%------------------------------------------------------------------------------
% 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/0002_Assumptions_and_Approximations/prob_00280_011909__16078692_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1393 ( 660 unt; 120 typ;   0 def)
%            Number of atoms       : 3282 (1086 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 10663 ( 313   ~;  86   |;  88   &;8884   @)
%                                         (   0 <=>;1292  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   6 avg)
%            Number of types       :   15 (  14 usr)
%            Number of type conns  :  327 ( 327   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  109 ( 106 usr;  21 con; 0-3 aty)
%            Number of variables   : 3252 ( 199   ^;3029   !;  24   ?;3252   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:47:19.200
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    numera4273646738625120315l_num1: $tType ).

thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    set_Ex3793607809372303086nnreal: $tType ).

thf(ty_n_t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
    numera6367994245245682809l_num1: $tType ).

thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
    numera2417102609627094330l_num1: $tType ).

thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
    extend8495563244428889912nnreal: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

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

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

thf(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

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

thf(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (106)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

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_OL0_H_H,type,
    assumptions_and_L03: 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_Oeps,type,
    assumptions_and_eps: real ).

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_Osecond__assumptions__axioms,type,
    assump8934899134041091456axioms: nat > nat > $o ).

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

thf(sy_c_Assumptions__and__Approximations_Othird__assumptions__axioms,type,
    assump4853309720620433339axioms: nat > nat > nat > $o ).

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

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
    minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
    minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

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__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    minus_5410813661909488930l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
    one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    one_on7795324986448017462l_num1: numera4273646738625120315l_num1 ).

thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
    one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).

thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
    one_on7819281148064737470l_num1: numera6367994245245682809l_num1 ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
    plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

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

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > num > num ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    plus_p1441664204671982194l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
    plus_p2313304076027620419l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: real > real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
    times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: num > num > num ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    times_2938166955517408246l_num1: numera4273646738625120315l_num1 > numera4273646738625120315l_num1 > numera4273646738625120315l_num1 ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    times_4022348038934646771nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
    times_times_set_int: set_int > set_int > set_int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
    times_times_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Num__Onum_J,type,
    times_times_set_num: set_num > set_num > set_num ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
    times_times_set_real: set_real > set_real > set_real ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nonnegative____Real__Oennreal,type,
    semiri6283507881447550617nnreal: nat > extend8495563244428889912nnreal ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: nat > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    semiri5667362542588693146l_num1: nat > numera4273646738625120315l_num1 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
    semiri1795386414920522267l_num1: nat > numera2417102609627094330l_num1 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

thf(sy_c_NthRoot_Oroot,type,
    root: nat > real > real ).

thf(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nonnegative____Real__Oennreal,type,
    numera4658534427948366547nnreal: num > extend8495563244428889912nnreal ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: num > int ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    numera7754357348821619680l_num1: num > numera4273646738625120315l_num1 ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
    numera2161328050825114965l_num1: num > numera2417102609627094330l_num1 ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit1_It__Numeral____Type__Onum1_J,type,
    numera6112219686443703444l_num1: num > numera6367994245245682809l_num1 ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

thf(sy_c_Num_Opow,type,
    pow: num > num > num ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
    ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: int > int > $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__Num__Onum,type,
    ord_less_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
    ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: int > int > $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__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).

thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nonnegative____Real__Oennreal,type,
    power_6007165696250533058nnreal: extend8495563244428889912nnreal > nat > extend8495563244428889912nnreal ).

thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > nat > int ).

thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > nat > nat ).

thf(sy_c_Power_Opower__class_Opower_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
    power_1002146276965246001l_num1: numera4273646738625120315l_num1 > nat > numera4273646738625120315l_num1 ).

thf(sy_c_Power_Opower__class_Opower_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
    power_7402600760894073284l_num1: numera2417102609627094330l_num1 > nat > numera2417102609627094330l_num1 ).

thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: real > nat > real ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Nonnegative____Real__Oennreal,type,
    divide4826598186094686858nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: int > int > int ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: nat > nat > nat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Transcendental_Olog,type,
    log: real > real > real ).

thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
    member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int: int > set_int > $o ).

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

thf(sy_c_member_001t__Num__Onum,type,
    member_num: num > set_num > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_v_k,type,
    k: nat ).

thf(sy_v_l,type,
    l: nat ).

thf(sy_v_p,type,
    p: nat ).

% Relevant facts (1269)
thf(fact_0__092_060open_062root_A8_A_Ireal_Al_A_094_A8_J_A_061_Aroot_A8_A_Ireal_Am_J_092_060close_062,axiom,
    ( ( root @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ l ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).

% \<open>root 8 (real l ^ 8) = root 8 (real m)\<close>
thf(fact_1_km__id,axiom,
    ( ( semiri5074537144036343181t_real @ k )
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).

% km_id
thf(fact_2__092_060open_062real_Ak_A_094_A4_A_061_Areal_Am_092_060close_062,axiom,
    ( ( power_power_real @ ( semiri5074537144036343181t_real @ k ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ).

% \<open>real k ^ 4 = real m\<close>
thf(fact_3__092_060open_062root_A4_A_Ireal_Ak_A_094_A4_J_A_061_Aroot_A4_A_Ireal_Am_J_092_060close_062,axiom,
    ( ( root @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ k ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).

% \<open>root 4 (real k ^ 4) = root 4 (real m)\<close>
thf(fact_4__092_060open_062real_Al_A_094_A8_A_061_Areal_Am_092_060close_062,axiom,
    ( ( power_power_real @ ( semiri5074537144036343181t_real @ l ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ).

% \<open>real l ^ 8 = real m\<close>
thf(fact_5_first__assumptions_Om_Ocong,axiom,
    assump1710595444109740334irst_m = assump1710595444109740334irst_m ).

% first_assumptions.m.cong
thf(fact_6_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5667362542588693146l_num1 @ ( numeral_numeral_nat @ N ) )
      = ( numera7754357348821619680l_num1 @ N ) ) ).

% of_nat_numeral
thf(fact_7_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N ) )
      = ( numera1916890842035813515d_enat @ N ) ) ).

% of_nat_numeral
thf(fact_8_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_9_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_10_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_11_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ N ) )
      = ( numera4658534427948366547nnreal @ N ) ) ).

% of_nat_numeral
thf(fact_12_kl2,axiom,
    ( k
    = ( power_power_nat @ l @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% kl2
thf(fact_13_m2,axiom,
    ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ k ) ).

% m2
thf(fact_14_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_15_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_16_m__def,axiom,
    ( ( assump1710595444109740334irst_m @ k )
    = ( power_power_nat @ k @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% m_def
thf(fact_17_l8,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ l ).

% l8
thf(fact_18_l2,axiom,
    ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ l ).

% l2
thf(fact_19_k2,axiom,
    ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ k ).

% k2
thf(fact_20_k,axiom,
    ord_less_nat @ l @ k ).

% k
thf(fact_21_km,axiom,
    ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).

% km
thf(fact_22_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_23_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_24_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri6283507881447550617nnreal @ M )
        = ( semiri6283507881447550617nnreal @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_25_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_26_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri4216267220026989637d_enat @ M )
        = ( semiri4216267220026989637d_enat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_27_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera1916890842035813515d_enat @ M )
        = ( numera1916890842035813515d_enat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_28_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_29_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_30_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_31_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera4658534427948366547nnreal @ M )
        = ( numera4658534427948366547nnreal @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_32_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_33_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_34_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_35_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_36_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_37_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_38_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_39_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_40_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_41_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_42_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_43_L0_H,axiom,
    ord_less_eq_nat @ assumptions_and_L02 @ l ).

% L0'
thf(fact_44_L0,axiom,
    ord_less_eq_nat @ assumptions_and_L0 @ l ).

% L0
thf(fact_45_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_46_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_47_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_48_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_49_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_50_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_51_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_52_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_53_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_54_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_55_M0_H,axiom,
    ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ k ) ).

% M0'
thf(fact_56_M0,axiom,
    ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ k ) ).

% M0
thf(fact_57_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ I ) @ ( semiri6283507881447550617nnreal @ J ) ) ) ).

% of_nat_mono
thf(fact_58_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ I ) @ ( semiri4216267220026989637d_enat @ J ) ) ) ).

% of_nat_mono
thf(fact_59_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_60_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_61_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_62_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_63_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_64_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_65_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_66_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_67_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_68_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_69_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_70_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_71_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_72_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_73_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_74_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_75_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_76_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( M2 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_77_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_78_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_79_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_80_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_81_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_82_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_83_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_84_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_86_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_87_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_88_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_nat @ M2 @ N2 )
          | ( M2 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_89_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_90_linorder__neqE__nat,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
     => ( ~ ( ord_less_nat @ X2 @ Y )
       => ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_91_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y3: nat] :
                ( ( P @ Y3 )
               => ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_92_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_93_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_94_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_95_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_96_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_97_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_98_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_99_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_100_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_101_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_102_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_103_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_104_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_105_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_106_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X2: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_107_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X2: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_108_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X2: num,N: nat] :
      ( ( ( semiri6283507881447550617nnreal @ Y )
        = ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X2 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_109_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X2: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_110_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X2: num,N: nat] :
      ( ( ( semiri4216267220026989637d_enat @ Y )
        = ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_111_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_112_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_113_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: nat] :
      ( ( ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X2 ) @ N )
        = ( semiri6283507881447550617nnreal @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_114_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_115_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,Y: nat] :
      ( ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ N )
        = ( semiri4216267220026989637d_enat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_116_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_117_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_118_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_119_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_120_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_121_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_122_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_123_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_124_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_125_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_126_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_127_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_128_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X2 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_129_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X2 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_130_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri6283507881447550617nnreal @ X2 )
        = ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_131_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X2 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_132_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri4216267220026989637d_enat @ X2 )
        = ( power_8040749407984259932d_enat @ ( semiri4216267220026989637d_enat @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_133_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_134_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_135_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W )
        = ( semiri6283507881447550617nnreal @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_136_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_137_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_8040749407984259932d_enat @ ( semiri4216267220026989637d_enat @ B ) @ W )
        = ( semiri4216267220026989637d_enat @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_138_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_139_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_140_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_141_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_142_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% of_nat_power
thf(fact_143_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% of_nat_power
thf(fact_144_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri6283507881447550617nnreal @ ( power_power_nat @ M @ N ) )
      = ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ M ) @ N ) ) ).

% of_nat_power
thf(fact_145_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_146_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4216267220026989637d_enat @ ( power_power_nat @ M @ N ) )
      = ( power_8040749407984259932d_enat @ ( semiri4216267220026989637d_enat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_147_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5667362542588693146l_num1 @ ( power_power_nat @ M @ N ) )
      = ( power_1002146276965246001l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ N ) ) ).

% of_nat_power
thf(fact_148_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_149_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_150_le__num__One__iff,axiom,
    ! [X2: num] :
      ( ( ord_less_eq_num @ X2 @ one )
      = ( X2 = one ) ) ).

% le_num_One_iff
thf(fact_151_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_152_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_153_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_154_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_155_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_156_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_157_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_158_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_159_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_160_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_161_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_162_kml,axiom,
    ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).

% kml
thf(fact_163_second__assumptions__axioms_Ointro,axiom,
    ! [K: nat,L: nat] :
      ( ( K
        = ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L )
       => ( assump8934899134041091456axioms @ L @ K ) ) ) ).

% second_assumptions_axioms.intro
thf(fact_164_second__assumptions__axioms__def,axiom,
    ( assump8934899134041091456axioms
    = ( ^ [L2: nat,K2: nat] :
          ( ( K2
            = ( power_power_nat @ L2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          & ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L2 ) ) ) ) ).

% second_assumptions_axioms_def
thf(fact_165_Multiseries__Expansion_Ointyness__simps_I3_J,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_real @ ( semiri5074537144036343181t_real @ A ) @ N )
      = ( semiri5074537144036343181t_real @ ( power_power_nat @ A @ N ) ) ) ).

% Multiseries_Expansion.intyness_simps(3)
thf(fact_166_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_167_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X4: real] : ( member_real @ X4 @ S2 )
     => ( ? [Z: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S2 )
           => ( ord_less_eq_real @ X3 @ Z ) )
       => ? [Y2: real] :
            ( ! [X4: real] :
                ( ( member_real @ X4 @ S2 )
               => ( ord_less_eq_real @ X4 @ Y2 ) )
            & ! [Z: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S2 )
                   => ( ord_less_eq_real @ X3 @ Z ) )
               => ( ord_less_eq_real @ Y2 @ Z ) ) ) ) ) ).

% complete_real
thf(fact_168_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_real @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% less_eq_real_def
thf(fact_169_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M3 @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_170_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_171_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_172_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_173_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_174_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5667362542588693146l_num1 @ ( minus_minus_nat @ M @ N ) )
        = ( minus_5410813661909488930l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ ( semiri5667362542588693146l_num1 @ N ) ) ) ) ).

% of_nat_diff
thf(fact_175_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_176_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_177_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_178_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_179_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_180_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_181_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_182_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_183_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_184_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_185_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_186_Multiseries__Expansion_Ointyness__of__nat,axiom,
    ! [N: nat] :
      ( ( N = N )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( semiri5074537144036343181t_real @ N ) ) ) ).

% Multiseries_Expansion.intyness_of_nat
thf(fact_187_power2__commute,axiom,
    ! [X2: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_188_power2__commute,axiom,
    ! [X2: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_189_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_190_Multiseries__Expansion_Ointyness__numeral,axiom,
    ! [Num: num] :
      ( ( Num = Num )
     => ( ( numeral_numeral_real @ Num )
        = ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num ) ) ) ) ).

% Multiseries_Expansion.intyness_numeral
thf(fact_191_Multiseries__Expansion_Ointyness__simps_I6_J,axiom,
    ( numeral_numeral_real
    = ( ^ [Num2: num] : ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num2 ) ) ) ) ).

% Multiseries_Expansion.intyness_simps(6)
thf(fact_192_seq__mono__lemma,axiom,
    ! [M: nat,D: nat > real,E: nat > real] :
      ( ! [N3: nat] :
          ( ( ord_less_eq_nat @ M @ N3 )
         => ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ M @ N3 )
           => ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M ) ) )
       => ! [N4: nat] :
            ( ( ord_less_eq_nat @ M @ N4 )
           => ( ord_less_real @ ( D @ N4 ) @ ( E @ M ) ) ) ) ) ).

% seq_mono_lemma
thf(fact_193_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_1002146276965246001l_num1 @ ( numera7754357348821619680l_num1 @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera7754357348821619680l_num1 @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_194_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera1916890842035813515d_enat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_195_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_196_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_197_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_198_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera4658534427948366547nnreal @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_199_local_Omp,axiom,
    ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).

% local.mp
thf(fact_200_p,axiom,
    ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).

% p
thf(fact_201_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_202_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_203_order__refl,axiom,
    ! [X2: extended_enat] : ( ord_le2932123472753598470d_enat @ X2 @ X2 ) ).

% order_refl
thf(fact_204_order__refl,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).

% order_refl
thf(fact_205_order__refl,axiom,
    ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).

% order_refl
thf(fact_206_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_207_order__refl,axiom,
    ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).

% order_refl
thf(fact_208_dual__order_Orefl,axiom,
    ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).

% dual_order.refl
thf(fact_209_dual__order_Orefl,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% dual_order.refl
thf(fact_210_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_211_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_212_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_213_kp,axiom,
    ord_less_nat @ p @ k ).

% kp
thf(fact_214_pl,axiom,
    ord_less_nat @ l @ p ).

% pl
thf(fact_215_third__assumptions__axioms,axiom,
    assump2119784843035796504ptions @ l @ p @ k ).

% third_assumptions_axioms
thf(fact_216_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M4: nat,N3: nat] :
          ( Z2
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_diff_cases
thf(fact_217_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( semiri1314217659103216013at_int @ A3 )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_218_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_219_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_220_pow_Osimps_I1_J,axiom,
    ! [X2: num] :
      ( ( pow @ X2 @ one )
      = X2 ) ).

% pow.simps(1)
thf(fact_221_order__antisym__conv,axiom,
    ! [Y: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X2 )
     => ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_222_order__antisym__conv,axiom,
    ! [Y: real,X2: real] :
      ( ( ord_less_eq_real @ Y @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_223_order__antisym__conv,axiom,
    ! [Y: num,X2: num] :
      ( ( ord_less_eq_num @ Y @ X2 )
     => ( ( ord_less_eq_num @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_224_order__antisym__conv,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_225_order__antisym__conv,axiom,
    ! [Y: int,X2: int] :
      ( ( ord_less_eq_int @ Y @ X2 )
     => ( ( ord_less_eq_int @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_226_linorder__le__cases,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ X2 @ Y )
     => ( ord_le2932123472753598470d_enat @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_227_linorder__le__cases,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_eq_real @ X2 @ Y )
     => ( ord_less_eq_real @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_228_linorder__le__cases,axiom,
    ! [X2: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X2 @ Y )
     => ( ord_less_eq_num @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_229_linorder__le__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_230_linorder__le__cases,axiom,
    ! [X2: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X2 @ Y )
     => ( ord_less_eq_int @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_231_ord__le__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_232_ord__le__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_233_ord__le__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_234_ord__le__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_235_ord__le__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_236_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_237_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_238_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_239_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_240_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_241_ord__eq__le__subst,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_242_ord__eq__le__subst,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_243_ord__eq__le__subst,axiom,
    ! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_244_ord__eq__le__subst,axiom,
    ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_245_ord__eq__le__subst,axiom,
    ! [A: int,F: extended_enat > int,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_246_ord__eq__le__subst,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_247_ord__eq__le__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_248_ord__eq__le__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_249_ord__eq__le__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_250_ord__eq__le__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_251_linorder__linear,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
      | ( ord_le2932123472753598470d_enat @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_252_linorder__linear,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
      | ( ord_less_eq_real @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_253_linorder__linear,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
      | ( ord_less_eq_num @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_254_linorder__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_255_linorder__linear,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
      | ( ord_less_eq_int @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_256_verit__la__disequality,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = B )
      | ~ ( ord_le2932123472753598470d_enat @ A @ B )
      | ~ ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_257_verit__la__disequality,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
      | ~ ( ord_less_eq_real @ A @ B )
      | ~ ( ord_less_eq_real @ B @ A ) ) ).

% verit_la_disequality
thf(fact_258_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_259_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_260_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_261_order__eq__refl,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( X2 = Y )
     => ( ord_le2932123472753598470d_enat @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_262_order__eq__refl,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 = Y )
     => ( ord_less_eq_real @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_263_order__eq__refl,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 = Y )
     => ( ord_less_eq_num @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_264_order__eq__refl,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_265_order__eq__refl,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 = Y )
     => ( ord_less_eq_int @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_266_order__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_267_order__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_268_order__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_269_order__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_270_order__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_271_order__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_272_order__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_273_order__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_274_order__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_275_order__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_276_order__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_277_order__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_278_order__subst1,axiom,
    ! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_eq_num @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_279_order__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_280_order__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_281_order__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_282_order__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_283_order__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_eq_num @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_284_order__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_285_order__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_286_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: extended_enat,Z3: extended_enat] : ( Y5 = Z3 ) )
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
          & ( ord_le2932123472753598470d_enat @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_287_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_288_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_289_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_290_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_291_antisym,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_292_antisym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_293_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_294_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_295_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_296_dual__order_Otrans,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ B )
       => ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_297_dual__order_Otrans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_298_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_299_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_300_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_301_dual__order_Oantisym,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_302_dual__order_Oantisym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_303_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_304_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_305_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_306_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: extended_enat,Z3: extended_enat] : ( Y5 = Z3 ) )
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
          & ( ord_le2932123472753598470d_enat @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_307_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_308_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_309_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_310_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_311_linorder__wlog,axiom,
    ! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
      ( ! [A4: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: extended_enat,B3: extended_enat] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_312_linorder__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A4: real,B3: real] :
          ( ( ord_less_eq_real @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: real,B3: real] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_313_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( ( ord_less_eq_num @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: num,B3: num] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_314_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_315_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: int,B3: int] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_316_order__trans,axiom,
    ! [X2: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
     => ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
       => ( ord_le2932123472753598470d_enat @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_317_order__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_318_order__trans,axiom,
    ! [X2: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_eq_num @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_319_order__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_320_order__trans,axiom,
    ! [X2: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_321_order_Otrans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).

% order.trans
thf(fact_322_order_Otrans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% order.trans
thf(fact_323_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_324_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_325_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_326_order__antisym,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
     => ( ( ord_le2932123472753598470d_enat @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_327_order__antisym,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_328_order__antisym,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
     => ( ( ord_less_eq_num @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_329_order__antisym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_330_order__antisym,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
     => ( ( ord_less_eq_int @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_331_ord__le__eq__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( B = C )
       => ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_332_ord__le__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_333_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_334_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_335_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_336_ord__eq__le__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A = B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_337_ord__eq__le__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_338_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_339_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_340_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_341_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: extended_enat,Z3: extended_enat] : ( Y5 = Z3 ) )
    = ( ^ [X: extended_enat,Y4: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X @ Y4 )
          & ( ord_le2932123472753598470d_enat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_342_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_eq_real @ X @ Y4 )
          & ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_343_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_eq_num @ X @ Y4 )
          & ( ord_less_eq_num @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_344_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_345_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_eq_int @ X @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_346_le__cases3,axiom,
    ! [X2: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
       => ~ ( ord_le2932123472753598470d_enat @ Y @ Z2 ) )
     => ( ( ( ord_le2932123472753598470d_enat @ Y @ X2 )
         => ~ ( ord_le2932123472753598470d_enat @ X2 @ Z2 ) )
       => ( ( ( ord_le2932123472753598470d_enat @ X2 @ Z2 )
           => ~ ( ord_le2932123472753598470d_enat @ Z2 @ Y ) )
         => ( ( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
             => ~ ( ord_le2932123472753598470d_enat @ Y @ X2 ) )
           => ( ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
               => ~ ( ord_le2932123472753598470d_enat @ Z2 @ X2 ) )
             => ~ ( ( ord_le2932123472753598470d_enat @ Z2 @ X2 )
                 => ~ ( ord_le2932123472753598470d_enat @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_347_le__cases3,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ( ord_less_eq_real @ X2 @ Y )
       => ~ ( ord_less_eq_real @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_real @ Y @ X2 )
         => ~ ( ord_less_eq_real @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_real @ X2 @ Z2 )
           => ~ ( ord_less_eq_real @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_real @ Z2 @ Y )
             => ~ ( ord_less_eq_real @ Y @ X2 ) )
           => ( ( ( ord_less_eq_real @ Y @ Z2 )
               => ~ ( ord_less_eq_real @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_real @ Z2 @ X2 )
                 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_348_le__cases3,axiom,
    ! [X2: num,Y: num,Z2: num] :
      ( ( ( ord_less_eq_num @ X2 @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_num @ Y @ X2 )
         => ~ ( ord_less_eq_num @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_num @ X2 @ Z2 )
           => ~ ( ord_less_eq_num @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_num @ Z2 @ Y )
             => ~ ( ord_less_eq_num @ Y @ X2 ) )
           => ( ( ( ord_less_eq_num @ Y @ Z2 )
               => ~ ( ord_less_eq_num @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_num @ Z2 @ X2 )
                 => ~ ( ord_less_eq_num @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_349_le__cases3,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_350_le__cases3,axiom,
    ! [X2: int,Y: int,Z2: int] :
      ( ( ( ord_less_eq_int @ X2 @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_int @ Y @ X2 )
         => ~ ( ord_less_eq_int @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_int @ X2 @ Z2 )
           => ~ ( ord_less_eq_int @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z2 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X2 ) )
           => ( ( ( ord_less_eq_int @ Y @ Z2 )
               => ~ ( ord_less_eq_int @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_int @ Z2 @ X2 )
                 => ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_351_nle__le,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) )
      = ( ( ord_le2932123472753598470d_enat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_352_nle__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_eq_real @ A @ B ) )
      = ( ( ord_less_eq_real @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_353_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_354_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_355_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_356_verit__comp__simplify1_I2_J,axiom,
    ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_357_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_358_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_359_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_360_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_361_order__less__imp__not__less,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_362_order__less__imp__not__less,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ~ ( ord_less_real @ Y @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_363_order__less__imp__not__less,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ~ ( ord_less_num @ Y @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_364_order__less__imp__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ~ ( ord_less_nat @ Y @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_365_order__less__imp__not__less,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ~ ( ord_less_int @ Y @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_366_order__less__imp__not__eq2,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_367_order__less__imp__not__eq2,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_368_order__less__imp__not__eq2,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_369_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_370_order__less__imp__not__eq2,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_371_order__less__imp__not__eq,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_372_order__less__imp__not__eq,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_373_order__less__imp__not__eq,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_374_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_375_order__less__imp__not__eq,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_376_linorder__less__linear,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
      | ( X2 = Y )
      | ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ).

% linorder_less_linear
thf(fact_377_linorder__less__linear,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
      | ( X2 = Y )
      | ( ord_less_real @ Y @ X2 ) ) ).

% linorder_less_linear
thf(fact_378_linorder__less__linear,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
      | ( X2 = Y )
      | ( ord_less_num @ Y @ X2 ) ) ).

% linorder_less_linear
thf(fact_379_linorder__less__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
      | ( X2 = Y )
      | ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_less_linear
thf(fact_380_linorder__less__linear,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
      | ( X2 = Y )
      | ( ord_less_int @ Y @ X2 ) ) ).

% linorder_less_linear
thf(fact_381_order__less__imp__triv,axiom,
    ! [X2: extended_enat,Y: extended_enat,P: $o] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_382_order__less__imp__triv,axiom,
    ! [X2: real,Y: real,P: $o] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ( ord_less_real @ Y @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_383_order__less__imp__triv,axiom,
    ! [X2: num,Y: num,P: $o] :
      ( ( ord_less_num @ X2 @ Y )
     => ( ( ord_less_num @ Y @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_384_order__less__imp__triv,axiom,
    ! [X2: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_385_order__less__imp__triv,axiom,
    ! [X2: int,Y: int,P: $o] :
      ( ( ord_less_int @ X2 @ Y )
     => ( ( ord_less_int @ Y @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_386_order__less__not__sym,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ).

% order_less_not_sym
thf(fact_387_order__less__not__sym,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ~ ( ord_less_real @ Y @ X2 ) ) ).

% order_less_not_sym
thf(fact_388_order__less__not__sym,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ~ ( ord_less_num @ Y @ X2 ) ) ).

% order_less_not_sym
thf(fact_389_order__less__not__sym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ~ ( ord_less_nat @ Y @ X2 ) ) ).

% order_less_not_sym
thf(fact_390_order__less__not__sym,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ~ ( ord_less_int @ Y @ X2 ) ) ).

% order_less_not_sym
thf(fact_391_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_392_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_393_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_394_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_395_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_396_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_397_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_398_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_399_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_400_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_401_order__less__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_402_order__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_403_order__less__subst1,axiom,
    ! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_404_order__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_405_order__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_406_order__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_407_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_408_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_409_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_410_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_411_order__less__irrefl,axiom,
    ! [X2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_412_order__less__irrefl,axiom,
    ! [X2: real] :
      ~ ( ord_less_real @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_413_order__less__irrefl,axiom,
    ! [X2: num] :
      ~ ( ord_less_num @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_414_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_415_order__less__irrefl,axiom,
    ! [X2: int] :
      ~ ( ord_less_int @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_416_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_417_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_418_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_419_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_420_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_421_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_422_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_423_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_424_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_425_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_426_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_427_ord__eq__less__subst,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_428_ord__eq__less__subst,axiom,
    ! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_429_ord__eq__less__subst,axiom,
    ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_430_ord__eq__less__subst,axiom,
    ! [A: int,F: extended_enat > int,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_431_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_432_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_433_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_434_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_435_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_436_order__less__trans,axiom,
    ! [X2: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z2 )
       => ( ord_le72135733267957522d_enat @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_437_order__less__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_438_order__less__trans,axiom,
    ! [X2: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_439_order__less__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_440_order__less__trans,axiom,
    ! [X2: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_441_order__less__asym_H,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).

% order_less_asym'
thf(fact_442_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_443_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_444_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_445_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_446_linorder__neq__iff,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( X2 != Y )
      = ( ( ord_le72135733267957522d_enat @ X2 @ Y )
        | ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_447_linorder__neq__iff,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 != Y )
      = ( ( ord_less_real @ X2 @ Y )
        | ( ord_less_real @ Y @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_448_linorder__neq__iff,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 != Y )
      = ( ( ord_less_num @ X2 @ Y )
        | ( ord_less_num @ Y @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_449_linorder__neq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
      = ( ( ord_less_nat @ X2 @ Y )
        | ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_450_linorder__neq__iff,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 != Y )
      = ( ( ord_less_int @ X2 @ Y )
        | ( ord_less_int @ Y @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_451_order__less__asym,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ).

% order_less_asym
thf(fact_452_order__less__asym,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ~ ( ord_less_real @ Y @ X2 ) ) ).

% order_less_asym
thf(fact_453_order__less__asym,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ~ ( ord_less_num @ Y @ X2 ) ) ).

% order_less_asym
thf(fact_454_order__less__asym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ~ ( ord_less_nat @ Y @ X2 ) ) ).

% order_less_asym
thf(fact_455_order__less__asym,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ~ ( ord_less_int @ Y @ X2 ) ) ).

% order_less_asym
thf(fact_456_linorder__neqE,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( X2 != Y )
     => ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y )
       => ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ) ).

% linorder_neqE
thf(fact_457_linorder__neqE,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 != Y )
     => ( ~ ( ord_less_real @ X2 @ Y )
       => ( ord_less_real @ Y @ X2 ) ) ) ).

% linorder_neqE
thf(fact_458_linorder__neqE,axiom,
    ! [X2: num,Y: num] :
      ( ( X2 != Y )
     => ( ~ ( ord_less_num @ X2 @ Y )
       => ( ord_less_num @ Y @ X2 ) ) ) ).

% linorder_neqE
thf(fact_459_linorder__neqE,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
     => ( ~ ( ord_less_nat @ X2 @ Y )
       => ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_neqE
thf(fact_460_linorder__neqE,axiom,
    ! [X2: int,Y: int] :
      ( ( X2 != Y )
     => ( ~ ( ord_less_int @ X2 @ Y )
       => ( ord_less_int @ Y @ X2 ) ) ) ).

% linorder_neqE
thf(fact_461_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_462_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_463_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_464_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_465_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_466_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_467_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_468_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_469_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_470_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_471_dual__order_Ostrict__trans,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_472_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_473_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_474_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_475_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_476_not__less__iff__gr__or__eq,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Y @ X2 )
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_477_not__less__iff__gr__or__eq,axiom,
    ! [X2: real,Y: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y ) )
      = ( ( ord_less_real @ Y @ X2 )
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_478_not__less__iff__gr__or__eq,axiom,
    ! [X2: num,Y: num] :
      ( ( ~ ( ord_less_num @ X2 @ Y ) )
      = ( ( ord_less_num @ Y @ X2 )
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_479_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y ) )
      = ( ( ord_less_nat @ Y @ X2 )
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_480_not__less__iff__gr__or__eq,axiom,
    ! [X2: int,Y: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y ) )
      = ( ( ord_less_int @ Y @ X2 )
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_481_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_482_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_483_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_484_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_485_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_486_linorder__less__wlog,axiom,
    ! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
      ( ! [A4: extended_enat,B3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: extended_enat] : ( P @ A4 @ A4 )
       => ( ! [A4: extended_enat,B3: extended_enat] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_487_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A4: real,B3: real] :
          ( ( ord_less_real @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: real] : ( P @ A4 @ A4 )
       => ( ! [A4: real,B3: real] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_488_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( ( ord_less_num @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: num] : ( P @ A4 @ A4 )
       => ( ! [A4: num,B3: num] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_489_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_490_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( ( ord_less_int @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B3: int] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_491_exists__least__iff,axiom,
    ( ( ^ [P2: extended_enat > $o] :
        ? [X5: extended_enat] : ( P2 @ X5 ) )
    = ( ^ [P3: extended_enat > $o] :
        ? [N2: extended_enat] :
          ( ( P3 @ N2 )
          & ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
             => ~ ( P3 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_492_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ~ ( P3 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_493_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% dual_order.irrefl
thf(fact_494_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_495_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_496_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_497_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_498_dual__order_Oasym,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ).

% dual_order.asym
thf(fact_499_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_500_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_501_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_502_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_503_linorder__cases,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ( X2 != Y )
       => ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ) ).

% linorder_cases
thf(fact_504_linorder__cases,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_real @ X2 @ Y )
     => ( ( X2 != Y )
       => ( ord_less_real @ Y @ X2 ) ) ) ).

% linorder_cases
thf(fact_505_linorder__cases,axiom,
    ! [X2: num,Y: num] :
      ( ~ ( ord_less_num @ X2 @ Y )
     => ( ( X2 != Y )
       => ( ord_less_num @ Y @ X2 ) ) ) ).

% linorder_cases
thf(fact_506_linorder__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ( X2 != Y )
       => ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_cases
thf(fact_507_linorder__cases,axiom,
    ! [X2: int,Y: int] :
      ( ~ ( ord_less_int @ X2 @ Y )
     => ( ( X2 != Y )
       => ( ord_less_int @ Y @ X2 ) ) ) ).

% linorder_cases
thf(fact_508_antisym__conv3,axiom,
    ! [Y: extended_enat,X2: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ Y @ X2 )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv3
thf(fact_509_antisym__conv3,axiom,
    ! [Y: real,X2: real] :
      ( ~ ( ord_less_real @ Y @ X2 )
     => ( ( ~ ( ord_less_real @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv3
thf(fact_510_antisym__conv3,axiom,
    ! [Y: num,X2: num] :
      ( ~ ( ord_less_num @ Y @ X2 )
     => ( ( ~ ( ord_less_num @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv3
thf(fact_511_antisym__conv3,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv3
thf(fact_512_antisym__conv3,axiom,
    ! [Y: int,X2: int] :
      ( ~ ( ord_less_int @ Y @ X2 )
     => ( ( ~ ( ord_less_int @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv3
thf(fact_513_less__induct,axiom,
    ! [P: extended_enat > $o,A: extended_enat] :
      ( ! [X3: extended_enat] :
          ( ! [Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ Y3 @ X3 )
             => ( P @ Y3 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_514_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y3: nat] :
              ( ( ord_less_nat @ Y3 @ X3 )
             => ( P @ Y3 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_515_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( B = C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_516_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_517_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_518_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_519_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_520_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A = B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_521_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_522_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_523_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_524_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_525_order_Oasym,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).

% order.asym
thf(fact_526_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_527_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_528_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_529_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_530_less__imp__neq,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( X2 != Y ) ) ).

% less_imp_neq
thf(fact_531_less__imp__neq,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( X2 != Y ) ) ).

% less_imp_neq
thf(fact_532_less__imp__neq,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ( X2 != Y ) ) ).

% less_imp_neq
thf(fact_533_less__imp__neq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( X2 != Y ) ) ).

% less_imp_neq
thf(fact_534_less__imp__neq,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ( X2 != Y ) ) ).

% less_imp_neq
thf(fact_535_dense,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ? [Z4: real] :
          ( ( ord_less_real @ X2 @ Z4 )
          & ( ord_less_real @ Z4 @ Y ) ) ) ).

% dense
thf(fact_536_gt__ex,axiom,
    ! [X2: real] :
    ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).

% gt_ex
thf(fact_537_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_538_gt__ex,axiom,
    ! [X2: int] :
    ? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).

% gt_ex
thf(fact_539_lt__ex,axiom,
    ! [X2: real] :
    ? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).

% lt_ex
thf(fact_540_lt__ex,axiom,
    ! [X2: int] :
    ? [Y2: int] : ( ord_less_int @ Y2 @ X2 ) ).

% lt_ex
thf(fact_541_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_542_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_543_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_544_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_545_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_546_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_547_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_548_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_549_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_550_order__le__imp__less__or__eq,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
     => ( ( ord_le72135733267957522d_enat @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_551_order__le__imp__less__or__eq,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_real @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_552_order__le__imp__less__or__eq,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
     => ( ( ord_less_num @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_553_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_nat @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_554_order__le__imp__less__or__eq,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
     => ( ( ord_less_int @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_555_linorder__le__less__linear,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
      | ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_556_linorder__le__less__linear,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
      | ( ord_less_real @ Y @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_557_linorder__le__less__linear,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
      | ( ord_less_num @ Y @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_558_linorder__le__less__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_559_linorder__le__less__linear,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
      | ( ord_less_int @ Y @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_560_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_561_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_562_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_563_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_564_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > extended_enat,C: extended_enat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_565_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_566_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_567_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_568_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_569_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_570_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_571_order__less__le__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_572_order__less__le__subst1,axiom,
    ! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_573_order__less__le__subst1,axiom,
    ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_574_order__less__le__subst1,axiom,
    ! [A: int,F: extended_enat > int,B: extended_enat,C: extended_enat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_575_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_576_order__less__le__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_577_order__less__le__subst1,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_578_order__less__le__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_579_order__less__le__subst1,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_580_order__le__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_581_order__le__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_582_order__le__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_583_order__le__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_584_order__le__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_585_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_586_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_587_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_588_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_589_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_eq_real @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_590_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_591_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_592_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_593_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_594_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_595_order__le__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_596_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y2: real] :
              ( ( ord_less_real @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_597_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_598_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_599_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_600_order__less__le__trans,axiom,
    ! [X2: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
       => ( ord_le72135733267957522d_enat @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_601_order__less__le__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_real @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_602_order__less__le__trans,axiom,
    ! [X2: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_num @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_603_order__less__le__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_604_order__less__le__trans,axiom,
    ! [X2: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_int @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_605_order__le__less__trans,axiom,
    ! [X2: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z2 )
       => ( ord_le72135733267957522d_enat @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_606_order__le__less__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_607_order__le__less__trans,axiom,
    ! [X2: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_608_order__le__less__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_609_order__le__less__trans,axiom,
    ! [X2: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_610_order__neq__le__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != B )
     => ( ( ord_le2932123472753598470d_enat @ A @ B )
       => ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_611_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_612_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_613_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_614_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_615_order__le__neq__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( A != B )
       => ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_616_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_617_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_618_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_619_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_620_order__less__imp__le,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ord_le2932123472753598470d_enat @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_621_order__less__imp__le,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ord_less_eq_real @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_622_order__less__imp__le,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_num @ X2 @ Y )
     => ( ord_less_eq_num @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_623_order__less__imp__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_624_order__less__imp__le,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_int @ X2 @ Y )
     => ( ord_less_eq_int @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_625_linorder__not__less,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y ) )
      = ( ord_le2932123472753598470d_enat @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_626_linorder__not__less,axiom,
    ! [X2: real,Y: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y ) )
      = ( ord_less_eq_real @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_627_linorder__not__less,axiom,
    ! [X2: num,Y: num] :
      ( ( ~ ( ord_less_num @ X2 @ Y ) )
      = ( ord_less_eq_num @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_628_linorder__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_629_linorder__not__less,axiom,
    ! [X2: int,Y: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y ) )
      = ( ord_less_eq_int @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_630_linorder__not__le,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ X2 @ Y ) )
      = ( ord_le72135733267957522d_enat @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_631_linorder__not__le,axiom,
    ! [X2: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
      = ( ord_less_real @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_632_linorder__not__le,axiom,
    ! [X2: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X2 @ Y ) )
      = ( ord_less_num @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_633_linorder__not__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
      = ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_634_linorder__not__le,axiom,
    ! [X2: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
      = ( ord_less_int @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_635_order__less__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X: extended_enat,Y4: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_636_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_eq_real @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_637_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_eq_num @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_638_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_639_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_eq_int @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_640_order__le__less,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [X: extended_enat,Y4: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_641_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_real @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_642_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_num @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_643_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_nat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_644_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_int @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_645_dual__order_Ostrict__implies__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_646_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_647_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_648_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_649_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_650_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ord_le2932123472753598470d_enat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_651_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_652_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_653_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_654_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_655_dual__order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
          & ~ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_656_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_657_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_658_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_659_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_660_dual__order_Ostrict__trans2,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_661_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_662_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_663_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_664_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_665_dual__order_Ostrict__trans1,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_666_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_667_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_668_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_669_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_670_dual__order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_671_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_672_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_673_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_674_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_675_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_676_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_real @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_677_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_num @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_678_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_nat @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_679_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_int @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_680_dense__le__bounded,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X2 @ W2 )
           => ( ( ord_less_real @ W2 @ Y )
             => ( ord_less_eq_real @ W2 @ Z2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_681_dense__ge__bounded,axiom,
    ! [Z2: real,X2: real,Y: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z2 @ W2 )
           => ( ( ord_less_real @ W2 @ X2 )
             => ( ord_less_eq_real @ Y @ W2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_682_order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
          & ~ ( ord_le2932123472753598470d_enat @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_683_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_684_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_685_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_686_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_687_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_688_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_689_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_690_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_691_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_692_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_693_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_694_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_695_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_696_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_697_order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_698_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_699_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_700_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_701_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_702_order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_703_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_real @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_704_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_num @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_705_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_706_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_int @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_707_not__le__imp__less,axiom,
    ! [Y: extended_enat,X2: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ Y @ X2 )
     => ( ord_le72135733267957522d_enat @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_708_not__le__imp__less,axiom,
    ! [Y: real,X2: real] :
      ( ~ ( ord_less_eq_real @ Y @ X2 )
     => ( ord_less_real @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_709_not__le__imp__less,axiom,
    ! [Y: num,X2: num] :
      ( ~ ( ord_less_eq_num @ Y @ X2 )
     => ( ord_less_num @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_710_not__le__imp__less,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X2 )
     => ( ord_less_nat @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_711_not__le__imp__less,axiom,
    ! [Y: int,X2: int] :
      ( ~ ( ord_less_eq_int @ Y @ X2 )
     => ( ord_less_int @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_712_less__le__not__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X: extended_enat,Y4: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X @ Y4 )
          & ~ ( ord_le2932123472753598470d_enat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_713_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_eq_real @ X @ Y4 )
          & ~ ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_714_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_eq_num @ X @ Y4 )
          & ~ ( ord_less_eq_num @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_715_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_716_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_eq_int @ X @ Y4 )
          & ~ ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_717_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y )
         => ( ord_less_eq_real @ X3 @ Z2 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_le
thf(fact_718_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z2 @ X3 )
         => ( ord_less_eq_real @ Y @ X3 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_719_antisym__conv2,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_720_antisym__conv2,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ~ ( ord_less_real @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_721_antisym__conv2,axiom,
    ! [X2: num,Y: num] :
      ( ( ord_less_eq_num @ X2 @ Y )
     => ( ( ~ ( ord_less_num @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_722_antisym__conv2,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_723_antisym__conv2,axiom,
    ! [X2: int,Y: int] :
      ( ( ord_less_eq_int @ X2 @ Y )
     => ( ( ~ ( ord_less_int @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_724_antisym__conv1,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_725_antisym__conv1,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_726_antisym__conv1,axiom,
    ! [X2: num,Y: num] :
      ( ~ ( ord_less_num @ X2 @ Y )
     => ( ( ord_less_eq_num @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_727_antisym__conv1,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_728_antisym__conv1,axiom,
    ! [X2: int,Y: int] :
      ( ~ ( ord_less_int @ X2 @ Y )
     => ( ( ord_less_eq_int @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_729_nless__le,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_730_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_731_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_732_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_733_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_734_leI,axiom,
    ! [X2: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X2 @ Y )
     => ( ord_le2932123472753598470d_enat @ Y @ X2 ) ) ).

% leI
thf(fact_735_leI,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_real @ X2 @ Y )
     => ( ord_less_eq_real @ Y @ X2 ) ) ).

% leI
thf(fact_736_leI,axiom,
    ! [X2: num,Y: num] :
      ( ~ ( ord_less_num @ X2 @ Y )
     => ( ord_less_eq_num @ Y @ X2 ) ) ).

% leI
thf(fact_737_leI,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% leI
thf(fact_738_leI,axiom,
    ! [X2: int,Y: int] :
      ( ~ ( ord_less_int @ X2 @ Y )
     => ( ord_less_eq_int @ Y @ X2 ) ) ).

% leI
thf(fact_739_leD,axiom,
    ! [Y: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X2 )
     => ~ ( ord_le72135733267957522d_enat @ X2 @ Y ) ) ).

% leD
thf(fact_740_leD,axiom,
    ! [Y: real,X2: real] :
      ( ( ord_less_eq_real @ Y @ X2 )
     => ~ ( ord_less_real @ X2 @ Y ) ) ).

% leD
thf(fact_741_leD,axiom,
    ! [Y: num,X2: num] :
      ( ( ord_less_eq_num @ Y @ X2 )
     => ~ ( ord_less_num @ X2 @ Y ) ) ).

% leD
thf(fact_742_leD,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y ) ) ).

% leD
thf(fact_743_leD,axiom,
    ! [Y: int,X2: int] :
      ( ( ord_less_eq_int @ Y @ X2 )
     => ~ ( ord_less_int @ X2 @ Y ) ) ).

% leD
thf(fact_744_verit__comp__simplify1_I3_J,axiom,
    ! [B4: extended_enat,A5: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ B4 @ A5 ) )
      = ( ord_le72135733267957522d_enat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_745_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A5: real] :
      ( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
      = ( ord_less_real @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_746_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A5: num] :
      ( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
      = ( ord_less_num @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_747_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
      = ( ord_less_nat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_748_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
      = ( ord_less_int @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_749_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_750_second__assumptions__axioms,axiom,
    assump2881078719466019805ptions @ l @ p @ k ).

% second_assumptions_axioms
thf(fact_751_Lm,axiom,
    ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).

% Lm
thf(fact_752_first__assumptions__axioms,axiom,
    assump5453534214990993103ptions @ l @ p @ k ).

% first_assumptions_axioms
thf(fact_753_Lp,axiom,
    ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).

% Lp
thf(fact_754_pllog_I1_J,axiom,
    ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ l ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ) @ ( semiri5074537144036343181t_real @ p ) ).

% pllog(1)
thf(fact_755_first__assumptions_Om__def,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ( assump1710595444109740334irst_m @ K )
        = ( power_power_nat @ K @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% first_assumptions.m_def
thf(fact_756_first__assumptions_Om2,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.m2
thf(fact_757_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K3 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_758_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: numera4273646738625120315l_num1] :
      ( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z2 ) )
      = ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_759_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_760_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_761_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_762_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_763_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z2 ) )
      = ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_764_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
      = ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_765_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_766_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_767_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_768_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_769_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
      = ( numera4658534427948366547nnreal @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_770_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_771_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_772_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri6283507881447550617nnreal @ ( times_times_nat @ M @ N ) )
      = ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) ) ) ).

% of_nat_mult
thf(fact_773_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_774_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4216267220026989637d_enat @ ( times_times_nat @ M @ N ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) ) ) ).

% of_nat_mult
thf(fact_775_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5667362542588693146l_num1 @ ( times_times_nat @ M @ N ) )
      = ( times_2938166955517408246l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ ( semiri5667362542588693146l_num1 @ N ) ) ) ).

% of_nat_mult
thf(fact_776_left__diff__distrib__numeral,axiom,
    ! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,V: num] :
      ( ( times_2938166955517408246l_num1 @ ( minus_5410813661909488930l_num1 @ A @ B ) @ ( numera7754357348821619680l_num1 @ V ) )
      = ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ B @ ( numera7754357348821619680l_num1 @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_777_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_778_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_779_right__diff__distrib__numeral,axiom,
    ! [V: num,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
      ( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( minus_5410813661909488930l_num1 @ B @ C ) )
      = ( minus_5410813661909488930l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_780_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_781_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_782_verit__la__generic,axiom,
    ! [A: int,X2: int] :
      ( ( ord_less_eq_int @ A @ X2 )
      | ( A = X2 )
      | ( ord_less_eq_int @ X2 @ A ) ) ).

% verit_la_generic
thf(fact_783_third__assumptions_Oaxioms_I1_J,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( assump2881078719466019805ptions @ L @ P4 @ K ) ) ).

% third_assumptions.axioms(1)
thf(fact_784_second__assumptions_Oaxioms_I1_J,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( assump5453534214990993103ptions @ L @ P4 @ K ) ) ).

% second_assumptions.axioms(1)
thf(fact_785_first__assumptions_OL_Ocong,axiom,
    assump1710595444109740301irst_L = assump1710595444109740301irst_L ).

% first_assumptions.L.cong
thf(fact_786_second__assumptions_OLp,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( ord_less_nat @ P4 @ ( assump1710595444109740301irst_L @ L @ P4 ) ) ) ).

% second_assumptions.Lp
thf(fact_787_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_788_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_789_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_790_power__commutes,axiom,
    ! [A: extend8495563244428889912nnreal,N: nat] :
      ( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ N ) @ A )
      = ( times_1893300245718287421nnreal @ A @ ( power_6007165696250533058nnreal @ A @ N ) ) ) ).

% power_commutes
thf(fact_791_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_792_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_793_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_794_power__mult__distrib,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,N: nat] :
      ( ( power_6007165696250533058nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ N )
      = ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ N ) @ ( power_6007165696250533058nnreal @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_795_power__commuting__commutes,axiom,
    ! [X2: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X2 @ Y )
        = ( times_times_real @ Y @ X2 ) )
     => ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X2 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_796_power__commuting__commutes,axiom,
    ! [X2: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X2 @ Y )
        = ( times_times_nat @ Y @ X2 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X2 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_797_power__commuting__commutes,axiom,
    ! [X2: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X2 @ Y )
        = ( times_times_int @ Y @ X2 ) )
     => ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X2 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_798_power__commuting__commutes,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N: nat] :
      ( ( ( times_1893300245718287421nnreal @ X2 @ Y )
        = ( times_1893300245718287421nnreal @ Y @ X2 ) )
     => ( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ X2 @ N ) @ Y )
        = ( times_1893300245718287421nnreal @ Y @ ( power_6007165696250533058nnreal @ X2 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_799_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_800_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_801_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ X2 ) @ Y )
      = ( times_1893300245718287421nnreal @ Y @ ( semiri6283507881447550617nnreal @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_802_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_803_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ X2 ) @ Y )
      = ( times_7803423173614009249d_enat @ Y @ ( semiri4216267220026989637d_enat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_804_mult__of__nat__commute,axiom,
    ! [X2: nat,Y: numera4273646738625120315l_num1] :
      ( ( times_2938166955517408246l_num1 @ ( semiri5667362542588693146l_num1 @ X2 ) @ Y )
      = ( times_2938166955517408246l_num1 @ Y @ ( semiri5667362542588693146l_num1 @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_805_second__assumptions_Ointro,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ( assump8934899134041091456axioms @ L @ K )
       => ( assump2881078719466019805ptions @ L @ P4 @ K ) ) ) ).

% second_assumptions.intro
thf(fact_806_second__assumptions__def,axiom,
    ( assump2881078719466019805ptions
    = ( ^ [L2: nat,P5: nat,K2: nat] :
          ( ( assump5453534214990993103ptions @ L2 @ P5 @ K2 )
          & ( assump8934899134041091456axioms @ L2 @ K2 ) ) ) ) ).

% second_assumptions_def
thf(fact_807_second__assumptions_OLm,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K ) @ ( assump1710595444109740301irst_L @ L @ P4 ) ) ) ).

% second_assumptions.Lm
thf(fact_808_first__assumptions_Ok,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ L @ K ) ) ).

% first_assumptions.k
thf(fact_809_first__assumptions_Okp,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ P4 @ K ) ) ).

% first_assumptions.kp
thf(fact_810_first__assumptions_Opl,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ L @ P4 ) ) ).

% first_assumptions.pl
thf(fact_811_third__assumptions_Opllog_I1_J,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K ) ) ) ) @ ( semiri5074537144036343181t_real @ P4 ) ) ) ).

% third_assumptions.pllog(1)
thf(fact_812_mult__numeral__1__right,axiom,
    ! [A: numera4273646738625120315l_num1] :
      ( ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_813_mult__numeral__1__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_814_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_815_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_816_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_817_mult__numeral__1__right,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ ( numera4658534427948366547nnreal @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_818_mult__numeral__1,axiom,
    ! [A: numera4273646738625120315l_num1] :
      ( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_819_mult__numeral__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_820_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_821_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_822_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_823_mult__numeral__1,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_824_second__assumptions_Oaxioms_I2_J,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( assump8934899134041091456axioms @ L @ K ) ) ).

% second_assumptions.axioms(2)
thf(fact_825_first__assumptions_Omp,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ P4 @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.mp
thf(fact_826_first__assumptions_Okm,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.km
thf(fact_827_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_828_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_829_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_830_power2__eq__square,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_1893300245718287421nnreal @ A @ A ) ) ).

% power2_eq_square
thf(fact_831_power4__eq__xxxx,axiom,
    ! [X2: real] :
      ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_832_power4__eq__xxxx,axiom,
    ! [X2: nat] :
      ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_833_power4__eq__xxxx,axiom,
    ! [X2: int] :
      ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_834_power4__eq__xxxx,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( power_6007165696250533058nnreal @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_835_third__assumptions_OL0,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ assumptions_and_L0 @ L ) ) ).

% third_assumptions.L0
thf(fact_836_third__assumptions_OL0_H,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ assumptions_and_L02 @ L ) ) ).

% third_assumptions.L0'
thf(fact_837_first__assumptions_Okml,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).

% first_assumptions.kml
thf(fact_838_first__assumptions__def,axiom,
    ( assump5453534214990993103ptions
    = ( ^ [L2: nat,P5: nat,K2: nat] :
          ( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L2 )
          & ( ord_less_nat @ L2 @ P5 )
          & ( ord_less_nat @ P5 @ K2 ) ) ) ) ).

% first_assumptions_def
thf(fact_839_first__assumptions_Ointro,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L )
     => ( ( ord_less_nat @ L @ P4 )
       => ( ( ord_less_nat @ P4 @ K )
         => ( assump5453534214990993103ptions @ L @ P4 @ K ) ) ) ) ).

% first_assumptions.intro
thf(fact_840_first__assumptions_Ol2,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) ) ).

% first_assumptions.l2
thf(fact_841_first__assumptions_Ok2,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) ) ).

% first_assumptions.k2
thf(fact_842_first__assumptions_Op,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 ) ) ).

% first_assumptions.p
thf(fact_843_third__assumptions_OM0_H,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% third_assumptions.M0'
thf(fact_844_third__assumptions_OM0,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% third_assumptions.M0
thf(fact_845_second__assumptions_Ol8,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L ) ) ).

% second_assumptions.l8
thf(fact_846_second__assumptions_Okl2,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( K
        = ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% second_assumptions.kl2
thf(fact_847_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_848_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_849_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_850_L3,axiom,
    ord_less_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( assump1710595444109740301irst_L @ l @ p ) ).

% L3
thf(fact_851_four__x__squared,axiom,
    ! [X2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_852_pllog_I2_J,axiom,
    ord_less_eq_real @ ( semiri5074537144036343181t_real @ p ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ l ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ) @ one_one_real ) ).

% pllog(2)
thf(fact_853_set__times__intro,axiom,
    ! [A: real,C2: set_real,B: real,D2: set_real] :
      ( ( member_real @ A @ C2 )
     => ( ( member_real @ B @ D2 )
       => ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D2 ) ) ) ) ).

% set_times_intro
thf(fact_854_set__times__intro,axiom,
    ! [A: num,C2: set_num,B: num,D2: set_num] :
      ( ( member_num @ A @ C2 )
     => ( ( member_num @ B @ D2 )
       => ( member_num @ ( times_times_num @ A @ B ) @ ( times_times_set_num @ C2 @ D2 ) ) ) ) ).

% set_times_intro
thf(fact_855_set__times__intro,axiom,
    ! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
      ( ( member_nat @ A @ C2 )
     => ( ( member_nat @ B @ D2 )
       => ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).

% set_times_intro
thf(fact_856_set__times__intro,axiom,
    ! [A: int,C2: set_int,B: int,D2: set_int] :
      ( ( member_int @ A @ C2 )
     => ( ( member_int @ B @ D2 )
       => ( member_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C2 @ D2 ) ) ) ) ).

% set_times_intro
thf(fact_857_set__times__intro,axiom,
    ! [A: extend8495563244428889912nnreal,C2: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,D2: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ C2 )
     => ( ( member7908768830364227535nnreal @ B @ D2 )
       => ( member7908768830364227535nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ ( times_4022348038934646771nnreal @ C2 @ D2 ) ) ) ) ).

% set_times_intro
thf(fact_858_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_859_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_860_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
      = ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_861_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_862_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_863_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_864_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_865_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
      = ( numera4658534427948366547nnreal @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_866_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: numera4273646738625120315l_num1] :
      ( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z2 ) )
      = ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_867_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_868_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_869_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_870_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_871_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z2 ) )
      = ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_872_power__one,axiom,
    ! [N: nat] :
      ( ( power_7402600760894073284l_num1 @ one_on3868389512446148991l_num1 @ N )
      = one_on3868389512446148991l_num1 ) ).

% power_one
thf(fact_873_power__one,axiom,
    ! [N: nat] :
      ( ( power_1002146276965246001l_num1 @ one_on7795324986448017462l_num1 @ N )
      = one_on7795324986448017462l_num1 ) ).

% power_one
thf(fact_874_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_875_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_876_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_877_power__one,axiom,
    ! [N: nat] :
      ( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ N )
      = one_on2969667320475766781nnreal ) ).

% power_one
thf(fact_878_of__nat__1,axiom,
    ( ( semiri1795386414920522267l_num1 @ one_one_nat )
    = one_on3868389512446148991l_num1 ) ).

% of_nat_1
thf(fact_879_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_880_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_881_of__nat__1,axiom,
    ( ( semiri6283507881447550617nnreal @ one_one_nat )
    = one_on2969667320475766781nnreal ) ).

% of_nat_1
thf(fact_882_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_883_of__nat__1,axiom,
    ( ( semiri4216267220026989637d_enat @ one_one_nat )
    = one_on7984719198319812577d_enat ) ).

% of_nat_1
thf(fact_884_of__nat__1,axiom,
    ( ( semiri5667362542588693146l_num1 @ one_one_nat )
    = one_on7795324986448017462l_num1 ) ).

% of_nat_1
thf(fact_885_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_886_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_887_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_on2969667320475766781nnreal
        = ( semiri6283507881447550617nnreal @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_888_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_889_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_on7984719198319812577d_enat
        = ( semiri4216267220026989637d_enat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_890_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_891_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_892_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri6283507881447550617nnreal @ N )
        = one_on2969667320475766781nnreal )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_893_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_894_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4216267220026989637d_enat @ N )
        = one_on7984719198319812577d_enat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_895_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_896_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_897_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri6283507881447550617nnreal @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p1859984266308609217nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) ) ) ).

% of_nat_add
thf(fact_898_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_899_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4216267220026989637d_enat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) ) ) ).

% of_nat_add
thf(fact_900_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5667362542588693146l_num1 @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p1441664204671982194l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ ( semiri5667362542588693146l_num1 @ N ) ) ) ).

% of_nat_add
thf(fact_901_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_902_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_903_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_904_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_905_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_906_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_907_distrib__left__numeral,axiom,
    ! [V: num,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
      ( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ B @ C ) )
      = ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ B ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_908_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_909_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_910_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_911_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_912_distrib__left__numeral,axiom,
    ! [V: num,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( plus_p1859984266308609217nnreal @ B @ C ) )
      = ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ B ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_913_distrib__right__numeral,axiom,
    ! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,V: num] :
      ( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ ( numera7754357348821619680l_num1 @ V ) )
      = ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ V ) ) @ ( times_2938166955517408246l_num1 @ B @ ( numera7754357348821619680l_num1 @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_914_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_915_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_916_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_917_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_918_distrib__right__numeral,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,V: num] :
      ( ( times_1893300245718287421nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( numera4658534427948366547nnreal @ V ) )
      = ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ ( numera4658534427948366547nnreal @ V ) ) @ ( times_1893300245718287421nnreal @ B @ ( numera4658534427948366547nnreal @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_919_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera1916890842035813515d_enat @ N )
        = one_on7984719198319812577d_enat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_920_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_921_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_922_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_923_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera4658534427948366547nnreal @ N )
        = one_on2969667320475766781nnreal )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_924_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on7984719198319812577d_enat
        = ( numera1916890842035813515d_enat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_925_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_926_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_927_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_928_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on2969667320475766781nnreal
        = ( numera4658534427948366547nnreal @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_929_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_930_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_931_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_932_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_933_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_934_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_935_power__mult__numeral,axiom,
    ! [A: extend8495563244428889912nnreal,M: num,N: num] :
      ( ( power_6007165696250533058nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_936_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_937_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_938_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_939_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_940_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ N ) )
      = ( numera2161328050825114965l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_941_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ N ) )
      = ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_942_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_943_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_944_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_945_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_946_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
      = ( numera4658534427948366547nnreal @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_947_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N ) @ one_on3868389512446148991l_num1 )
      = ( numera2161328050825114965l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_948_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ one_on7795324986448017462l_num1 )
      = ( numera7754357348821619680l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_949_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_950_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_951_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_952_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_953_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N ) @ one_on2969667320475766781nnreal )
      = ( numera4658534427948366547nnreal @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_954_power__strict__increasing__iff,axiom,
    ! [B: real,X2: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X2 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_955_power__strict__increasing__iff,axiom,
    ! [B: nat,X2: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X2 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_956_power__strict__increasing__iff,axiom,
    ! [B: int,X2: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X2 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_957_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_958_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_959_one__add__one,axiom,
    ( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
    = ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_960_one__add__one,axiom,
    ( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
    = ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_961_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_962_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_963_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_964_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_965_one__add__one,axiom,
    ( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal )
    = ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_966_power__increasing__iff,axiom,
    ! [B: real,X2: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X2 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_967_power__increasing__iff,axiom,
    ! [B: nat,X2: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X2 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_968_power__increasing__iff,axiom,
    ! [B: int,X2: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X2 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_969_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N ) @ one_on2969667320475766781nnreal )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_970_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_971_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_972_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_973_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_974_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_975_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_976_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_977_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_978_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_979_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_980_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X2 )
      = ( root @ M @ ( root @ N @ X2 ) ) ) ).

% real_root_mult_exp
thf(fact_981_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_982_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_983_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ X2 ) )
      = ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ X2 ) @ one_on3868389512446148991l_num1 ) ) ).

% one_plus_numeral_commute
thf(fact_984_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ X2 ) )
      = ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X2 ) @ one_on7795324986448017462l_num1 ) ) ).

% one_plus_numeral_commute
thf(fact_985_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_986_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_987_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_988_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_989_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ X2 ) )
      = ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ X2 ) @ one_on2969667320475766781nnreal ) ) ).

% one_plus_numeral_commute
thf(fact_990_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_Bit1
thf(fact_991_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_992_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_993_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_994_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera4658534427948366547nnreal @ ( bit1 @ N ) )
      = ( plus_p1859984266308609217nnreal @ ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N ) @ ( numera4658534427948366547nnreal @ N ) ) @ one_on2969667320475766781nnreal ) ) ).

% numeral_Bit1
thf(fact_995_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_996_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_997_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_998_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_999_mult__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 @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_1000_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_1001_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_1002_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_1003_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_1004_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_1005_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_1006_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% nat_less_real_le
thf(fact_1007_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_1008_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_1009_Multiseries__Expansion_Ointyness__simps_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
      = ( semiri5074537144036343181t_real @ ( times_times_nat @ A @ B ) ) ) ).

% Multiseries_Expansion.intyness_simps(2)
thf(fact_1010_real__arch__pow,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X2 @ N3 ) ) ) ).

% real_arch_pow
thf(fact_1011_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_1012_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_1013_real__root__commute,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( ( root @ M @ ( root @ N @ X2 ) )
      = ( root @ N @ ( root @ M @ X2 ) ) ) ).

% real_root_commute
thf(fact_1014_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_1015_third__assumptions_Opllog_I2_J,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ P4 ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K ) ) ) ) @ one_one_real ) ) ) ).

% third_assumptions.pllog(2)
thf(fact_1016_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_1017_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_1018_second__assumptions_OL3,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( ord_less_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( assump1710595444109740301irst_L @ L @ P4 ) ) ) ).

% second_assumptions.L3
thf(fact_1019_real__root__mult,axiom,
    ! [N: nat,X2: real,Y: real] :
      ( ( root @ N @ ( times_times_real @ X2 @ Y ) )
      = ( times_times_real @ ( root @ N @ X2 ) @ ( root @ N @ Y ) ) ) ).

% real_root_mult
thf(fact_1020_L0_H_H,axiom,
    ! [L: nat] :
      ( ( ord_less_eq_nat @ assumptions_and_L03 @ L )
     => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L0''
thf(fact_1021_third__assumptions__axioms__def,axiom,
    ( assump4853309720620433339axioms
    = ( ^ [L2: nat,P5: nat,K2: nat] :
          ( ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ P5 ) )
          & ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ P5 ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) @ one_one_real ) )
          & ( ord_less_eq_nat @ assumptions_and_L0 @ L2 )
          & ( ord_less_eq_nat @ assumptions_and_L02 @ L2 )
          & ( ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ K2 ) )
          & ( ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ K2 ) ) ) ) ) ).

% third_assumptions_axioms_def
thf(fact_1022_third__assumptions__axioms_Ointro,axiom,
    ! [L: nat,K: nat,P4: nat] :
      ( ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K ) ) ) ) @ ( semiri5074537144036343181t_real @ P4 ) )
     => ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ P4 ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K ) ) ) ) @ one_one_real ) )
       => ( ( ord_less_eq_nat @ assumptions_and_L0 @ L )
         => ( ( ord_less_eq_nat @ assumptions_and_L02 @ L )
           => ( ( ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ K ) )
             => ( ( ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ K ) )
               => ( assump4853309720620433339axioms @ L @ P4 @ K ) ) ) ) ) ) ) ).

% third_assumptions_axioms.intro
thf(fact_1023_landau__product__preprocess_I17_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% landau_product_preprocess(17)
thf(fact_1024_landau__product__preprocess_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% landau_product_preprocess(16)
thf(fact_1025_lm,axiom,
    ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).

% lm
thf(fact_1026_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_1027_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1028_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_1029_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1030_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1031_zle__add1__eq__le,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z2 ) ) ).

% zle_add1_eq_le
thf(fact_1032_landau__product__preprocess_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% landau_product_preprocess(8)
thf(fact_1033_landau__product__preprocess_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% landau_product_preprocess(15)
thf(fact_1034_landau__product__preprocess_I14_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% landau_product_preprocess(14)
thf(fact_1035_landau__product__preprocess_I13_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% landau_product_preprocess(13)
thf(fact_1036_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_1037_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_1038_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_1039_landau__product__preprocess_I4_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% landau_product_preprocess(4)
thf(fact_1040_landau__product__preprocess_I11_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% landau_product_preprocess(11)
thf(fact_1041_landau__product__preprocess_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% landau_product_preprocess(9)
thf(fact_1042_zle__diff1__eq,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
      = ( ord_less_int @ W @ Z2 ) ) ).

% zle_diff1_eq
thf(fact_1043_landau__product__preprocess_I12_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% landau_product_preprocess(12)
thf(fact_1044_landau__product__preprocess_I10_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% landau_product_preprocess(10)
thf(fact_1045_landau__product__preprocess_I7_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% landau_product_preprocess(7)
thf(fact_1046_landau__product__preprocess_I6_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% landau_product_preprocess(6)
thf(fact_1047_landau__product__preprocess_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% landau_product_preprocess(5)
thf(fact_1048_landau__product__preprocess_I18_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% landau_product_preprocess(18)
thf(fact_1049_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_1050_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_1051_zless__add1__eq,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z2 )
        | ( W = Z2 ) ) ) ).

% zless_add1_eq
thf(fact_1052_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_1053_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_1054_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_1055_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_1056_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_1057_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_1058_add1__zle__eq,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
      = ( ord_less_int @ W @ Z2 ) ) ).

% add1_zle_eq
thf(fact_1059_zless__imp__add1__zle,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_int @ W @ Z2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).

% zless_imp_add1_zle
thf(fact_1060_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_1061_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_1062_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_1063_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_1064_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_1065_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_1066_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_1067_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_1068_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_1069_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_1070_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_1071_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_1072_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1073_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_1074_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_1075_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1076_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1077_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_1078_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_1079_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_1080_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_1081_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1082_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1083_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_1084_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_1085_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_1086_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_1087_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_1088_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1089_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1090_add__diff__assoc__enat,axiom,
    ! [Z2: extended_enat,Y: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y ) @ Z2 ) ) ) ).

% add_diff_assoc_enat
thf(fact_1091_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_1092_first__assumptions_Olm,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).

% first_assumptions.lm
thf(fact_1093_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: 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 @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1101_Multiseries__Expansion_Ointyness__simps_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
      = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).

% Multiseries_Expansion.intyness_simps(1)
thf(fact_1102_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z5: int] :
        ? [N2: nat] :
          ( Z5
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_1103_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_1104_Multiseries__Expansion_Ointyness__1,axiom,
    ( one_one_real
    = ( semiri5074537144036343181t_real @ one_one_nat ) ) ).

% Multiseries_Expansion.intyness_1
thf(fact_1105_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_1106_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_1107_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_1108_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_1109_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_1110_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_1111_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_1112_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_1113_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_1114_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_1115_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_1116_third__assumptions_Oaxioms_I2_J,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( assump4853309720620433339axioms @ L @ P4 @ K ) ) ).

% third_assumptions.axioms(2)
thf(fact_1117_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_1118_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_1119_third__assumptions_Ointro,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump2881078719466019805ptions @ L @ P4 @ K )
     => ( ( assump4853309720620433339axioms @ L @ P4 @ K )
       => ( assump2119784843035796504ptions @ L @ P4 @ K ) ) ) ).

% third_assumptions.intro
thf(fact_1120_third__assumptions__def,axiom,
    ( assump2119784843035796504ptions
    = ( ^ [L2: nat,P5: nat,K2: nat] :
          ( ( assump2881078719466019805ptions @ L2 @ P5 @ K2 )
          & ( assump4853309720620433339axioms @ L2 @ P5 @ K2 ) ) ) ) ).

% third_assumptions_def
thf(fact_1121_approximation3,axiom,
    ! [S: nat] :
      ( ( ord_less_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ S @ ( times_times_nat @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_nat @ p @ one_one_nat ) ) ) )
     => ( ord_less_real @ ( divide_divide_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ) @ ( semiri5074537144036343181t_real @ k ) ) @ l ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( semiri5074537144036343181t_real @ S ) ) ) ).

% approximation3
thf(fact_1122_one__less__numeral,axiom,
    ! [N: num] :
      ( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral
thf(fact_1123_L__def,axiom,
    ( ( assump1710595444109740301irst_L @ l @ p )
    = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ l ) @ ( power_power_nat @ ( minus_minus_nat @ p @ one_one_nat ) @ l ) ) ) ).

% L_def
thf(fact_1124_real__of__nat__ge__one__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ one_one_nat @ N ) ) ).

% real_of_nat_ge_one_iff
thf(fact_1125_nat__add__1__add__1,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ one_one_nat )
      = ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% nat_add_1_add_1
thf(fact_1126_real__divide__square__eq,axiom,
    ! [R: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
      = ( divide_divide_real @ A @ R ) ) ).

% real_divide_square_eq
thf(fact_1127_real__root__divide,axiom,
    ! [N: nat,X2: real,Y: real] :
      ( ( root @ N @ ( divide_divide_real @ X2 @ Y ) )
      = ( divide_divide_real @ ( root @ N @ X2 ) @ ( root @ N @ Y ) ) ) ).

% real_root_divide
thf(fact_1128_numeral__eq__of__nat,axiom,
    ( numera4658534427948366547nnreal
    = ( ^ [A3: num] : ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ A3 ) ) ) ) ).

% numeral_eq_of_nat
thf(fact_1129_first__assumptions_OL__def,axiom,
    ! [L: nat,P4: nat,K: nat] :
      ( ( assump5453534214990993103ptions @ L @ P4 @ K )
     => ( ( assump1710595444109740301irst_L @ L @ P4 )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ L ) @ ( power_power_nat @ ( minus_minus_nat @ P4 @ one_one_nat ) @ L ) ) ) ) ).

% first_assumptions.L_def
thf(fact_1130_third__assumptions_Oapproximation3,axiom,
    ! [L: nat,P4: nat,K: nat,S: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ( ord_less_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ S @ ( times_times_nat @ ( power_power_nat @ ( assump1710595444109740301irst_L @ L @ P4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_nat @ P4 @ one_one_nat ) ) ) )
       => ( ord_less_real @ ( divide_divide_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) @ ( semiri5074537144036343181t_real @ K ) ) @ L ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ L @ P4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( semiri5074537144036343181t_real @ S ) ) ) ) ).

% third_assumptions.approximation3
thf(fact_1131_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_1132_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_1133_approximation2,axiom,
    ! [S: nat] :
      ( ( ord_less_eq_nat @ ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) @ ( times_times_nat @ ( times_times_nat @ S @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( binomial @ ( minus_minus_nat @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ k @ l ) @ one_one_nat ) ) ) )
     => ( ord_less_real @ ( divide_divide_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ) @ ( semiri5074537144036343181t_real @ k ) ) @ l ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( semiri5074537144036343181t_real @ S ) ) ) ).

% approximation2
thf(fact_1134_int__ops_I8_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(8)
thf(fact_1135_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K4 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_1136_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_1137_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_1138_real__of__nat__div4,axiom,
    ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% real_of_nat_div4
thf(fact_1139_real__of__nat__div3,axiom,
    ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_1140_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_1141_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).

% binomial_mono
thf(fact_1142_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K4 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_1143_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).

% binomial_strict_mono
thf(fact_1144_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_1145_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_1146_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_1147_third__assumptions_Oapproximation2,axiom,
    ! [L: nat,P4: nat,K: nat,S: nat] :
      ( ( assump2119784843035796504ptions @ L @ P4 @ K )
     => ( ( ord_less_eq_nat @ ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) @ ( times_times_nat @ ( times_times_nat @ S @ ( power_power_nat @ ( assump1710595444109740301irst_L @ L @ P4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( binomial @ ( minus_minus_nat @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ K @ L ) @ one_one_nat ) ) ) )
       => ( ord_less_real @ ( divide_divide_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) @ ( semiri5074537144036343181t_real @ K ) ) @ L ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ L @ P4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( semiri5074537144036343181t_real @ S ) ) ) ) ).

% third_assumptions.approximation2
thf(fact_1148_neq__4k1__k43,axiom,
    ! [M: nat,N: nat,M5: nat,N5: nat] :
      ( ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) )
     != ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N5 ) ) ) ) ).

% neq_4k1_k43
thf(fact_1149_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_1150_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_1151_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_1152_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_1153_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_1154_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).

% div_mult2_eq
thf(fact_1155_zdiv__int,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% zdiv_int
thf(fact_1156_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_1157_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_1158_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_1159_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_1160_fact__approx__add,axiom,
    ! [L: nat,N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ L @ N ) ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ L ) @ ( power_power_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ L ) @ ( semiri5074537144036343181t_real @ N ) ) @ N ) ) ) ).

% fact_approx_add
thf(fact_1161_fact__approx__minus,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_eq_nat @ N @ K )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ K ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ K @ N ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ K ) @ N ) ) ) ) ).

% fact_approx_minus
thf(fact_1162_choose__mono,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ M @ K ) ) ) ).

% choose_mono
thf(fact_1163_div__mult__le,axiom,
    ! [A: nat,B: nat,C: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) @ ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ).

% div_mult_le
thf(fact_1164_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_1165_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_1166_div__mult__pow__le,axiom,
    ! [A: nat,B: nat,N: nat] : ( ord_less_eq_nat @ ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N ) @ ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% div_mult_pow_le
thf(fact_1167_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).

% binomial_le_pow
thf(fact_1168_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_1169_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_1170_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_1171_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_1172_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_1173_half__bounded__equal,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( times_times_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real )
        = ( X2
          = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% half_bounded_equal
thf(fact_1174_eps__def,axiom,
    ( assumptions_and_eps
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% eps_def
thf(fact_1175_sum__le__prod1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ B @ one_one_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% sum_le_prod1
thf(fact_1176_sum__of__squares__ge__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ A ) @ B ) @ ( plus_p1859984266308609217nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_6007165696250533058nnreal @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_of_squares_ge_ennreal
thf(fact_1177_square__bound__lemma,axiom,
    ! [X2: real] : ( ord_less_real @ X2 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ).

% square_bound_lemma
thf(fact_1178_add__diff__eq__iff__ennreal,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ X2 ) )
        = Y )
      = ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).

% add_diff_eq_iff_ennreal
thf(fact_1179_divide__right__mono__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ord_le3935885782089961368nnreal @ ( divide4826598186094686858nnreal @ A @ C ) @ ( divide4826598186094686858nnreal @ B @ C ) ) ) ).

% divide_right_mono_ennreal
thf(fact_1180_power__mono__ennreal,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N: nat] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ord_le3935885782089961368nnreal @ ( power_6007165696250533058nnreal @ X2 @ N ) @ ( power_6007165696250533058nnreal @ Y @ N ) ) ) ).

% power_mono_ennreal
thf(fact_1181_diff__diff__ennreal_H,axiom,
    ! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
     => ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) @ X2 )
       => ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) )
          = ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Z2 ) @ Y ) ) ) ) ).

% diff_diff_ennreal'
thf(fact_1182_add__diff__eq__ennreal,axiom,
    ! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
     => ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) )
        = ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Y ) @ Z2 ) ) ) ).

% add_diff_eq_ennreal
thf(fact_1183_add__diff__le__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C ) @ ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ C ) ) ) ).

% add_diff_le_ennreal
thf(fact_1184_add__diff__self__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ A @ B )
       => ( ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ A ) )
          = B ) )
      & ( ~ ( ord_le3935885782089961368nnreal @ A @ B )
       => ( ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ A ) )
          = A ) ) ) ).

% add_diff_self_ennreal
thf(fact_1185_diff__add__self__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ A @ B )
       => ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A ) @ A )
          = B ) )
      & ( ~ ( ord_le3935885782089961368nnreal @ A @ B )
       => ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A ) @ A )
          = A ) ) ) ).

% diff_add_self_ennreal
thf(fact_1186_ennreal__ineq__diff__add,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ A )
     => ( A
        = ( plus_p1859984266308609217nnreal @ B @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ) ).

% ennreal_ineq_diff_add
thf(fact_1187_ennreal__diff__add__assoc,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ C @ B ) @ A )
        = ( plus_p1859984266308609217nnreal @ C @ ( minus_8429688780609304081nnreal @ B @ A ) ) ) ) ).

% ennreal_diff_add_assoc
thf(fact_1188_diff__add__assoc2__ennreal,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ A )
     => ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
        = ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ B ) ) ) ).

% diff_add_assoc2_ennreal
thf(fact_1189_diff__add__cancel__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A ) @ A )
        = B ) ) ).

% diff_add_cancel_ennreal
thf(fact_1190_add__diff__inverse__ennreal,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ X2 ) )
        = Y ) ) ).

% add_diff_inverse_ennreal
thf(fact_1191_diff__add__eq__diff__diff__swap__ennreal,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( minus_8429688780609304081nnreal @ X2 @ ( plus_p1859984266308609217nnreal @ Y @ Z2 ) )
      = ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ X2 @ Y ) @ Z2 ) ) ).

% diff_add_eq_diff_diff_swap_ennreal
thf(fact_1192_ennreal__diff__le__mono__left,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).

% ennreal_diff_le_mono_left
thf(fact_1193_diff__le__self__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ A ) ).

% diff_le_self_ennreal
thf(fact_1194_ennreal__mono__minus,axiom,
    ! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ C @ B )
     => ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).

% ennreal_mono_minus
thf(fact_1195_ennreal__minus__mono,axiom,
    ! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ C )
     => ( ( ord_le3935885782089961368nnreal @ D @ B )
       => ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ C @ D ) ) ) ) ).

% ennreal_minus_mono
thf(fact_1196_ennreal__times__divide,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ ( divide4826598186094686858nnreal @ B @ C ) )
      = ( divide4826598186094686858nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C ) ) ).

% ennreal_times_divide
thf(fact_1197_ennreal__divide__times,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( divide4826598186094686858nnreal @ A @ B ) @ C )
      = ( times_1893300245718287421nnreal @ A @ ( divide4826598186094686858nnreal @ C @ B ) ) ) ).

% ennreal_divide_times
thf(fact_1198_exhaust__2,axiom,
    ! [X2: numera2417102609627094330l_num1] :
      ( ( X2 = one_on3868389512446148991l_num1 )
      | ( X2
        = ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).

% exhaust_2
thf(fact_1199_forall__2,axiom,
    ( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
        ! [X5: numera2417102609627094330l_num1] : ( P2 @ X5 ) )
    = ( ^ [P3: numera2417102609627094330l_num1 > $o] :
          ( ( P3 @ one_on3868389512446148991l_num1 )
          & ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).

% forall_2
thf(fact_1200_forall__4,axiom,
    ( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
        ! [X5: numera4273646738625120315l_num1] : ( P2 @ X5 ) )
    = ( ^ [P3: numera4273646738625120315l_num1 > $o] :
          ( ( P3 @ one_on7795324986448017462l_num1 )
          & ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
          & ( P3 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
          & ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% forall_4
thf(fact_1201_exhaust__4,axiom,
    ! [X2: numera4273646738625120315l_num1] :
      ( ( X2 = one_on7795324986448017462l_num1 )
      | ( X2
        = ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
      | ( X2
        = ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
      | ( X2
        = ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% exhaust_4
thf(fact_1202_forall__3,axiom,
    ( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
        ! [X5: numera6367994245245682809l_num1] : ( P2 @ X5 ) )
    = ( ^ [P3: numera6367994245245682809l_num1 > $o] :
          ( ( P3 @ one_on7819281148064737470l_num1 )
          & ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
          & ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).

% forall_3
thf(fact_1203_exhaust__3,axiom,
    ! [X2: numera6367994245245682809l_num1] :
      ( ( X2 = one_on7819281148064737470l_num1 )
      | ( X2
        = ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
      | ( X2
        = ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).

% exhaust_3
thf(fact_1204_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_1205_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_1206_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_1207_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1208_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1209_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1210_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1211_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1212_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1213_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1214_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1215_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1216_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1217_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1218_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1219_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1220_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1221_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1222_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1223_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1224_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1225_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_1226_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1227_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1228_nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1229_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1230_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_1231_real__root__eq__iff,axiom,
    ! [N: nat,X2: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X2 )
          = ( root @ N @ Y ) )
        = ( X2 = Y ) ) ) ).

% real_root_eq_iff
thf(fact_1232_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1233_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_1234_real__root__le__iff,axiom,
    ! [N: nat,X2: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X2 @ Y ) ) ) ).

% real_root_le_iff
thf(fact_1235_real__root__less__iff,axiom,
    ! [N: nat,X2: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X2 @ Y ) ) ) ).

% real_root_less_iff
thf(fact_1236_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_1237_real__root__eq__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X2 )
          = one_one_real )
        = ( X2 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_1238_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_1239_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_1240_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_1241_real__root__le__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_1242_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_1243_real__root__lt__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_1244_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_1245_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1246_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1247_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1248_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1249_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1250_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1251_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1252_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_1253_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1254_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1255_diff__diff__commute__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
      = ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).

% diff_diff_commute_ennreal
thf(fact_1256_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1257_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1258_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1259_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1260_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1261_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1262_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N3 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_1263_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K3 )
               => ~ ( P @ I3 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1264_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1265_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1266_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_1267_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_1268_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( semiri5074537144036343181t_real @ l )
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).

%------------------------------------------------------------------------------