: monotone_mformula_a,Va2: monotone_mformula_a] :
                      ( X
                      = ( monotone_Disj_a @ V2 @ Va2 ) )
                 => ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).

% approx_pos.elims
thf(fact_630_approx__neg_Oelims,axiom,
    ! [X: monotone_mformula_a,Y: set_nat_nat] :
      ( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X )
        = Y )
     => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
            ( ( X
              = ( monotone_Conj_a @ Phi3 @ Psi3 ) )
           => ( Y
             != ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
       => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
              ( ( X
                = ( monotone_Disj_a @ Phi3 @ Psi3 ) )
             => ( Y
               != ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
         => ( ( ( X = monotone_TRUE_a )
             => ( Y != bot_bot_set_nat_nat ) )
           => ( ( ( X = monotone_FALSE_a )
               => ( Y != bot_bot_set_nat_nat ) )
             => ~ ( ? [V2: a] :
                      ( X
                      = ( monotone_Var_a @ V2 ) )
                 => ( Y != bot_bot_set_nat_nat ) ) ) ) ) ) ) ).

% approx_neg.elims
thf(fact_631_no__deviation_I4_J,axiom,
    ! [X: a] :
      ( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
      = bot_bot_set_nat_nat ) ).

% no_deviation(4)
thf(fact_632_no__deviation_I2_J,axiom,
    ( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ monotone_FALSE_a )
    = bot_bot_set_nat_nat ) ).

% no_deviation(2)
thf(fact_633_no__deviation_I3_J,axiom,
    ! [X: a] :
      ( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
      = bot_bo7198184520161983622et_nat ) ).

% no_deviation(3)
thf(fact_634_no__deviation_I1_J,axiom,
    ( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
    = bot_bo7198184520161983622et_nat ) ).

% no_deviation(1)
thf(fact_635_psubsetD,axiom,
    ! [A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a,C2: monotone_mformula_a] :
      ( ( ord_le3562860513790772192mula_a @ A @ B )
     => ( ( member535913909593306477mula_a @ C2 @ A )
       => ( member535913909593306477mula_a @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_636_psubsetD,axiom,
    ! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le52856854838348540et_nat @ A @ B )
     => ( ( member2946998982187404937et_nat @ C2 @ A )
       => ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_637_psubsetD,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C2: nat > nat] :
      ( ( ord_less_set_nat_nat @ A @ B )
     => ( ( member_nat_nat @ C2 @ A )
       => ( member_nat_nat @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_638_psubsetD,axiom,
    ! [A: set_a,B: set_a,C2: a] :
      ( ( ord_less_set_a @ A @ B )
     => ( ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_639_psubsetD,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A @ B )
     => ( ( member_set_set_nat @ C2 @ A )
       => ( member_set_set_nat @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_640_psubset__trans,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A @ B )
     => ( ( ord_le152980574450754630et_nat @ B @ C )
       => ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).

% psubset_trans
thf(fact_641_first__assumptions_OACC__cf_Ocong,axiom,
    clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).

% first_assumptions.ACC_cf.cong
thf(fact_642_first__assumptions_OL_Ocong,axiom,
    assump1710595444109740301irst_L = assump1710595444109740301irst_L ).

% first_assumptions.L.cong
thf(fact_643_bot__set__def,axiom,
    ( bot_bo7198184520161983622et_nat
    = ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).

% bot_set_def
thf(fact_644_bot__set__def,axiom,
    ( bot_bot_set_nat_nat
    = ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).

% bot_set_def
thf(fact_645_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_646_first__assumptions_OACC__cf__empty,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
        = bot_bot_set_nat_nat ) ) ).

% first_assumptions.ACC_cf_empty
thf(fact_647_second__assumptions_OLp,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ord_less_nat @ P2 @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).

% second_assumptions.Lp
thf(fact_648_first__assumptions_Ofinite__ACC,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) ) ) ).

% first_assumptions.finite_ACC
thf(fact_649_first__assumptions_OACC__cf__union,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) )
        = ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y5 ) ) ) ) ).

% first_assumptions.ACC_cf_union
thf(fact_650_first__assumptions_OACC__cf__mono,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
       => ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y5 ) ) ) ) ).

% first_assumptions.ACC_cf_mono
thf(fact_651_first__assumptions_OACC__cf___092_060F_062,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).

% first_assumptions.ACC_cf_\<F>
thf(fact_652_first__assumptions_Opl,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_nat @ L @ P2 ) ) ).

% first_assumptions.pl
thf(fact_653_first__assumptions_Okp,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_nat @ P2 @ K ) ) ).

% first_assumptions.kp
thf(fact_654_first__assumptions_Ok,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_nat @ L @ K ) ) ).

% first_assumptions.k
thf(fact_655_second__assumptions_Oaxioms_I1_J,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ).

% second_assumptions.axioms(1)
thf(fact_656_third__assumptions_Oaxioms_I1_J,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P2 @ K )
     => ( assump2881078719466019805ptions @ L @ P2 @ K ) ) ).

% third_assumptions.axioms(1)
thf(fact_657_empty__CLIQUE,axiom,
    ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).

% empty_CLIQUE
thf(fact_658_deviate__neg__cap__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique1591571987438064265eg_cap @ l @ p @ k @ X5 @ Y5 )
      = ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X5 @ Y5 ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) ) ) ) ).

% deviate_neg_cap_def
thf(fact_659_deviate__neg__cup__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique1591571987439376245eg_cup @ l @ p @ k @ X5 @ Y5 )
      = ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X5 @ Y5 ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) ) ).

% deviate_neg_cup_def
thf(fact_660_tf__mformula_Osimps,axiom,
    ! [A2: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ A2 @ monoto4877036962378694605mula_a )
      = ( ( A2 = monotone_FALSE_a )
        | ? [X3: a] :
            ( A2
            = ( monotone_Var_a @ X3 ) )
        | ? [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
            ( ( A2
              = ( monotone_Disj_a @ Phi5 @ Psi5 ) )
            & ( member535913909593306477mula_a @ Phi5 @ monoto4877036962378694605mula_a )
            & ( member535913909593306477mula_a @ Psi5 @ monoto4877036962378694605mula_a ) )
        | ? [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
            ( ( A2
              = ( monotone_Conj_a @ Phi5 @ Psi5 ) )
            & ( member535913909593306477mula_a @ Phi5 @ monoto4877036962378694605mula_a )
            & ( member535913909593306477mula_a @ Psi5 @ monoto4877036962378694605mula_a ) ) ) ) ).

% tf_mformula.simps
thf(fact_661_tf__mformula_Ocases,axiom,
    ! [A2: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ A2 @ monoto4877036962378694605mula_a )
     => ( ( A2 != monotone_FALSE_a )
       => ( ! [X2: a] :
              ( A2
             != ( monotone_Var_a @ X2 ) )
         => ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
                ( ( A2
                  = ( monotone_Disj_a @ Phi4 @ Psi4 ) )
               => ( ( member535913909593306477mula_a @ Phi4 @ monoto4877036962378694605mula_a )
                 => ~ ( member535913909593306477mula_a @ Psi4 @ monoto4877036962378694605mula_a ) ) )
           => ~ ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
                  ( ( A2
                    = ( monotone_Conj_a @ Phi4 @ Psi4 ) )
                 => ( ( member535913909593306477mula_a @ Phi4 @ monoto4877036962378694605mula_a )
                   => ~ ( member535913909593306477mula_a @ Psi4 @ monoto4877036962378694605mula_a ) ) ) ) ) ) ) ).

% tf_mformula.cases
thf(fact_662_SUB_Ocases,axiom,
    ! [X: monotone_mformula_a] :
      ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
          ( X
         != ( monotone_Conj_a @ Phi4 @ Psi4 ) )
     => ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
            ( X
           != ( monotone_Disj_a @ Phi4 @ Psi4 ) )
       => ( ! [X2: a] :
              ( X
             != ( monotone_Var_a @ X2 ) )
         => ( ( X != monotone_FALSE_a )
           => ( X = monotone_TRUE_a ) ) ) ) ) ).

% SUB.cases
thf(fact_663_Diff__idemp,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ B )
      = ( minus_8121590178497047118at_nat @ A @ B ) ) ).

% Diff_idemp
thf(fact_664_Diff__idemp,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ B )
      = ( minus_2447799839930672331et_nat @ A @ B ) ) ).

% Diff_idemp
thf(fact_665_Diff__iff,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ ( minus_3028096444314564325mula_a @ A @ B ) )
      = ( ( member535913909593306477mula_a @ C2 @ A )
        & ~ ( member535913909593306477mula_a @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_666_Diff__iff,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ ( minus_3113942175840221057et_nat @ A @ B ) )
      = ( ( member2946998982187404937et_nat @ C2 @ A )
        & ~ ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_667_Diff__iff,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
      = ( ( member_a @ C2 @ A )
        & ~ ( member_a @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_668_Diff__iff,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( minus_8121590178497047118at_nat @ A @ B ) )
      = ( ( member_nat_nat @ C2 @ A )
        & ~ ( member_nat_nat @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_669_Diff__iff,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A @ B ) )
      = ( ( member_set_set_nat @ C2 @ A )
        & ~ ( member_set_set_nat @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_670_DiffI,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ A )
     => ( ~ ( member535913909593306477mula_a @ C2 @ B )
       => ( member535913909593306477mula_a @ C2 @ ( minus_3028096444314564325mula_a @ A @ B ) ) ) ) ).

% DiffI
thf(fact_671_DiffI,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ A )
     => ( ~ ( member2946998982187404937et_nat @ C2 @ B )
       => ( member2946998982187404937et_nat @ C2 @ ( minus_3113942175840221057et_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_672_DiffI,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ A )
     => ( ~ ( member_a @ C2 @ B )
       => ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ).

% DiffI
thf(fact_673_DiffI,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ A )
     => ( ~ ( member_nat_nat @ C2 @ B )
       => ( member_nat_nat @ C2 @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_674_DiffI,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ A )
     => ( ~ ( member_set_set_nat @ C2 @ B )
       => ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_675_mformula_Oinject_I2_J,axiom,
    ! [X41: monotone_mformula_a,X42: monotone_mformula_a,Y41: monotone_mformula_a,Y42: monotone_mformula_a] :
      ( ( ( monotone_Conj_a @ X41 @ X42 )
        = ( monotone_Conj_a @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% mformula.inject(2)
thf(fact_676_mformula_Oinject_I3_J,axiom,
    ! [X51: monotone_mformula_a,X52: monotone_mformula_a,Y51: monotone_mformula_a,Y52: monotone_mformula_a] :
      ( ( ( monotone_Disj_a @ X51 @ X52 )
        = ( monotone_Disj_a @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% mformula.inject(3)
thf(fact_677_mformula_Oinject_I1_J,axiom,
    ! [X32: a,Y32: a] :
      ( ( ( monotone_Var_a @ X32 )
        = ( monotone_Var_a @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% mformula.inject(1)
thf(fact_678_Diff__cancel,axiom,
    ! [A: set_set_nat] :
      ( ( minus_2163939370556025621et_nat @ A @ A )
      = bot_bot_set_set_nat ) ).

% Diff_cancel
thf(fact_679_Diff__cancel,axiom,
    ! [A: set_nat_nat] :
      ( ( minus_8121590178497047118at_nat @ A @ A )
      = bot_bot_set_nat_nat ) ).

% Diff_cancel
thf(fact_680_Diff__cancel,axiom,
    ! [A: set_set_set_nat] :
      ( ( minus_2447799839930672331et_nat @ A @ A )
      = bot_bo7198184520161983622et_nat ) ).

% Diff_cancel
thf(fact_681_empty__Diff,axiom,
    ! [A: set_set_nat] :
      ( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A )
      = bot_bot_set_set_nat ) ).

% empty_Diff
thf(fact_682_empty__Diff,axiom,
    ! [A: set_nat_nat] :
      ( ( minus_8121590178497047118at_nat @ bot_bot_set_nat_nat @ A )
      = bot_bot_set_nat_nat ) ).

% empty_Diff
thf(fact_683_empty__Diff,axiom,
    ! [A: set_set_set_nat] :
      ( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A )
      = bot_bo7198184520161983622et_nat ) ).

% empty_Diff
thf(fact_684_Diff__empty,axiom,
    ! [A: set_set_nat] :
      ( ( minus_2163939370556025621et_nat @ A @ bot_bot_set_set_nat )
      = A ) ).

% Diff_empty
thf(fact_685_Diff__empty,axiom,
    ! [A: set_nat_nat] :
      ( ( minus_8121590178497047118at_nat @ A @ bot_bot_set_nat_nat )
      = A ) ).

% Diff_empty
thf(fact_686_Diff__empty,axiom,
    ! [A: set_set_set_nat] :
      ( ( minus_2447799839930672331et_nat @ A @ bot_bo7198184520161983622et_nat )
      = A ) ).

% Diff_empty
thf(fact_687_finite__Diff,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).

% finite_Diff
thf(fact_688_finite__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).

% finite_Diff
thf(fact_689_finite__Diff,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).

% finite_Diff
thf(fact_690_finite__Diff,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A )
     => ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).

% finite_Diff
thf(fact_691_finite__Diff2,axiom,
    ! [B: set_set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B )
     => ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) )
        = ( finite1152437895449049373et_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_692_finite__Diff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_693_finite__Diff2,axiom,
    ! [B: set_nat_nat,A: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ B )
     => ( ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) )
        = ( finite2115694454571419734at_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_694_finite__Diff2,axiom,
    ! [B: set_set_set_nat,A: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ B )
     => ( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) )
        = ( finite6739761609112101331et_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_695_Un__Diff__cancel2,axiom,
    ! [B: set_set_nat,A: set_set_nat] :
      ( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B @ A ) @ A )
      = ( sup_sup_set_set_nat @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_696_Un__Diff__cancel2,axiom,
    ! [B: set_nat_nat,A: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ B @ A ) @ A )
      = ( sup_sup_set_nat_nat @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_697_Un__Diff__cancel2,axiom,
    ! [B: set_set_set_nat,A: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ B @ A ) @ A )
      = ( sup_su4213647025997063966et_nat @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_698_Un__Diff__cancel,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
      = ( sup_sup_set_set_nat @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_699_Un__Diff__cancel,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
      = ( sup_sup_set_nat_nat @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_700_Un__Diff__cancel,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( sup_su4213647025997063966et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ A ) )
      = ( sup_su4213647025997063966et_nat @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_701_Diff__eq__empty__iff,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ( minus_2447799839930672331et_nat @ A @ B )
        = bot_bo7198184520161983622et_nat )
      = ( ord_le9131159989063066194et_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_702_Diff__eq__empty__iff,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ( minus_8121590178497047118at_nat @ A @ B )
        = bot_bot_set_nat_nat )
      = ( ord_le9059583361652607317at_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_703_Diff__eq__empty__iff,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ( minus_2163939370556025621et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ord_le6893508408891458716et_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_704_deviate__neg__def,axiom,
    ! [Phi: monotone_mformula_a] :
      ( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi )
      = ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique8961599393750669800f_mf_a @ k @ pi @ Phi ) ) ) ).

% deviate_neg_def
thf(fact_705_first__assumptions_OCLIQUE_Ocong,axiom,
    clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).

% first_assumptions.CLIQUE.cong
thf(fact_706_DiffD2,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ ( minus_3028096444314564325mula_a @ A @ B ) )
     => ~ ( member535913909593306477mula_a @ C2 @ B ) ) ).

% DiffD2
thf(fact_707_DiffD2,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ ( minus_3113942175840221057et_nat @ A @ B ) )
     => ~ ( member2946998982187404937et_nat @ C2 @ B ) ) ).

% DiffD2
thf(fact_708_DiffD2,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
     => ~ ( member_a @ C2 @ B ) ) ).

% DiffD2
thf(fact_709_DiffD2,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( minus_8121590178497047118at_nat @ A @ B ) )
     => ~ ( member_nat_nat @ C2 @ B ) ) ).

% DiffD2
thf(fact_710_DiffD2,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A @ B ) )
     => ~ ( member_set_set_nat @ C2 @ B ) ) ).

% DiffD2
thf(fact_711_DiffD1,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ ( minus_3028096444314564325mula_a @ A @ B ) )
     => ( member535913909593306477mula_a @ C2 @ A ) ) ).

% DiffD1
thf(fact_712_DiffD1,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ ( minus_3113942175840221057et_nat @ A @ B ) )
     => ( member2946998982187404937et_nat @ C2 @ A ) ) ).

% DiffD1
thf(fact_713_DiffD1,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
     => ( member_a @ C2 @ A ) ) ).

% DiffD1
thf(fact_714_DiffD1,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( minus_8121590178497047118at_nat @ A @ B ) )
     => ( member_nat_nat @ C2 @ A ) ) ).

% DiffD1
thf(fact_715_DiffD1,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A @ B ) )
     => ( member_set_set_nat @ C2 @ A ) ) ).

% DiffD1
thf(fact_716_DiffE,axiom,
    ! [C2: monotone_mformula_a,A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( member535913909593306477mula_a @ C2 @ ( minus_3028096444314564325mula_a @ A @ B ) )
     => ~ ( ( member535913909593306477mula_a @ C2 @ A )
         => ( member535913909593306477mula_a @ C2 @ B ) ) ) ).

% DiffE
thf(fact_717_DiffE,axiom,
    ! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ C2 @ ( minus_3113942175840221057et_nat @ A @ B ) )
     => ~ ( ( member2946998982187404937et_nat @ C2 @ A )
         => ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).

% DiffE
thf(fact_718_DiffE,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
     => ~ ( ( member_a @ C2 @ A )
         => ( member_a @ C2 @ B ) ) ) ).

% DiffE
thf(fact_719_DiffE,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( minus_8121590178497047118at_nat @ A @ B ) )
     => ~ ( ( member_nat_nat @ C2 @ A )
         => ( member_nat_nat @ C2 @ B ) ) ) ).

% DiffE
thf(fact_720_DiffE,axiom,
    ! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
      ( ( member_set_set_nat @ C2 @ ( minus_2447799839930672331et_nat @ A @ B ) )
     => ~ ( ( member_set_set_nat @ C2 @ A )
         => ( member_set_set_nat @ C2 @ B ) ) ) ).

% DiffE
thf(fact_721_double__diff,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( ord_le9131159989063066194et_nat @ B @ C )
       => ( ( minus_2447799839930672331et_nat @ B @ ( minus_2447799839930672331et_nat @ C @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_722_double__diff,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( ord_le9059583361652607317at_nat @ B @ C )
       => ( ( minus_8121590178497047118at_nat @ B @ ( minus_8121590178497047118at_nat @ C @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_723_double__diff,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( ord_le6893508408891458716et_nat @ B @ C )
       => ( ( minus_2163939370556025621et_nat @ B @ ( minus_2163939370556025621et_nat @ C @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_724_Diff__subset,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_725_Diff__subset,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_726_Diff__subset,axiom,
    ! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_727_Diff__mono,axiom,
    ! [A: set_set_set_nat,C: set_set_set_nat,D2: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ C )
     => ( ( ord_le9131159989063066194et_nat @ D2 @ B )
       => ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ C @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_728_Diff__mono,axiom,
    ! [A: set_nat_nat,C: set_nat_nat,D2: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ C )
     => ( ( ord_le9059583361652607317at_nat @ D2 @ B )
       => ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ C @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_729_Diff__mono,axiom,
    ! [A: set_set_nat,C: set_set_nat,D2: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ C )
     => ( ( ord_le6893508408891458716et_nat @ D2 @ B )
       => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ C @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_730_Diff__infinite__finite,axiom,
    ! [T2: set_set_nat,S: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ T2 )
     => ( ~ ( finite1152437895449049373et_nat @ S )
       => ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_731_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_732_Diff__infinite__finite,axiom,
    ! [T2: set_nat_nat,S: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ T2 )
     => ( ~ ( finite2115694454571419734at_nat @ S )
       => ~ ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_733_Diff__infinite__finite,axiom,
    ! [T2: set_set_set_nat,S: set_set_set_nat] :
      ( ( finite6739761609112101331et_nat @ T2 )
     => ( ~ ( finite6739761609112101331et_nat @ S )
       => ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_734_Un__Diff,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( minus_2163939370556025621et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
      = ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ C ) @ ( minus_2163939370556025621et_nat @ B @ C ) ) ) ).

% Un_Diff
thf(fact_735_Un__Diff,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( minus_8121590178497047118at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
      = ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ C ) @ ( minus_8121590178497047118at_nat @ B @ C ) ) ) ).

% Un_Diff
thf(fact_736_Un__Diff,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( minus_2447799839930672331et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
      = ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A @ C ) @ ( minus_2447799839930672331et_nat @ B @ C ) ) ) ).

% Un_Diff
thf(fact_737_psubset__imp__ex__mem,axiom,
    ! [A: set_Mo2626137824023173004mula_a,B: set_Mo2626137824023173004mula_a] :
      ( ( ord_le3562860513790772192mula_a @ A @ B )
     => ? [B7: monotone_mformula_a] : ( member535913909593306477mula_a @ B7 @ ( minus_3028096444314564325mula_a @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_738_psubset__imp__ex__mem,axiom,
    ! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
      ( ( ord_le52856854838348540et_nat @ A @ B )
     => ? [B7: set_set_set_nat] : ( member2946998982187404937et_nat @ B7 @ ( minus_3113942175840221057et_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_739_psubset__imp__ex__mem,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_set_a @ A @ B )
     => ? [B7: a] : ( member_a @ B7 @ ( minus_minus_set_a @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_740_psubset__imp__ex__mem,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A @ B )
     => ? [B7: nat > nat] : ( member_nat_nat @ B7 @ ( minus_8121590178497047118at_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_741_psubset__imp__ex__mem,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A @ B )
     => ? [B7: set_set_nat] : ( member_set_set_nat @ B7 @ ( minus_2447799839930672331et_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_742_Diff__subset__conv,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ C )
      = ( ord_le9131159989063066194et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) ) ) ).

% Diff_subset_conv
thf(fact_743_Diff__subset__conv,axiom,
    ! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ C )
      = ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) ) ) ).

% Diff_subset_conv
thf(fact_744_Diff__subset__conv,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ C )
      = ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).

% Diff_subset_conv
thf(fact_745_Diff__partition,axiom,
    ! [A: set_set_set_nat,B: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A @ B )
     => ( ( sup_su4213647025997063966et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_746_Diff__partition,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A @ B )
     => ( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_747_Diff__partition,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_748_first__assumptions_Oempty__CLIQUE,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).

% first_assumptions.empty_CLIQUE
thf(fact_749_second__assumptions_Odeviate__neg__cup__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( clique1591571987439376245eg_cup @ L @ P2 @ K @ X5 @ Y5 )
        = ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X5 @ Y5 ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) ) ) ).

% second_assumptions.deviate_neg_cup_def
thf(fact_750_second__assumptions_Odeviate__neg__cap__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( clique1591571987438064265eg_cap @ L @ P2 @ K @ X5 @ Y5 )
        = ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X5 @ Y5 ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) ) ) ) ) ).

% second_assumptions.deviate_neg_cap_def
thf(fact_751_mformula_Odistinct_I19_J,axiom,
    ! [X41: monotone_mformula_a,X42: monotone_mformula_a,X51: monotone_mformula_a,X52: monotone_mformula_a] :
      ( ( monotone_Conj_a @ X41 @ X42 )
     != ( monotone_Disj_a @ X51 @ X52 ) ) ).

% mformula.distinct(19)
thf(fact_752_mformula_Odistinct_I15_J,axiom,
    ! [X32: a,X41: monotone_mformula_a,X42: monotone_mformula_a] :
      ( ( monotone_Var_a @ X32 )
     != ( monotone_Conj_a @ X41 @ X42 ) ) ).

% mformula.distinct(15)
thf(fact_753_mformula_Odistinct_I17_J,axiom,
    ! [X32: a,X51: monotone_mformula_a,X52: monotone_mformula_a] :
      ( ( monotone_Var_a @ X32 )
     != ( monotone_Disj_a @ X51 @ X52 ) ) ).

% mformula.distinct(17)
thf(fact_754_mformula_Odistinct_I11_J,axiom,
    ! [X41: monotone_mformula_a,X42: monotone_mformula_a] :
      ( monotone_FALSE_a
     != ( monotone_Conj_a @ X41 @ X42 ) ) ).

% mformula.distinct(11)
thf(fact_755_mformula_Odistinct_I13_J,axiom,
    ! [X51: monotone_mformula_a,X52: monotone_mformula_a] :
      ( monotone_FALSE_a
     != ( monotone_Disj_a @ X51 @ X52 ) ) ).

% mformula.distinct(13)
thf(fact_756_mformula_Odistinct_I9_J,axiom,
    ! [X32: a] :
      ( monotone_FALSE_a
     != ( monotone_Var_a @ X32 ) ) ).

% mformula.distinct(9)
thf(fact_757_mformula_Odistinct_I5_J,axiom,
    ! [X41: monotone_mformula_a,X42: monotone_mformula_a] :
      ( monotone_TRUE_a
     != ( monotone_Conj_a @ X41 @ X42 ) ) ).

% mformula.distinct(5)
thf(fact_758_mformula_Odistinct_I7_J,axiom,
    ! [X51: monotone_mformula_a,X52: monotone_mformula_a] :
      ( monotone_TRUE_a
     != ( monotone_Disj_a @ X51 @ X52 ) ) ).

% mformula.distinct(7)
thf(fact_759_mformula_Odistinct_I3_J,axiom,
    ! [X32: a] :
      ( monotone_TRUE_a
     != ( monotone_Var_a @ X32 ) ) ).

% mformula.distinct(3)
thf(fact_760_mformula_Odistinct_I1_J,axiom,
    monotone_TRUE_a != monotone_FALSE_a ).

% mformula.distinct(1)
thf(fact_761_tf__mformula_Otf__Conj,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
     => ( ( member535913909593306477mula_a @ Psi @ monoto4877036962378694605mula_a )
       => ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ monoto4877036962378694605mula_a ) ) ) ).

% tf_mformula.tf_Conj
thf(fact_762_tf__mformula_Otf__Disj,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
     => ( ( member535913909593306477mula_a @ Psi @ monoto4877036962378694605mula_a )
       => ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ monoto4877036962378694605mula_a ) ) ) ).

% tf_mformula.tf_Disj
thf(fact_763_tf__mformula_Otf__Var,axiom,
    ! [X: a] : ( member535913909593306477mula_a @ ( monotone_Var_a @ X ) @ monoto4877036962378694605mula_a ) ).

% tf_mformula.tf_Var
thf(fact_764_tf__mformula_Otf__False,axiom,
    member535913909593306477mula_a @ monotone_FALSE_a @ monoto4877036962378694605mula_a ).

% tf_mformula.tf_False
thf(fact_765_to__tf__formula_Ocases,axiom,
    ! [X: monotone_mformula_a] :
      ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
          ( X
         != ( monotone_Disj_a @ Phi3 @ Psi3 ) )
     => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
            ( X
           != ( monotone_Conj_a @ Phi3 @ Psi3 ) )
       => ( ( X != monotone_TRUE_a )
         => ( ( X != monotone_FALSE_a )
           => ~ ! [V2: a] :
                  ( X
                 != ( monotone_Var_a @ V2 ) ) ) ) ) ) ).

% to_tf_formula.cases
thf(fact_766_mformula_Oexhaust,axiom,
    ! [Y: monotone_mformula_a] :
      ( ( Y != monotone_TRUE_a )
     => ( ( Y != monotone_FALSE_a )
       => ( ! [X33: a] :
              ( Y
             != ( monotone_Var_a @ X33 ) )
         => ( ! [X412: monotone_mformula_a,X422: monotone_mformula_a] :
                ( Y
               != ( monotone_Conj_a @ X412 @ X422 ) )
           => ~ ! [X512: monotone_mformula_a,X522: monotone_mformula_a] :
                  ( Y
                 != ( monotone_Disj_a @ X512 @ X522 ) ) ) ) ) ) ).

% mformula.exhaust
thf(fact_767_vars_Ocases,axiom,
    ! [X: monotone_mformula_a] :
      ( ! [X2: a] :
          ( X
         != ( monotone_Var_a @ X2 ) )
     => ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
            ( X
           != ( monotone_Conj_a @ Phi4 @ Psi4 ) )
       => ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
              ( X
             != ( monotone_Disj_a @ Phi4 @ Psi4 ) )
         => ( ( X != monotone_FALSE_a )
           => ( X = monotone_TRUE_a ) ) ) ) ) ).

% vars.cases
thf(fact_768_POS__sub__CLIQUE,axiom,
    ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).

% POS_sub_CLIQUE
thf(fact_769_approx__neg_Opelims,axiom,
    ! [X: monotone_mformula_a,Y: set_nat_nat] :
      ( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X )
        = Y )
     => ( ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ X )
       => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
              ( ( X
                = ( monotone_Conj_a @ Phi3 @ Psi3 ) )
             => ( ( Y
                  = ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
               => ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
         => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
                ( ( X
                  = ( monotone_Disj_a @ Phi3 @ Psi3 ) )
               => ( ( Y
                    = ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
                 => ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
           => ( ( ( X = monotone_TRUE_a )
               => ( ( Y = bot_bot_set_nat_nat )
                 => ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_TRUE_a ) ) )
             => ( ( ( X = monotone_FALSE_a )
                 => ( ( Y = bot_bot_set_nat_nat )
                   => ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_FALSE_a ) ) )
               => ~ ! [V2: a] :
                      ( ( X
                        = ( monotone_Var_a @ V2 ) )
                     => ( ( Y = bot_bot_set_nat_nat )
                       => ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Var_a @ V2 ) ) ) ) ) ) ) ) ) ) ).

% approx_neg.pelims
thf(fact_770_Lm,axiom,
    ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).

% Lm
thf(fact_771_approx__pos_Opelims,axiom,
    ! [X: monotone_mformula_a,Y: set_set_set_nat] :
      ( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X )
        = Y )
     => ( ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ X )
       => ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
              ( ( X
                = ( monotone_Conj_a @ Phi3 @ Psi3 ) )
             => ( ( Y
                  = ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
               => ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
         => ( ( ( X = monotone_TRUE_a )
             => ( ( Y = bot_bo7198184520161983622et_nat )
               => ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_TRUE_a ) ) )
           => ( ( ( X = monotone_FALSE_a )
               => ( ( Y = bot_bo7198184520161983622et_nat )
                 => ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_FALSE_a ) ) )
             => ( ! [V2: a] :
                    ( ( X
                      = ( monotone_Var_a @ V2 ) )
                   => ( ( Y = bot_bo7198184520161983622et_nat )
                     => ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Var_a @ V2 ) ) ) )
               => ~ ! [V2: monotone_mformula_a,Va2: monotone_mformula_a] :
                      ( ( X
                        = ( monotone_Disj_a @ V2 @ Va2 ) )
                     => ( ( Y = bot_bo7198184520161983622et_nat )
                       => ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Disj_a @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).

% approx_pos.pelims
thf(fact_772_POS__CLIQUE,axiom,
    ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).

% POS_CLIQUE
thf(fact_773_ACC__cf__I,axiom,
    ! [F2: nat > nat,X5: set_set_set_nat] :
      ( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
     => ( ( clique3686358387679108662ccepts @ X5 @ ( clique5033774636164728462irst_C @ k @ F2 ) )
       => ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ k @ X5 ) ) ) ) ).

% ACC_cf_I
thf(fact_774_km,axiom,
    ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).

% km
thf(fact_775_order__refl,axiom,
    ! [X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ X ) ).

% order_refl
thf(fact_776_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_777_order__refl,axiom,
    ! [X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ X ) ).

% order_refl
thf(fact_778_order__refl,axiom,
    ! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).

% order_refl
thf(fact_779_order__refl,axiom,
    ! [X: nat > nat] : ( ord_less_eq_nat_nat @ X @ X ) ).

% order_refl
thf(fact_780_dual__order_Orefl,axiom,
    ! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_781_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_782_dual__order_Orefl,axiom,
    ! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_783_dual__order_Orefl,axiom,
    ! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_784_dual__order_Orefl,axiom,
    ! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_785_kml,axiom,
    ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).

% kml
thf(fact_786_local_Omp,axiom,
    ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).

% local.mp
thf(fact_787_M0,axiom,
    ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ k ) ).

% M0
thf(fact_788_M0_H,axiom,
    ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ k ) ).

% M0'
thf(fact_789_finite__POS__NEG,axiom,
    finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).

% finite_POS_NEG
thf(fact_790_first__assumptions_OC_Ocong,axiom,
    clique5033774636164728462irst_C = clique5033774636164728462irst_C ).

% first_assumptions.C.cong
thf(fact_791_first__assumptions_O_092_060K_062_Ocong,axiom,
    clique3326749438856946062irst_K = clique3326749438856946062irst_K ).

% first_assumptions.\<K>.cong
thf(fact_792_forth__assumptions_OACC__cf__mf_Ocong,axiom,
    clique8961599393750669800f_mf_a = clique8961599393750669800f_mf_a ).

% forth_assumptions.ACC_cf_mf.cong
thf(fact_793_first__assumptions_Om_Ocong,axiom,
    assump1710595444109740334irst_m = assump1710595444109740334irst_m ).

% first_assumptions.m.cong
thf(fact_794_first__assumptions_Okml,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).

% first_assumptions.kml
thf(fact_795_first__assumptions_Okm,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.km
thf(fact_796_first__assumptions_Omp,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_nat @ P2 @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.mp
thf(fact_797_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_798_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_799_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_set_set_nat,Z2: set_set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ X3 @ Y3 )
          & ( ord_le9131159989063066194et_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_800_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_801_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_nat_nat,Z2: set_nat_nat] : ( Y2 = Z2 ) )
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
          & ( ord_le9059583361652607317at_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_802_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_set_nat,Z2: set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ X3 @ Y3 )
          & ( ord_le6893508408891458716et_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_803_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat > nat,Z2: nat > nat] : ( Y2 = Z2 ) )
    = ( ^ [X3: nat > nat,Y3: nat > nat] :
          ( ( ord_less_eq_nat_nat @ X3 @ Y3 )
          & ( ord_less_eq_nat_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_804_ord__eq__le__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
       => ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_805_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_806_ord__eq__le__trans,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( A2 = B2 )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
       => ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_807_ord__eq__le__trans,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_808_ord__eq__le__trans,axiom,
    ! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat_nat @ B2 @ C2 )
       => ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_809_ord__le__eq__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_810_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_811_ord__le__eq__trans,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_812_ord__le__eq__trans,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_813_ord__le__eq__trans,axiom,
    ! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_814_order__antisym,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ Y )
     => ( ( ord_le9131159989063066194et_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_815_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_816_order__antisym,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ Y )
     => ( ( ord_le9059583361652607317at_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_817_order__antisym,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( ord_le6893508408891458716et_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_818_order__antisym,axiom,
    ! [X: nat > nat,Y: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ Y )
     => ( ( ord_less_eq_nat_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_819_order_Otrans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
       => ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_820_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_821_order_Otrans,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
       => ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_822_order_Otrans,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_823_order_Otrans,axiom,
    ! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat_nat @ B2 @ C2 )
       => ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_824_order__trans,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ Y )
     => ( ( ord_le9131159989063066194et_nat @ Y @ Z )
       => ( ord_le9131159989063066194et_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_825_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_826_order__trans,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ Y )
     => ( ( ord_le9059583361652607317at_nat @ Y @ Z )
       => ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_827_order__trans,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( ord_le6893508408891458716et_nat @ Y @ Z )
       => ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_828_order__trans,axiom,
    ! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ Y )
     => ( ( ord_less_eq_nat_nat @ Y @ Z )
       => ( ord_less_eq_nat_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_829_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
         => ( P @ A7 @ B7 ) )
     => ( ! [A7: nat,B7: nat] :
            ( ( P @ B7 @ A7 )
           => ( P @ A7 @ B7 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_830_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_set_set_nat,Z2: set_set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ B4 @ A4 )
          & ( ord_le9131159989063066194et_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_831_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_832_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_nat_nat,Z2: set_nat_nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
          & ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_833_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: set_set_nat,Z2: set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
          & ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_834_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat > nat,Z2: nat > nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: nat > nat,B4: nat > nat] :
          ( ( ord_less_eq_nat_nat @ B4 @ A4 )
          & ( ord_less_eq_nat_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_835_dual__order_Oantisym,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
     => ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_836_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_837_dual__order_Oantisym,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
     => ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_838_dual__order_Oantisym,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_839_dual__order_Oantisym,axiom,
    ! [B2: nat > nat,A2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_840_dual__order_Otrans,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
     => ( ( ord_le9131159989063066194et_nat @ C2 @ B2 )
       => ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_841_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_842_dual__order_Otrans,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
     => ( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
       => ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_843_dual__order_Otrans,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ C2 @ B2 )
       => ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_844_dual__order_Otrans,axiom,
    ! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat_nat @ C2 @ B2 )
       => ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_845_antisym,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_846_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_847_antisym,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_848_antisym,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_849_antisym,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_850_le__funD,axiom,
    ! [F: nat > nat,G3: nat > nat,X: nat] :
      ( ( ord_less_eq_nat_nat @ F @ G3 )
     => ( ord_less_eq_nat @ ( F @ X ) @ ( G3 @ X ) ) ) ).

% le_funD
thf(fact_851_le__funE,axiom,
    ! [F: nat > nat,G3: nat > nat,X: nat] :
      ( ( ord_less_eq_nat_nat @ F @ G3 )
     => ( ord_less_eq_nat @ ( F @ X ) @ ( G3 @ X ) ) ) ).

% le_funE
thf(fact_852_le__funI,axiom,
    ! [F: nat > nat,G3: nat > nat] :
      ( ! [X2: nat] : ( ord_less_eq_nat @ ( F @ X2 ) @ ( G3 @ X2 ) )
     => ( ord_less_eq_nat_nat @ F @ G3 ) ) ).

% le_funI
thf(fact_853_le__fun__def,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [F3: nat > nat,G4: nat > nat] :
        ! [X3: nat] : ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( G4 @ X3 ) ) ) ) ).

% le_fun_def
thf(fact_854_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_set_set_nat,Z2: set_set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
          & ( ord_le9131159989063066194et_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_855_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_856_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_nat_nat,Z2: set_nat_nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
          & ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_857_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: set_set_nat,Z2: set_set_nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
          & ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_858_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat > nat,Z2: nat > nat] : ( Y2 = Z2 ) )
    = ( ^ [A4: nat > nat,B4: nat > nat] :
          ( ( ord_less_eq_nat_nat @ A4 @ B4 )
          & ( ord_less_eq_nat_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_859_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_860_order__subst1,axiom,
    ! [A2: nat,F: set_set_nat > nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_861_order__subst1,axiom,
    ! [A2: nat,F: ( nat > nat ) > nat,B2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat_nat @ B2 @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_862_order__subst1,axiom,
    ! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C2: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_863_order__subst1,axiom,
    ! [A2: nat > nat,F: nat > nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_864_order__subst1,axiom,
    ! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C2: nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9131159989063066194et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9131159989063066194et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_865_order__subst1,axiom,
    ! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le9131159989063066194et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_866_order__subst1,axiom,
    ! [A2: nat,F: set_nat_nat > nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
       => ( ! [X2: set_nat_nat,Y4: set_nat_nat] :
              ( ( ord_le9059583361652607317at_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_867_order__subst1,axiom,
    ! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C2: nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9059583361652607317at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_868_order__subst1,axiom,
    ! [A2: set_set_nat,F: set_set_nat > set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_869_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_870_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_set_nat,C2: set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_871_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_872_order__subst2,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C2: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_873_order__subst2,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat,C2: nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_874_order__subst2,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C2: nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le9131159989063066194et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_875_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le9131159989063066194et_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9131159989063066194et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9131159989063066194et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_876_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_nat_nat,C2: set_nat_nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le9059583361652607317at_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9059583361652607317at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_877_order__subst2,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C2: nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_nat_nat,Y4: set_nat_nat] :
              ( ( ord_le9059583361652607317at_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_878_order__subst2,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_879_order__eq__refl,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( X = Y )
     => ( ord_le9131159989063066194et_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_880_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_881_order__eq__refl,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( X = Y )
     => ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_882_order__eq__refl,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( X = Y )
     => ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_883_order__eq__refl,axiom,
    ! [X: nat > nat,Y: nat > nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_884_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_885_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_886_ord__eq__le__subst,axiom,
    ! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_887_ord__eq__le__subst,axiom,
    ! [A2: nat > nat,F: nat > nat > nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_888_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_set_nat > nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_889_ord__eq__le__subst,axiom,
    ! [A2: nat,F: ( nat > nat ) > nat,B2: nat > nat,C2: nat > nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat_nat @ B2 @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_890_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le9131159989063066194et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_891_ord__eq__le__subst,axiom,
    ! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9131159989063066194et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9131159989063066194et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_892_ord__eq__le__subst,axiom,
    ! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9059583361652607317at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_893_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_nat_nat > nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
       => ( ! [X2: set_nat_nat,Y4: set_nat_nat] :
              ( ( ord_le9059583361652607317at_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_894_ord__eq__le__subst,axiom,
    ! [A2: set_set_nat,F: set_set_nat > set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_895_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_896_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_set_nat,C2: set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_897_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_898_ord__le__eq__subst,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C2: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_899_ord__le__eq__subst,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat,C2: nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_900_ord__le__eq__subst,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C2: nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le9131159989063066194et_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_901_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9131159989063066194et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9131159989063066194et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_902_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_nat_nat,C2: set_nat_nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le9059583361652607317at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_903_ord__le__eq__subst,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C2: nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_nat_nat,Y4: set_nat_nat] :
              ( ( ord_le9059583361652607317at_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_904_ord__le__eq__subst,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_nat,Y4: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_905_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_906_order__antisym__conv,axiom,
    ! [Y: set_set_set_nat,X: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ Y @ X )
     => ( ( ord_le9131159989063066194et_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_907_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_908_order__antisym__conv,axiom,
    ! [Y: set_nat_nat,X: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ Y @ X )
     => ( ( ord_le9059583361652607317at_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_909_order__antisym__conv,axiom,
    ! [Y: set_set_nat,X: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ Y @ X )
     => ( ( ord_le6893508408891458716et_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_910_order__antisym__conv,axiom,
    ! [Y: nat > nat,X: nat > nat] :
      ( ( ord_less_eq_nat_nat @ Y @ X )
     => ( ( ord_less_eq_nat_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_911_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_912_order__less__imp__not__less,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ~ ( ord_le152980574450754630et_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_913_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_914_order__less__imp__not__eq2,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_915_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_916_order__less__imp__not__eq,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_917_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_918_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_919_order__less__imp__triv,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,P: $o] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( ( ord_le152980574450754630et_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_920_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_921_order__less__not__sym,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ~ ( ord_le152980574450754630et_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_922_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_923_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le152980574450754630et_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_924_order__less__subst2,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C2: nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_925_order__less__subst2,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( ord_le152980574450754630et_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_926_order__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_927_order__less__subst1,axiom,
    ! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_928_order__less__subst1,axiom,
    ! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C2: nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_929_order__less__subst1,axiom,
    ! [A2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_930_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_931_order__less__irrefl,axiom,
    ! [X: set_set_set_nat] :
      ~ ( ord_le152980574450754630et_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_932_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_933_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_934_ord__less__eq__subst,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C2: nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_935_ord__less__eq__subst,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_936_ord__eq__less__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_937_ord__eq__less__subst,axiom,
    ! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_938_ord__eq__less__subst,axiom,
    ! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_939_ord__eq__less__subst,axiom,
    ! [A2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_940_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_941_order__less__trans,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( ( ord_le152980574450754630et_nat @ Y @ Z )
       => ( ord_le152980574450754630et_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_942_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_943_order__less__asym_H,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ~ ( ord_le152980574450754630et_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_944_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_945_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_946_order__less__asym,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ~ ( ord_le152980574450754630et_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_947_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_948_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_949_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_950_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_951_order_Ostrict__implies__not__eq,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_952_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_953_dual__order_Ostrict__trans,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
     => ( ( ord_le152980574450754630et_nat @ C2 @ B2 )
       => ( ord_le152980574450754630et_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_954_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_955_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_956_order_Ostrict__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_957_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A7: nat,B7: nat] :
          ( ( ord_less_nat @ A7 @ B7 )
         => ( P @ A7 @ B7 ) )
     => ( ! [A7: nat] : ( P @ A7 @ A7 )
       => ( ! [A7: nat,B7: nat] :
              ( ( P @ B7 @ A7 )
             => ( P @ A7 @ B7 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_958_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [N: nat] :
          ( ( P4 @ N )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N )
             => ~ ( P4 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_959_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_960_dual__order_Oirrefl,axiom,
    ! [A2: set_set_set_nat] :
      ~ ( ord_le152980574450754630et_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_961_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_962_dual__order_Oasym,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
     => ~ ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_963_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_964_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_965_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X2: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X2 )
             => ( P @ Y6 ) )
         => ( P @ X2 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_966_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_967_ord__less__eq__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_968_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_969_ord__eq__less__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_970_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_971_order_Oasym,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ~ ( ord_le152980574450754630et_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_972_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_973_less__imp__neq,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_974_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_975_second__assumptions_OLm,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K ) @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).

% second_assumptions.Lm
thf(fact_976_first__assumptions_OPOS__CLIQUE,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).

% first_assumptions.POS_CLIQUE
thf(fact_977_first__assumptions_OPOS__sub__CLIQUE,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).

% first_assumptions.POS_sub_CLIQUE
thf(fact_978_leD,axiom,
    ! [Y: set_set_set_nat,X: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ Y @ X )
     => ~ ( ord_le152980574450754630et_nat @ X @ Y ) ) ).

% leD
thf(fact_979_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_980_leD,axiom,
    ! [Y: set_nat_nat,X: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ Y @ X )
     => ~ ( ord_less_set_nat_nat @ X @ Y ) ) ).

% leD
thf(fact_981_leD,axiom,
    ! [Y: set_set_nat,X: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ Y @ X )
     => ~ ( ord_less_set_set_nat @ X @ Y ) ) ).

% leD
thf(fact_982_leD,axiom,
    ! [Y: nat > nat,X: nat > nat] :
      ( ( ord_less_eq_nat_nat @ Y @ X )
     => ~ ( ord_less_nat_nat @ X @ Y ) ) ).

% leD
thf(fact_983_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_984_nless__le,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ~ ( ord_le152980574450754630et_nat @ A2 @ B2 ) )
      = ( ~ ( ord_le9131159989063066194et_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_985_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_986_nless__le,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ~ ( ord_less_set_nat_nat @ A2 @ B2 ) )
      = ( ~ ( ord_le9059583361652607317at_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_987_nless__le,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ~ ( ord_less_set_set_nat @ A2 @ B2 ) )
      = ( ~ ( ord_le6893508408891458716et_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_988_nless__le,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( ~ ( ord_less_nat_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_989_antisym__conv1,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ~ ( ord_le152980574450754630et_nat @ X @ Y )
     => ( ( ord_le9131159989063066194et_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_990_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_991_antisym__conv1,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ~ ( ord_less_set_nat_nat @ X @ Y )
     => ( ( ord_le9059583361652607317at_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_992_antisym__conv1,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ~ ( ord_less_set_set_nat @ X @ Y )
     => ( ( ord_le6893508408891458716et_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_993_antisym__conv1,axiom,
    ! [X: nat > nat,Y: nat > nat] :
      ( ~ ( ord_less_nat_nat @ X @ Y )
     => ( ( ord_less_eq_nat_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_994_antisym__conv2,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ Y )
     => ( ( ~ ( ord_le152980574450754630et_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_995_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_996_antisym__conv2,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_997_antisym__conv2,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_998_antisym__conv2,axiom,
    ! [X: nat > nat,Y: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_999_less__le__not__le,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ X3 @ Y3 )
          & ~ ( ord_le9131159989063066194et_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1000_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1001_less__le__not__le,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
          & ~ ( ord_le9059583361652607317at_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1002_less__le__not__le,axiom,
    ( ord_less_set_set_nat
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ X3 @ Y3 )
          & ~ ( ord_le6893508408891458716et_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1003_less__le__not__le,axiom,
    ( ord_less_nat_nat
    = ( ^ [X3: nat > nat,Y3: nat > nat] :
          ( ( ord_less_eq_nat_nat @ X3 @ Y3 )
          & ~ ( ord_less_eq_nat_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1004_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_1005_order_Oorder__iff__strict,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
          ( ( ord_le152980574450754630et_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1006_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1007_order_Oorder__iff__strict,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
          ( ( ord_less_set_nat_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1008_order_Oorder__iff__strict,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_less_set_set_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1009_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [A4: nat > nat,B4: nat > nat] :
          ( ( ord_less_nat_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1010_order_Ostrict__iff__order,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1011_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1012_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1013_order_Ostrict__iff__order,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1014_order_Ostrict__iff__order,axiom,
    ( ord_less_nat_nat
    = ( ^ [A4: nat > nat,B4: nat > nat] :
          ( ( ord_less_eq_nat_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1015_order_Ostrict__trans1,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1016_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1017_order_Ostrict__trans1,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat_nat @ B2 @ C2 )
       => ( ord_less_set_nat_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1018_order_Ostrict__trans1,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( ord_less_set_set_nat @ B2 @ C2 )
       => ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1019_order_Ostrict__trans1,axiom,
    ! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_nat_nat @ B2 @ C2 )
       => ( ord_less_nat_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1020_order_Ostrict__trans2,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
       => ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1021_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1022_order_Ostrict__trans2,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A2 @ B2 )
     => ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
       => ( ord_less_set_nat_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1023_order_Ostrict__trans2,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
       => ( ord_less_set_set_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1024_order_Ostrict__trans2,axiom,
    ! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
      ( ( ord_less_nat_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat_nat @ B2 @ C2 )
       => ( ord_less_nat_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1025_order_Ostrict__iff__not,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
          & ~ ( ord_le9131159989063066194et_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1026_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1027_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
          & ~ ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1028_order_Ostrict__iff__not,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
          & ~ ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1029_order_Ostrict__iff__not,axiom,
    ( ord_less_nat_nat
    = ( ^ [A4: nat > nat,B4: nat > nat] :
          ( ( ord_less_eq_nat_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1030_dual__order_Oorder__iff__strict,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
          ( ( ord_le152980574450754630et_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1031_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1032_dual__order_Oorder__iff__strict,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
          ( ( ord_less_set_nat_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1033_dual__order_Oorder__iff__strict,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [B4: set_set_nat,A4: set_set_nat] :
          ( ( ord_less_set_set_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1034_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [B4: nat > nat,A4: nat > nat] :
          ( ( ord_less_nat_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1035_dual__order_Ostrict__iff__order,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1036_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1037_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1038_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_set_nat
    = ( ^ [B4: set_set_nat,A4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1039_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat_nat
    = ( ^ [B4: nat > nat,A4: nat > nat] :
          ( ( ord_less_eq_nat_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1040_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
     => ( ( ord_le152980574450754630et_nat @ C2 @ B2 )
       => ( ord_le152980574450754630et_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1041_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1042_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
     => ( ( ord_less_set_nat_nat @ C2 @ B2 )
       => ( ord_less_set_nat_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1043_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ( ord_less_set_set_nat @ C2 @ B2 )
       => ( ord_less_set_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1044_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ B2 @ A2 )
     => ( ( ord_less_nat_nat @ C2 @ B2 )
       => ( ord_less_nat_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1045_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
     => ( ( ord_le9131159989063066194et_nat @ C2 @ B2 )
       => ( ord_le152980574450754630et_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1046_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1047_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ B2 @ A2 )
     => ( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
       => ( ord_less_set_nat_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1048_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
      ( ( ord_less_set_set_nat @ B2 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ C2 @ B2 )
       => ( ord_less_set_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1049_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
      ( ( ord_less_nat_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat_nat @ C2 @ B2 )
       => ( ord_less_nat_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1050_dual__order_Ostrict__iff__not,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ B4 @ A4 )
          & ~ ( ord_le9131159989063066194et_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1051_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1052_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
          & ~ ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1053_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_set_nat
    = ( ^ [B4: set_set_nat,A4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
          & ~ ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1054_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat_nat
    = ( ^ [B4: nat > nat,A4: nat > nat] :
          ( ( ord_less_eq_nat_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1055_order_Ostrict__implies__order,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
     => ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1056_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1057_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ A2 @ B2 )
     => ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1058_order_Ostrict__implies__order,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1059_order_Ostrict__implies__order,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( ord_less_nat_nat @ A2 @ B2 )
     => ( ord_less_eq_nat_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1060_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_set_set_nat,A2: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
     => ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1061_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1062_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat_nat,A2: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ B2 @ A2 )
     => ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1063_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( ord_less_set_set_nat @ B2 @ A2 )
     => ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1064_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat > nat,A2: nat > nat] :
      ( ( ord_less_nat_nat @ B2 @ A2 )
     => ( ord_less_eq_nat_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1065_order__le__less,axiom,
    ( ord_le9131159989063066194et_nat
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
          ( ( ord_le152980574450754630et_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1066_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1067_order__le__less,axiom,
    ( ord_le9059583361652607317at_nat
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
          ( ( ord_less_set_nat_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1068_order__le__less,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] :
          ( ( ord_less_set_set_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1069_order__le__less,axiom,
    ( ord_less_eq_nat_nat
    = ( ^ [X3: nat > nat,Y3: nat > nat] :
          ( ( ord_less_nat_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1070_order__less__le,axiom,
    ( ord_le152980574450754630et_nat
    = ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
          ( ( ord_le9131159989063066194et_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1071_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1072_order__less__le,axiom,
    ( ord_less_set_nat_nat
    = ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
          ( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1073_order__less__le,axiom,
    ( ord_less_set_set_nat
    = ( ^ [X3: set_set_nat,Y3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1074_order__less__le,axiom,
    ( ord_less_nat_nat
    = ( ^ [X3: nat > nat,Y3: nat > nat] :
          ( ( ord_less_eq_nat_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1075_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1076_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1077_order__less__imp__le,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( ord_le9131159989063066194et_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1078_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_1079_order__less__imp__le,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ X @ Y )
     => ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1080_order__less__imp__le,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( ord_less_set_set_nat @ X @ Y )
     => ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1081_order__less__imp__le,axiom,
    ! [X: nat > nat,Y: nat > nat] :
      ( ( ord_less_nat_nat @ X @ Y )
     => ( ord_less_eq_nat_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1082_order__le__neq__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1083_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1084_order__le__neq__trans,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1085_order__le__neq__trans,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1086_order__le__neq__trans,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1087_order__neq__le__trans,axiom,
    ! [A2: set_set_set_nat,B2: set_set_set_nat] :
      ( ( A2 != B2 )
     => ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
       => ( ord_le152980574450754630et_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1088_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1089_order__neq__le__trans,axiom,
    ! [A2: set_nat_nat,B2: set_nat_nat] :
      ( ( A2 != B2 )
     => ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
       => ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1090_order__neq__le__trans,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( A2 != B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ord_less_set_set_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1091_order__neq__le__trans,axiom,
    ! [A2: nat > nat,B2: nat > nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat_nat @ A2 @ B2 )
       => ( ord_less_nat_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1092_order__le__less__trans,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X @ Y )
     => ( ( ord_le152980574450754630et_nat @ Y @ Z )
       => ( ord_le152980574450754630et_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1093_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_1094_order__le__less__trans,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ X @ Y )
     => ( ( ord_less_set_nat_nat @ Y @ Z )
       => ( ord_less_set_nat_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1095_order__le__less__trans,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( ord_less_set_set_nat @ Y @ Z )
       => ( ord_less_set_set_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1096_order__le__less__trans,axiom,
    ! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
      ( ( ord_less_eq_nat_nat @ X @ Y )
     => ( ( ord_less_nat_nat @ Y @ Z )
       => ( ord_less_nat_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1097_order__less__le__trans,axiom,
    ! [X: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
      ( ( ord_le152980574450754630et_nat @ X @ Y )
     => ( ( ord_le9131159989063066194et_nat @ Y @ Z )
       => ( ord_le152980574450754630et_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1098_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_1099_order__less__le__trans,axiom,
    ! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
      ( ( ord_less_set_nat_nat @ X @ Y )
     => ( ( ord_le9059583361652607317at_nat @ Y @ Z )
       => ( ord_less_set_nat_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1100_order__less__le__trans,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
      ( ( ord_less_set_set_nat @ X @ Y )
     => ( ( ord_le6893508408891458716et_nat @ Y @ Z )
       => ( ord_less_set_set_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1101_order__less__le__trans,axiom,
    ! [X: nat > nat,Y: nat > nat,Z: nat > nat] :
      ( ( ord_less_nat_nat @ X @ Y )
     => ( ( ord_less_eq_nat_nat @ Y @ Z )
       => ( ord_less_nat_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1102_order__le__less__subst1,axiom,
    ! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C2: nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1103_order__le__less__subst1,axiom,
    ! [A2: set_set_set_nat,F: set_set_set_nat > set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_le152980574450754630et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1104_order__le__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1105_order__le__less__subst1,axiom,
    ! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1106_order__le__less__subst1,axiom,
    ! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C2: nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_set_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1107_order__le__less__subst1,axiom,
    ! [A2: set_nat_nat,F: set_set_set_nat > set_nat_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_set_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1108_order__le__less__subst1,axiom,
    ! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C2: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_set_set_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1109_order__le__less__subst1,axiom,
    ! [A2: set_set_nat,F: set_set_set_nat > set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_set_set_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1110_order__le__less__subst1,axiom,
    ! [A2: nat > nat,F: nat > nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1111_order__le__less__subst1,axiom,
    ! [A2: nat > nat,F: set_set_set_nat > nat > nat,B2: set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le152980574450754630et_nat @ B2 @ C2 )
       => ( ! [X2: set_set_set_nat,Y4: set_set_set_nat] :
              ( ( ord_le152980574450754630et_nat @ X2 @ Y4 )
             => ( ord_less_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1112_order__le__less__subst2,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > set_set_set_nat,C2: set_set_set_nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_le152980574450754630et_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_le9131159989063066194et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le152980574450754630et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1113_order__le__less__subst2,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat,C2: nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1114_order__le__less__subst2,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > set_nat_nat,C2: set_nat_nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_le9059583361652607317at_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_nat_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1115_order__le__less__subst2,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > set_set_nat,C2: set_set_nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_set_set_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1116_order__le__less__subst2,axiom,
    ! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat > nat,C2: nat > nat] :
      ( ( ord_less_eq_nat_nat @ A2 @ B2 )
     => ( ( ord_less_nat_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat > nat,Y4: nat > nat] :
              ( ( ord_less_eq_nat_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1117_first__assumptions_OACC__cf__I,axiom,
    ! [L: nat,P2: nat,K: nat,F2: nat > nat,X5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ K ) )
       => ( ( clique3686358387679108662ccepts @ X5 @ ( clique5033774636164728462irst_C @ K @ F2 ) )
         => ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ K @ X5 ) ) ) ) ) ).

% first_assumptions.ACC_cf_I
thf(fact_1118_ACC__cf__mf__def,axiom,
    ! [Phi: monotone_mformula_a] :
      ( ( clique8961599393750669800f_mf_a @ k @ pi @ Phi )
      = ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ).

% ACC_cf_mf_def
thf(fact_1119_SET_Osimps_I1_J,axiom,
    ( ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a )
    = bot_bo7198184520161983622et_nat ) ).

% SET.simps(1)
thf(fact_1120_SET_Osimps_I3_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) )
      = ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ).

% SET.simps(3)
thf(fact_1121_SET_Osimps_I4_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) )
      = ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ).

% SET.simps(4)
thf(fact_1122_ACC__cf__SET_I2_J,axiom,
    ( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
    = bot_bot_set_nat_nat ) ).

% ACC_cf_SET(2)
thf(fact_1123_ACC__cf__SET_I3_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
      = ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).

% ACC_cf_SET(3)
thf(fact_1124_forth__assumptions__axioms,axiom,
    clique8563529963003110213ions_a @ l @ p @ k @ v @ pi ).

% forth_assumptions_axioms
thf(fact_1125_first__assumptions_ONEG_Ocong,axiom,
    clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).

% first_assumptions.NEG.cong
thf(fact_1126_third__assumptions_OM0_H,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P2 @ K )
     => ( ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% third_assumptions.M0'
thf(fact_1127_third__assumptions_OM0,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P2 @ K )
     => ( ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% third_assumptions.M0
thf(fact_1128_first__assumptions_Ofinite__POS__NEG,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).

% first_assumptions.finite_POS_NEG
thf(fact_1129_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1130_CLIQUE__NEG,axiom,
    ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
    = bot_bo7198184520161983622et_nat ) ).

% CLIQUE_NEG
thf(fact_1131_local_ONEG__def,axiom,
    ( ( clique3210737375870294875st_NEG @ k )
    = ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).

% local.NEG_def
thf(fact_1132_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1133_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M2 @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_1134_phi_I2_J,axiom,
    member535913909593306477mula_a @ phi @ ( clique5987991184601036204th_A_a @ v ) ).

% phi(2)
thf(fact_1135_tf__Conj_Oprems,axiom,
    member535913909593306477mula_a @ ( monotone_Conj_a @ phi2 @ psi ) @ ( clique5987991184601036204th_A_a @ v ) ).

% tf_Conj.prems
thf(fact_1136_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_1137_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_1138_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_1139_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_1140_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_1141_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ? [X2: nat] :
            ( ( P @ X2 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1142_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_1143_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_1144_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_1145_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_1146_less__not__refl3,axiom,
    ! [S2: nat,T3: nat] :
      ( ( ord_less_nat @ S2 @ T3 )
     => ( S2 != T3 ) ) ).

% less_not_refl3
thf(fact_1147_less__not__refl2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( M2 != N2 ) ) ).

% less_not_refl2
thf(fact_1148_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_1149_nat__neq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 != N2 )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( ord_less_nat @ N2 @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_1150_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1151_first__assumptions_ONEG__def,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique3210737375870294875st_NEG @ K )
        = ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).

% first_assumptions.NEG_def
thf(fact_1152_first__assumptions_OCLIQUE__NEG,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
        = bot_bo7198184520161983622et_nat ) ) ).

% first_assumptions.CLIQUE_NEG
thf(fact_1153_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N: nat] :
          ( ( ord_less_eq_nat @ M @ N )
          & ( M != N ) ) ) ) ).

% nat_less_le
thf(fact_1154_less__imp__le__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_1155_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N: nat] :
          ( ( ord_less_nat @ M @ N )
          | ( M = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1156_less__or__eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_1157_le__neq__implies__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( M2 != N2 )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_1158_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1159_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M2 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_1160_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_1161_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1162_diff__le__mono,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_1163_diff__le__self,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).

% diff_le_self
thf(fact_1164_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_1165_diff__le__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1166_diff__less__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1167_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1168_APR,axiom,
    ! [Phi: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
     => ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
       => ( member2946998982187404937et_nat @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).

% APR
thf(fact_1169_sqcup__sub,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
     => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
       => ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X5 @ Y5 ) ) ) ) ) ).

% sqcup_sub
thf(fact_1170_deviate__pos__cup__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique3314026705536850673os_cup @ l @ p @ k @ X5 @ Y5 )
      = ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X5 @ Y5 ) ) ) ) ).

% deviate_pos_cup_def
thf(fact_1171_deviate__pos__cap__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique3314026705535538693os_cap @ l @ p @ k @ X5 @ Y5 )
      = ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X5 @ Y5 ) ) ) ) ).

% deviate_pos_cap_def
thf(fact_1172_ACC__union,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) )
      = ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X5 ) @ ( clique3210737319928189260st_ACC @ k @ Y5 ) ) ) ).

% ACC_union
thf(fact_1173_ACC__cf__odot,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) )
      = ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y5 ) ) ) ).

% ACC_cf_odot
thf(fact_1174__092_060A_062__simps_I1_J,axiom,
    member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ v ) ).

% \<A>_simps(1)
thf(fact_1175_ACC__odot,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) )
      = ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X5 ) @ ( clique3210737319928189260st_ACC @ k @ Y5 ) ) ) ).

% ACC_odot
thf(fact_1176__C_K_C_I1_J,axiom,
    member535913909593306477mula_a @ phi2 @ ( clique5987991184601036204th_A_a @ v ) ).

% "*"(1)
thf(fact_1177__C_K_C_I2_J,axiom,
    member535913909593306477mula_a @ psi @ ( clique5987991184601036204th_A_a @ v ) ).

% "*"(2)
thf(fact_1178_ACC__empty,axiom,
    ( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
    = bot_bo7198184520161983622et_nat ) ).

% ACC_empty
thf(fact_1179__092_060A_062__simps_I3_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ v ) )
      = ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
        & ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ v ) ) ) ) ).

% \<A>_simps(3)
thf(fact_1180__092_060A_062__simps_I4_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ v ) )
      = ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
        & ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ v ) ) ) ) ).

% \<A>_simps(4)
thf(fact_1181__092_060A_062__simps_I2_J,axiom,
    ! [X: a] :
      ( ( member535913909593306477mula_a @ ( monotone_Var_a @ X ) @ ( clique5987991184601036204th_A_a @ v ) )
      = ( member_a @ X @ v ) ) ).

% \<A>_simps(2)
thf(fact_1182_ACC__cf__SET_I4_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) )
      = ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).

% ACC_cf_SET(4)
thf(fact_1183_ACC__SET_I2_J,axiom,
    ( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
    = bot_bo7198184520161983622et_nat ) ).

% ACC_SET(2)
thf(fact_1184_ACC__mf__def,axiom,
    ! [Phi: monotone_mformula_a] :
      ( ( clique4708818501384062891C_mf_a @ k @ pi @ Phi )
      = ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ).

% ACC_mf_def
thf(fact_1185_ACC__SET_I3_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
      = ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).

% ACC_SET(3)
thf(fact_1186_ACC__SET_I4_J,axiom,
    ! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
      ( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) )
      = ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).

% ACC_SET(4)
thf(fact_1187_deviate__pos__def,axiom,
    ! [Phi: monotone_mformula_a] :
      ( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi )
      = ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique4708818501384062891C_mf_a @ k @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) ) ) ) ).

% deviate_pos_def
thf(fact_1188_first__assumptions_OACC_Ocong,axiom,
    clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).

% first_assumptions.ACC.cong
thf(fact_1189_first__assumptions_OACC__empty,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
        = bot_bo7198184520161983622et_nat ) ) ).

% first_assumptions.ACC_empty
thf(fact_1190_first__assumptions_OACC__union,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) )
        = ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X5 ) @ ( clique3210737319928189260st_ACC @ K @ Y5 ) ) ) ) ).

% first_assumptions.ACC_union
thf(fact_1191_first__assumptions_OACC__cf__odot,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) )
        = ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y5 ) ) ) ) ).

% first_assumptions.ACC_cf_odot
thf(fact_1192_first__assumptions_OACC__odot,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) )
        = ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X5 ) @ ( clique3210737319928189260st_ACC @ K @ Y5 ) ) ) ) ).

% first_assumptions.ACC_odot
thf(fact_1193_second__assumptions_Odeviate__pos__cap__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( clique3314026705535538693os_cap @ L @ P2 @ K @ X5 @ Y5 )
        = ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X5 @ Y5 ) ) ) ) ) ).

% second_assumptions.deviate_pos_cap_def
thf(fact_1194_second__assumptions_Odeviate__pos__cup__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( clique3314026705536850673os_cup @ L @ P2 @ K @ X5 @ Y5 )
        = ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X5 @ Y5 ) ) ) ) ) ).

% second_assumptions.deviate_pos_cup_def
thf(fact_1195_second__assumptions_Osqcup__sub,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( member2946998982187404937et_nat @ X5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
       => ( ( member2946998982187404937et_nat @ Y5 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
         => ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X5 @ Y5 ) ) ) ) ) ) ).

% second_assumptions.sqcup_sub
thf(fact_1196_odotl__def,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y5 )
      = ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).

% odotl_def
thf(fact_1197_first__assumptions_O_092_060G_062l_Ocong,axiom,
    clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).

% first_assumptions.\<G>l.cong
thf(fact_1198_first__assumptions_Oodotl__def,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y5 )
        = ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y5 ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).

% first_assumptions.odotl_def
thf(fact_1199_finite__v__gs__Gl,axiom,
    ! [X5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
     => ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ).

% finite_v_gs_Gl
thf(fact_1200_pointwise__minimal__pointwise__maximal_I2_J,axiom,
    ! [S2: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat_nat )
       => ( ! [X2: nat > nat] :
              ( ( member_nat_nat @ X2 @ S2 )
             => ! [Xa2: nat > nat] :
                  ( ( member_nat_nat @ Xa2 @ S2 )
                 => ( ( ord_less_eq_nat_nat @ X2 @ Xa2 )
                    | ( ord_less_eq_nat_nat @ Xa2 @ X2 ) ) ) )
         => ? [X2: nat > nat] :
              ( ( member_nat_nat @ X2 @ S2 )
              & ! [Xa: nat > nat] :
                  ( ( member_nat_nat @ Xa @ S2 )
                 => ( ord_less_eq_nat_nat @ Xa @ X2 ) ) ) ) ) ) ).

% pointwise_minimal_pointwise_maximal(2)
thf(fact_1201_pointwise__minimal__pointwise__maximal_I1_J,axiom,
    ! [S2: set_nat_nat] :
      ( ( finite2115694454571419734at_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat_nat )
       => ( ! [X2: nat > nat] :
              ( ( member_nat_nat @ X2 @ S2 )
             => ! [Xa2: nat > nat] :
                  ( ( member_nat_nat @ Xa2 @ S2 )
                 => ( ( ord_less_eq_nat_nat @ X2 @ Xa2 )
                    | ( ord_less_eq_nat_nat @ Xa2 @ X2 ) ) ) )
         => ? [X2: nat > nat] :
              ( ( member_nat_nat @ X2 @ S2 )
              & ! [Xa: nat > nat] :
                  ( ( member_nat_nat @ Xa @ S2 )
                 => ( ord_less_eq_nat_nat @ X2 @ Xa ) ) ) ) ) ) ).

% pointwise_minimal_pointwise_maximal(1)
thf(fact_1202_v__gs__union,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) )
      = ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ).

% v_gs_union
thf(fact_1203_v__gs__mono,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
     => ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ).

% v_gs_mono
thf(fact_1204_v__gs__empty,axiom,
    ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
    = bot_bot_set_set_nat ) ).

% v_gs_empty
thf(fact_1205_first__assumptions_Ov__gs__union,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X5 @ Y5 ) )
        = ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ) ).

% first_assumptions.v_gs_union
thf(fact_1206_first__assumptions_Ofinite__v__gs__Gl,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
       => ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ) ).

% first_assumptions.finite_v_gs_Gl
thf(fact_1207_first__assumptions_Ov__gs__empty,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
        = bot_bot_set_set_nat ) ) ).

% first_assumptions.v_gs_empty
thf(fact_1208_first__assumptions_Ov__gs__mono,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ Y5 )
       => ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y5 ) ) ) ) ).

% first_assumptions.v_gs_mono
thf(fact_1209_plucking__step_I3_J,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
     => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
       => ( ( Y5
            = ( clique4095374090462327202g_step @ p @ X5 ) )
         => ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X5 ) ) @ ( clique3210737319928189260st_ACC @ k @ Y5 ) ) ) ) ) ).

% plucking_step(3)
thf(fact_1210_plucking__step_I5_J,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
     => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
       => ( ( Y5
            = ( clique4095374090462327202g_step @ p @ X5 ) )
         => ( Y5 != bot_bo7198184520161983622et_nat ) ) ) ) ).

% plucking_step(5)
thf(fact_1211_plucking__step_I2_J,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
     => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
       => ( ( Y5
            = ( clique4095374090462327202g_step @ p @ X5 ) )
         => ( ord_le9131159989063066194et_nat @ Y5 @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ) ) ).

% plucking_step(2)
thf(fact_1212_first__assumptions_Oplucking__step_Ocong,axiom,
    clique4095374090462327202g_step = clique4095374090462327202g_step ).

% first_assumptions.plucking_step.cong
thf(fact_1213_second__assumptions_Oplucking__step_I2_J,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
       => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
         => ( ( Y5
              = ( clique4095374090462327202g_step @ P2 @ X5 ) )
           => ( ord_le9131159989063066194et_nat @ Y5 @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).

% second_assumptions.plucking_step(2)
thf(fact_1214_second__assumptions_Oplucking__step_I5_J,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
       => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
         => ( ( Y5
              = ( clique4095374090462327202g_step @ P2 @ X5 ) )
           => ( Y5 != bot_bo7198184520161983622et_nat ) ) ) ) ) ).

% second_assumptions.plucking_step(5)
thf(fact_1215_second__assumptions_Oplucking__step_I3_J,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
       => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
         => ( ( Y5
              = ( clique4095374090462327202g_step @ P2 @ X5 ) )
           => ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X5 ) ) @ ( clique3210737319928189260st_ACC @ K @ Y5 ) ) ) ) ) ) ).

% second_assumptions.plucking_step(3)
thf(fact_1216_PLU__main_Opinduct,axiom,
    ! [A0: set_set_set_nat,P: set_set_set_nat > $o] :
      ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ A0 )
     => ( ! [X7: set_set_set_nat] :
            ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ X7 )
           => ( ( ( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ l @ k ) )
                  & ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) ) )
               => ( P @ ( clique4095374090462327202g_step @ p @ X7 ) ) )
             => ( P @ X7 ) ) )
       => ( P @ A0 ) ) ) ).

% PLU_main.pinduct
thf(fact_1217_plucking__step_I1_J,axiom,
    ! [X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
     => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
       => ( ( Y5
            = ( clique4095374090462327202g_step @ p @ X5 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y5 ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ p ) @ one_one_nat ) ) ) ) ) ).

% plucking_step(1)
thf(fact_1218_lm,axiom,
    ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).

% lm
thf(fact_1219_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1220_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_1221_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1222_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1223_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1224_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1225_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_1226_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_1227_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1228_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1229_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_1230_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_1231_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_1232_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_1233_first__assumptions_Olm,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.lm
thf(fact_1234_diff__add__inverse2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse2
thf(fact_1235_diff__add__inverse,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse
thf(fact_1236_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_cancel2
thf(fact_1237_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_1238_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1239_add__diff__inverse__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_1240_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1241_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1242_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1243_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1244_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1245_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N: nat] :
        ? [K2: nat] :
          ( N
          = ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1246_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_1247_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_1248_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_1249_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_1250_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_1251_add__leD2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_1252_add__leD1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% add_leD1
thf(fact_1253_le__add2,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).

% le_add2
thf(fact_1254_le__add1,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).

% le_add1
thf(fact_1255_add__leE,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M2 @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_1256_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1257_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1258_second__assumptions_Oplucking__step_I1_J,axiom,
    ! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y5: set_set_set_nat] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
       => ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) )
         => ( ( Y5
              = ( clique4095374090462327202g_step @ P2 @ X5 ) )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y5 ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ P2 ) @ one_one_nat ) ) ) ) ) ) ).

% second_assumptions.plucking_step(1)
thf(fact_1259_second__assumptions_OPLU__main_Opinduct,axiom,
    ! [L: nat,P2: nat,K: nat,A0: set_set_set_nat,P: set_set_set_nat > $o] :
      ( ( assump2881078719466019805ptions @ L @ P2 @ K )
     => ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P2 @ K ) @ A0 )
       => ( ! [X7: set_set_set_nat] :
              ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P2 @ K ) @ X7 )
             => ( ( ( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ L @ K ) )
                    & ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) ) )
                 => ( P @ ( clique4095374090462327202g_step @ P2 @ X7 ) ) )
               => ( P @ X7 ) ) )
         => ( P @ A0 ) ) ) ) ).

% second_assumptions.PLU_main.pinduct
thf(fact_1260_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M5: nat] :
      ( ( P @ X )
     => ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( ord_less_eq_nat @ X2 @ M5 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X8: nat] :
                    ( ( P @ X8 )
                   => ( ord_less_eq_nat @ X8 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1261_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N4 )
         => ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1262_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N2: nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ N5 )
         => ( ord_less_nat @ X2 @ N2 ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1263_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N4 )
         => ( ord_less_nat @ X3 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1264_card__POS,axiom,
    ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
    = ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).

% card_POS
thf(fact_1265_first__assumptions_Ocard__POS,axiom,
    ! [L: nat,P2: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P2 @ K )
     => ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K ) )
        = ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) ) ) ).

% first_assumptions.card_POS
thf(fact_1266_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_1267_APR_Osimps_I2_J,axiom,
    ! [X: a] :
      ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
      = ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).

% APR.simps(2)
thf(fact_1268_inj__on___092_060pi_062,axiom,
    inj_on_a_set_nat @ pi @ v ).

% inj_on_\<pi>
thf(fact_1269_SET_Osimps_I2_J,axiom,
    ! [X: a] :
      ( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X ) )
      = ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).

% SET.simps(2)
thf(fact_1270__092_060pi_062__singleton_I2_J,axiom,
    ! [X: a] :
      ( ( member_a @ X @ v )
     => ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ).

% \<pi>_singleton(2)

% Conjectures (1)
thf(conj_0,conjecture,
    ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ phi2 @ psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ phi2 ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ psi ) ) ).

%------------------------------------------------------------------------------