TPTP Problem File: SLH0932^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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 : Quasi_Borel_Spaces/0000_StandardBorel/prob_00791_029885__15084690_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1374 ( 890 unt; 102 typ; 0 def)
% Number of atoms : 2670 (1551 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 9794 ( 235 ~; 78 |; 141 &;8599 @)
% ( 0 <=>; 741 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 203 ( 203 >; 0 *; 0 +; 0 <<)
% Number of symbols : 94 ( 91 usr; 22 con; 0-3 aty)
% Number of variables : 2526 ( 64 ^;2420 !; 42 ?;2526 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:07:34.876
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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__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__Extended____Real__Oereal,type,
extended_ereal: $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 (91)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
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__Extended____Real__Oereal,type,
one_on4623092294121504201_ereal: extended_ereal ).
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__Extended____Real__Oereal,type,
times_7703590493115627913_ereal: extended_ereal > extended_ereal > extended_ereal ).
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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
times_8498157372700349887l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J,type,
zero_z2241845390563828978l_num1: numera4273646738625120315l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
zero_z5982384998485459395l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: 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_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__Extended____Real__Oereal,type,
numera1204434989813589363_ereal: num > extended_ereal ).
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_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__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $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__Extended____Real__Oereal,type,
power_1054015426188190660_ereal: extended_ereal > nat > extended_ereal ).
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____Real__Oereal,type,
divide8893690120176169980_ereal: extended_ereal > extended_ereal > extended_ereal ).
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_Series_Osuminf_001t__Extended____Nonnegative____Real__Oennreal,type,
suminf7725996343205245138nnreal: ( nat > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).
thf(sy_c_Series_Osuminf_001t__Extended____Real__Oereal,type,
suminf4411151127299490740_ereal: ( nat > extended_ereal ) > extended_ereal ).
thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_StandardBorel_Or01__binary__expansion_H,type,
r01_binary_expansion: real > nat > nat ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_a,type,
a: nat > nat ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_m,type,
m: nat ).
% Relevant facts (1268)
thf(fact_0_assms_I3_J,axiom,
( ( a @ i )
= zero_zero_nat ) ).
% assms(3)
thf(fact_1__092_060open_062_I_092_060Sum_062n_O_Areal_A_Ia_A_In_A_L_Ai_J_J_A_K_A_I1_A_P_A2_J_A_094_ASuc_A_In_A_L_Ai_J_J_A_061_A_I_092_060Sum_062n_O_Areal_A_Ia_A_ISuc_An_A_L_Ai_J_J_A_K_A_I1_A_P_A2_J_A_094_ASuc_A_ISuc_An_A_L_Ai_J_J_092_060close_062,axiom,
( ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ N @ i ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ ( plus_plus_nat @ N @ i ) ) ) ) )
= ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ ( suc @ N ) @ i ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ ( plus_plus_nat @ ( suc @ N ) @ i ) ) ) ) ) ) ).
% \<open>(\<Sum>n. real (a (n + i)) * (1 / 2) ^ Suc (n + i)) = (\<Sum>n. real (a (Suc n + i)) * (1 / 2) ^ Suc (Suc n + i))\<close>
thf(fact_2__092_060open_062_I_092_060Sum_062n_O_Areal_A_Ia_A_ISuc_An_A_L_Ai_J_J_A_K_A_I1_A_P_A2_J_A_094_ASuc_A_ISuc_An_A_L_Ai_J_J_A_061_A_I_092_060Sum_062n_O_Areal_A_Ia_A_In_A_L_ASuc_Ai_J_J_A_K_A_I1_A_P_A2_J_A_094_A_ISuc_An_A_L_ASuc_Ai_J_J_092_060close_062,axiom,
( ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ ( suc @ N ) @ i ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ ( plus_plus_nat @ ( suc @ N ) @ i ) ) ) ) )
= ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ N @ ( suc @ i ) ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ ( suc @ N ) @ ( suc @ i ) ) ) ) ) ) ).
% \<open>(\<Sum>n. real (a (Suc n + i)) * (1 / 2) ^ Suc (Suc n + i)) = (\<Sum>n. real (a (n + Suc i)) * (1 / 2) ^ (Suc n + Suc i))\<close>
thf(fact_3_calculation,axiom,
( ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ N @ i ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ ( plus_plus_nat @ N @ i ) ) ) ) )
= ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ N @ ( suc @ i ) ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ ( suc @ N ) @ ( suc @ i ) ) ) ) ) ) ).
% calculation
thf(fact_4_half__sum,axiom,
! [K: nat] :
( ( suminf_real
@ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ ( plus_plus_nat @ N @ K ) ) ) )
= ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ K ) ) ).
% half_sum
thf(fact_5_one__add__one,axiom,
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_6_one__add__one,axiom,
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_7_one__add__one,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_8_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_9_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_10_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_11_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_12_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri4216267220026989637d_enat @ ( suc @ M ) )
= ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( semiri4216267220026989637d_enat @ M ) ) ) ).
% of_nat_Suc
thf(fact_13_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1795386414920522267l_num1 @ ( suc @ M ) )
= ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( semiri1795386414920522267l_num1 @ M ) ) ) ).
% of_nat_Suc
thf(fact_14_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5667362542588693146l_num1 @ ( suc @ M ) )
= ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( semiri5667362542588693146l_num1 @ M ) ) ) ).
% of_nat_Suc
thf(fact_15_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_16_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_17_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_18_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri6283507881447550617nnreal @ ( suc @ M ) )
= ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( semiri6283507881447550617nnreal @ M ) ) ) ).
% of_nat_Suc
thf(fact_19_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_20_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_21_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_22_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_23_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_24_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_25_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N2 ) @ one_on3868389512446148991l_num1 )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_26_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_27_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N2 ) @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_28_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_29_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_30_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_31_numeral__plus__one,axiom,
! [N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% numeral_plus_one
thf(fact_32_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ N2 ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_33_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_34_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_35_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_36_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_37_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_38_one__plus__numeral,axiom,
! [N2: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ N2 ) )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ one @ N2 ) ) ) ).
% one_plus_numeral
thf(fact_39_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_40_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_41_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_42_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_43_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_44_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_45_numeral__le__one__iff,axiom,
! [N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
= ( ord_less_eq_num @ N2 @ one ) ) ).
% numeral_le_one_iff
thf(fact_46_nat__neq__4k1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ M )
!= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% nat_neq_4k1
thf(fact_47_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri4216267220026989637d_enat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).
% of_nat_add
thf(fact_48_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_49_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_50_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% of_nat_add
thf(fact_51_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_52_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri5667362542588693146l_num1 @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_p1441664204671982194l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ ( semiri5667362542588693146l_num1 @ N2 ) ) ) ).
% of_nat_add
thf(fact_53_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1795386414920522267l_num1 @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_p2313304076027620419l_num1 @ ( semiri1795386414920522267l_num1 @ M ) @ ( semiri1795386414920522267l_num1 @ N2 ) ) ) ).
% of_nat_add
thf(fact_54_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N2 )
= ( semiri4216267220026989637d_enat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_55_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_56_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_57_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X ) @ N2 )
= ( semiri6283507881447550617nnreal @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_58_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_59_assms_I2_J,axiom,
ord_less_eq_nat @ m @ i ).
% assms(2)
thf(fact_60_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_61_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_62_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_63_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_64_numeral__eq__iff,axiom,
! [M: num,N2: num] :
( ( ( numera4658534427948366547nnreal @ M )
= ( numera4658534427948366547nnreal @ N2 ) )
= ( M = N2 ) ) ).
% numeral_eq_iff
thf(fact_65_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_66_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_67_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_68_power__one__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_69_power__one__right,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_70_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_71_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_72_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_73_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_74_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_75_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_76_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_77_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_78_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_79_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_80_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z ) )
= ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_81_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_82_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_83_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_84_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_85_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z ) )
= ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_86_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ W ) @ Z ) )
= ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_87_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N2 ) )
= ( numera7754357348821619680l_num1 @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_88_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_89_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_90_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_91_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_92_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_93_numeral__times__numeral,axiom,
! [M: num,N2: num] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N2 ) )
= ( numera2161328050825114965l_num1 @ ( times_times_num @ M @ N2 ) ) ) ).
% numeral_times_numeral
thf(fact_94_num__double,axiom,
! [N2: num] :
( ( times_times_num @ ( bit0 @ one ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_95_power__one,axiom,
! [N2: nat] :
( ( power_7402600760894073284l_num1 @ one_on3868389512446148991l_num1 @ N2 )
= one_on3868389512446148991l_num1 ) ).
% power_one
thf(fact_96_power__one,axiom,
! [N2: nat] :
( ( power_1002146276965246001l_num1 @ one_on7795324986448017462l_num1 @ N2 )
= one_on7795324986448017462l_num1 ) ).
% power_one
thf(fact_97_power__one,axiom,
! [N2: nat] :
( ( power_8040749407984259932d_enat @ one_on7984719198319812577d_enat @ N2 )
= one_on7984719198319812577d_enat ) ).
% power_one
thf(fact_98_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_99_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_100_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_101_power__one,axiom,
! [N2: nat] :
( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ N2 )
= one_on2969667320475766781nnreal ) ).
% power_one
thf(fact_102_power__one,axiom,
! [N2: nat] :
( ( power_1054015426188190660_ereal @ one_on4623092294121504201_ereal @ N2 )
= one_on4623092294121504201_ereal ) ).
% power_one
thf(fact_103_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_104_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_105_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( times_times_nat @ M @ N2 ) )
= ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% of_nat_mult
thf(fact_106_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_107_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5667362542588693146l_num1 @ ( times_times_nat @ M @ N2 ) )
= ( times_2938166955517408246l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ ( semiri5667362542588693146l_num1 @ N2 ) ) ) ).
% of_nat_mult
thf(fact_108_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1795386414920522267l_num1 @ ( times_times_nat @ M @ N2 ) )
= ( times_8498157372700349887l_num1 @ ( semiri1795386414920522267l_num1 @ M ) @ ( semiri1795386414920522267l_num1 @ N2 ) ) ) ).
% of_nat_mult
thf(fact_109_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri4216267220026989637d_enat @ N2 )
= one_on7984719198319812577d_enat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_110_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_111_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_112_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri6283507881447550617nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_113_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_114_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_on7984719198319812577d_enat
= ( semiri4216267220026989637d_enat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_115_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_116_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_117_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_on2969667320475766781nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_118_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_119_of__nat__1,axiom,
( ( semiri4216267220026989637d_enat @ one_one_nat )
= one_on7984719198319812577d_enat ) ).
% of_nat_1
thf(fact_120_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_121_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_122_of__nat__1,axiom,
( ( semiri6283507881447550617nnreal @ one_one_nat )
= one_on2969667320475766781nnreal ) ).
% of_nat_1
thf(fact_123_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_124_of__nat__1,axiom,
( ( semiri5667362542588693146l_num1 @ one_one_nat )
= one_on7795324986448017462l_num1 ) ).
% of_nat_1
thf(fact_125_of__nat__1,axiom,
( ( semiri1795386414920522267l_num1 @ one_one_nat )
= one_on3868389512446148991l_num1 ) ).
% of_nat_1
thf(fact_126_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_127_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N2 ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_128_power__mult__numeral,axiom,
! [A: real,M: num,N2: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_129_power__mult__numeral,axiom,
! [A: nat,M: num,N2: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_130_power__mult__numeral,axiom,
! [A: int,M: num,N2: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_131_power__mult__numeral,axiom,
! [A: extend8495563244428889912nnreal,M: num,N2: num] :
( ( power_6007165696250533058nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_132_power__mult__numeral,axiom,
! [A: extended_ereal,M: num,N2: num] :
( ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
= ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% power_mult_numeral
thf(fact_133_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_134_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_135_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_136_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_137_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_138_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_140_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ M @ ( suc @ N2 ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_141_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_142_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_143_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_144_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_145_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_146_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_147_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_148_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_149_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_150_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_151_distrib__right__numeral,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,V: num] :
( ( times_8498157372700349887l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ V ) ) @ ( times_8498157372700349887l_num1 @ B @ ( numera2161328050825114965l_num1 @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_152_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_153_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_154_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_155_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_156_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_157_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_158_distrib__left__numeral,axiom,
! [V: num,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( plus_p2313304076027620419l_num1 @ B @ C ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ B ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_159_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_160_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_real
= ( numeral_numeral_real @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_161_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_162_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_one_int
= ( numeral_numeral_int @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_163_one__eq__numeral__iff,axiom,
! [N2: num] :
( ( one_on2969667320475766781nnreal
= ( numera4658534427948366547nnreal @ N2 ) )
= ( one = N2 ) ) ).
% one_eq_numeral_iff
thf(fact_164_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera1916890842035813515d_enat @ N2 )
= one_on7984719198319812577d_enat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_165_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_real @ N2 )
= one_one_real )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_166_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_nat @ N2 )
= one_one_nat )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_167_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numeral_numeral_int @ N2 )
= one_one_int )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_168_numeral__eq__one__iff,axiom,
! [N2: num] :
( ( ( numera4658534427948366547nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one ) ) ).
% numeral_eq_one_iff
thf(fact_169_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_170_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_171_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_172_power__0__Suc,axiom,
! [N2: nat] :
( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ ( suc @ N2 ) )
= zero_z7100319975126383169nnreal ) ).
% power_0_Suc
thf(fact_173_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_174_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_175_of__nat__0,axiom,
( ( semiri6283507881447550617nnreal @ zero_zero_nat )
= zero_z7100319975126383169nnreal ) ).
% of_nat_0
thf(fact_176_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_177_of__nat__0,axiom,
( ( semiri5667362542588693146l_num1 @ zero_zero_nat )
= zero_z2241845390563828978l_num1 ) ).
% of_nat_0
thf(fact_178_of__nat__0,axiom,
( ( semiri1795386414920522267l_num1 @ zero_zero_nat )
= zero_z5982384998485459395l_num1 ) ).
% of_nat_0
thf(fact_179_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_180_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_181_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_z7100319975126383169nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_182_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_183_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_184_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_185_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_186_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_187_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_188_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_189_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_190_power__zero__numeral,axiom,
! [K: num] :
( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ ( numeral_numeral_nat @ K ) )
= zero_z7100319975126383169nnreal ) ).
% power_zero_numeral
thf(fact_191_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_192_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_193_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_194_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_195_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_196_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N2 ) )
= ( numera1916890842035813515d_enat @ N2 ) ) ).
% of_nat_numeral
thf(fact_197_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_real @ N2 ) ) ).
% of_nat_numeral
thf(fact_198_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% of_nat_numeral
thf(fact_199_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ N2 ) )
= ( numera4658534427948366547nnreal @ N2 ) ) ).
% of_nat_numeral
thf(fact_200_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% of_nat_numeral
thf(fact_201_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri5667362542588693146l_num1 @ ( numeral_numeral_nat @ N2 ) )
= ( numera7754357348821619680l_num1 @ N2 ) ) ).
% of_nat_numeral
thf(fact_202_of__nat__numeral,axiom,
! [N2: num] :
( ( semiri1795386414920522267l_num1 @ ( numeral_numeral_nat @ N2 ) )
= ( numera2161328050825114965l_num1 @ N2 ) ) ).
% of_nat_numeral
thf(fact_203_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_204_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_205_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_206_power__Suc0__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_207_power__Suc0__right,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_208_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_209_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_210_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_211_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_212_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_213_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri6283507881447550617nnreal @ X )
= ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_214_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_215_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_216_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_217_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ B ) @ W )
= ( semiri6283507881447550617nnreal @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_218_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_219_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_220_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_221_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( power_power_nat @ M @ N2 ) )
= ( power_6007165696250533058nnreal @ ( semiri6283507881447550617nnreal @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_222_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_223_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri5667362542588693146l_num1 @ ( power_power_nat @ M @ N2 ) )
= ( power_1002146276965246001l_num1 @ ( semiri5667362542588693146l_num1 @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_224_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1795386414920522267l_num1 @ ( power_power_nat @ M @ N2 ) )
= ( power_7402600760894073284l_num1 @ ( semiri1795386414920522267l_num1 @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_225_add__numeral__left,axiom,
! [V: num,W: num,Z: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_226_add__numeral__left,axiom,
! [V: num,W: num,Z: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_227_add__numeral__left,axiom,
! [V: num,W: num,Z: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_228_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_229_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_230_add__numeral__left,axiom,
! [V: num,W: num,Z: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ V ) @ ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ W ) @ Z ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_231_add__numeral__left,axiom,
! [V: num,W: num,Z: numera2417102609627094330l_num1] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ W ) @ Z ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_232_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N2 ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_233_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_234_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_235_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_236_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_237_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( numera4658534427948366547nnreal @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_238_numeral__plus__numeral,axiom,
! [M: num,N2: num] :
( ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N2 ) )
= ( numera2161328050825114965l_num1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% numeral_plus_numeral
thf(fact_239_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_240_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_241_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_242_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_243_numeral__le__iff,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% numeral_le_iff
thf(fact_244_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_245_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_246_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_247_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M ) @ zero_z5237406670263579293d_enat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_248_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_249_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_250_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_251_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_252_power__add__numeral2,axiom,
! [A: extended_ereal,M: num,N2: num,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_253_power__add__numeral2,axiom,
! [A: real,M: num,N2: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_254_power__add__numeral2,axiom,
! [A: nat,M: num,N2: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_255_power__add__numeral2,axiom,
! [A: int,M: num,N2: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_256_power__add__numeral2,axiom,
! [A: extend8495563244428889912nnreal,M: num,N2: num,B: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
= ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_257_power__add__numeral,axiom,
! [A: extended_ereal,M: num,N2: num] :
( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_258_power__add__numeral,axiom,
! [A: real,M: num,N2: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_259_power__add__numeral,axiom,
! [A: nat,M: num,N2: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_260_power__add__numeral,axiom,
! [A: int,M: num,N2: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_261_power__add__numeral,axiom,
! [A: extend8495563244428889912nnreal,M: num,N2: num] :
( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ N2 ) ) )
= ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% power_add_numeral
thf(fact_262_Suc__numeral,axiom,
! [N2: num] :
( ( suc @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% Suc_numeral
thf(fact_263_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_264_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_265_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_266_zero__eq__power2,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7100319975126383169nnreal )
= ( A = zero_z7100319975126383169nnreal ) ) ).
% zero_eq_power2
thf(fact_267_add__2__eq__Suc_H,axiom,
! [N2: nat] :
( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc'
thf(fact_268_add__2__eq__Suc,axiom,
! [N2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc
thf(fact_269_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_270_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_271_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_272_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_273_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_274_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_275_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri4216267220026989637d_enat @ Y )
= ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_276_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_277_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_278_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ Y )
= ( power_6007165696250533058nnreal @ ( numera4658534427948366547nnreal @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_279_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_280_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_281_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_282_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_283_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_284_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_285_sum__power2__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_286_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_287_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_288_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_289_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_290_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_291_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_292_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_293_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_294_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_295_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_296_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_297_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_298_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_299_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_300_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_301_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_302_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_303_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_304_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_305_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_306_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_307_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_308_power2__nat__le__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_309_power2__nat__le__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_310_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_311_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_312_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_313_power__mult,axiom,
! [A: real,M: nat,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_314_power__mult,axiom,
! [A: nat,M: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_315_power__mult,axiom,
! [A: int,M: nat,N2: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_316_power__mult,axiom,
! [A: extend8495563244428889912nnreal,M: nat,N2: nat] :
( ( power_6007165696250533058nnreal @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_6007165696250533058nnreal @ ( power_6007165696250533058nnreal @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_317_power__mult,axiom,
! [A: extended_ereal,M: nat,N2: nat] :
( ( power_1054015426188190660_ereal @ A @ ( times_times_nat @ M @ N2 ) )
= ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ M ) @ N2 ) ) ).
% power_mult
thf(fact_318_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_319_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_320_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_321_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_322_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_323_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X4 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_324_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_325_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_326_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_327_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_328_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_329_Suc__le__D,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_330_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_331_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_332_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_333_zero__neq__numeral,axiom,
! [N2: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_334_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N2 ) ) ).
% zero_neq_numeral
thf(fact_335_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N2 ) ) ).
% zero_neq_numeral
thf(fact_336_zero__neq__numeral,axiom,
! [N2: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N2 ) ) ).
% zero_neq_numeral
thf(fact_337_zero__neq__numeral,axiom,
! [N2: num] :
( zero_z7100319975126383169nnreal
!= ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_neq_numeral
thf(fact_338_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_339_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_340_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
? [K2: nat] :
( N
= ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_341_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_342_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_343_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_344_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_345_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_346_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_347_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_348_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_349_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_350_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_351_power__not__zero,axiom,
! [A: real,N2: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N2 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_352_power__not__zero,axiom,
! [A: nat,N2: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_353_power__not__zero,axiom,
! [A: int,N2: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N2 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_354_power__not__zero,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( A != zero_z7100319975126383169nnreal )
=> ( ( power_6007165696250533058nnreal @ A @ N2 )
!= zero_z7100319975126383169nnreal ) ) ).
% power_not_zero
thf(fact_355_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_356_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_357_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_358_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_359_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_360_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_361_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_362_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_363_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_364_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_365_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_366_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_367_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_368_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_369_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_370_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_7402600760894073284l_num1 @ zero_z5982384998485459395l_num1 @ N2 )
= one_on3868389512446148991l_num1 ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_7402600760894073284l_num1 @ zero_z5982384998485459395l_num1 @ N2 )
= zero_z5982384998485459395l_num1 ) ) ) ).
% power_0_left
thf(fact_371_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_1002146276965246001l_num1 @ zero_z2241845390563828978l_num1 @ N2 )
= one_on7795324986448017462l_num1 ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_1002146276965246001l_num1 @ zero_z2241845390563828978l_num1 @ N2 )
= zero_z2241845390563828978l_num1 ) ) ) ).
% power_0_left
thf(fact_372_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
= one_on7984719198319812577d_enat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
= zero_z5237406670263579293d_enat ) ) ) ).
% power_0_left
thf(fact_373_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_374_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_375_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_376_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N2 )
= one_on2969667320475766781nnreal ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N2 )
= zero_z7100319975126383169nnreal ) ) ) ).
% power_0_left
thf(fact_377_zero__le__even__power_H,axiom,
! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_378_zero__le__even__power_H,axiom,
! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% zero_le_even_power'
thf(fact_379_power__even__eq,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_380_power__even__eq,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_381_power__even__eq,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_382_power__even__eq,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( power_6007165696250533058nnreal @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_6007165696250533058nnreal @ ( power_6007165696250533058nnreal @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_383_power__even__eq,axiom,
! [A: extended_ereal,N2: nat] :
( ( power_1054015426188190660_ereal @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_384_Suc__nat__number__of__add,axiom,
! [V: num,N2: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).
% Suc_nat_number_of_add
thf(fact_385_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_386_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_387_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_388_zero__power2,axiom,
( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_z7100319975126383169nnreal ) ).
% zero_power2
thf(fact_389_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_390_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_391_odd__0__le__power__imp__0__le,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_392_odd__0__le__power__imp__0__le,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_393_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_le_zero
thf(fact_394_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_395_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_396_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_397_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_398_zero__le__numeral,axiom,
! [N2: num] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% zero_le_numeral
thf(fact_399_zero__le__numeral,axiom,
! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% zero_le_numeral
thf(fact_400_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% zero_le_numeral
thf(fact_401_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_le_numeral
thf(fact_402_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% zero_le_numeral
thf(fact_403_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_404_zero__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_405_zero__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_406_zero__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_le_power
thf(fact_407_power__mono,axiom,
! [A: real,B: real,N2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_408_power__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_409_power__mono,axiom,
! [A: int,B: int,N2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% power_mono
thf(fact_410_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_411_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_412_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_413_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_414_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_415_power__0,axiom,
! [A: numera2417102609627094330l_num1] :
( ( power_7402600760894073284l_num1 @ A @ zero_zero_nat )
= one_on3868389512446148991l_num1 ) ).
% power_0
thf(fact_416_power__0,axiom,
! [A: numera4273646738625120315l_num1] :
( ( power_1002146276965246001l_num1 @ A @ zero_zero_nat )
= one_on7795324986448017462l_num1 ) ).
% power_0
thf(fact_417_power__0,axiom,
! [A: extended_enat] :
( ( power_8040749407984259932d_enat @ A @ zero_zero_nat )
= one_on7984719198319812577d_enat ) ).
% power_0
thf(fact_418_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_419_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_420_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_421_power__0,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ zero_zero_nat )
= one_on2969667320475766781nnreal ) ).
% power_0
thf(fact_422_power__0,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ zero_zero_nat )
= one_on4623092294121504201_ereal ) ).
% power_0
thf(fact_423_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_424_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_425_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_426_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_427_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_428_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri6283507881447550617nnreal @ ( suc @ N2 ) )
!= zero_z7100319975126383169nnreal ) ).
% of_nat_neq_0
thf(fact_429_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_430_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_431_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_432_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_433_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_434_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_435_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_436_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_437_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_438_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_439_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_440_sum__power2__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_441_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_442_sum__power2__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_443_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_444_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_445_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_446_power__decreasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_447_power__decreasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_448_power__decreasing,axiom,
! [N2: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_449_power__le__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_450_power__le__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_451_power__le__one,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_452_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_453_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_454_power__le__imp__le__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_455_power__le__imp__le__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_456_power__le__imp__le__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_457_power__inject__base,axiom,
! [A: real,N2: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N2 ) )
= ( power_power_real @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_458_power__inject__base,axiom,
! [A: nat,N2: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( power_power_nat @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_459_power__inject__base,axiom,
! [A: int,N2: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N2 ) )
= ( power_power_int @ B @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_460_power__numeral__even,axiom,
! [Z: extended_ereal,W: num] :
( ( power_1054015426188190660_ereal @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_1054015426188190660_ereal @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_461_power__numeral__even,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_462_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_463_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_464_power__numeral__even,axiom,
! [Z: extend8495563244428889912nnreal,W: num] :
( ( power_6007165696250533058nnreal @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_6007165696250533058nnreal @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_465_power2__eq__square,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_7703590493115627913_ereal @ A @ A ) ) ).
% power2_eq_square
thf(fact_466_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_467_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_468_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_469_power2__eq__square,axiom,
! [A: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_1893300245718287421nnreal @ A @ A ) ) ).
% power2_eq_square
thf(fact_470_power4__eq__xxxx,axiom,
! [X: extended_ereal] :
( ( power_1054015426188190660_ereal @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_471_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_472_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_473_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_474_power4__eq__xxxx,axiom,
! [X: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_475_one__power2,axiom,
( ( power_7402600760894073284l_num1 @ one_on3868389512446148991l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on3868389512446148991l_num1 ) ).
% one_power2
thf(fact_476_one__power2,axiom,
( ( power_1002146276965246001l_num1 @ one_on7795324986448017462l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on7795324986448017462l_num1 ) ).
% one_power2
thf(fact_477_one__power2,axiom,
( ( power_8040749407984259932d_enat @ one_on7984719198319812577d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on7984719198319812577d_enat ) ).
% one_power2
thf(fact_478_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_479_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_480_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_481_one__power2,axiom,
( ( power_6007165696250533058nnreal @ one_on2969667320475766781nnreal @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_on2969667320475766781nnreal ) ).
% one_power2
thf(fact_482_power__Suc__le__self,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_483_power__Suc__le__self,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_484_power__Suc__le__self,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_485_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_486_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_487_is__num__normalize_I1_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C )
= ( plus_p1441664204671982194l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_488_is__num__normalize_I1_J,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ C )
= ( plus_p2313304076027620419l_num1 @ A @ ( plus_p2313304076027620419l_num1 @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_489_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_490_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_491_power__odd__eq,axiom,
! [A: extended_ereal,N2: nat] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_7703590493115627913_ereal @ A @ ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_492_power__odd__eq,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_493_power__odd__eq,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_494_power__odd__eq,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_495_power__odd__eq,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( power_6007165696250533058nnreal @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
= ( times_1893300245718287421nnreal @ A @ ( power_6007165696250533058nnreal @ ( power_6007165696250533058nnreal @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_496_le__numeral__extra_I4_J,axiom,
ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ).
% le_numeral_extra(4)
thf(fact_497_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_498_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_499_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_500_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_501_lift__Suc__antimono__le,axiom,
! [F: nat > extended_enat,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_le2932123472753598470d_enat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_502_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_real @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_503_lift__Suc__antimono__le,axiom,
! [F: nat > num,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_504_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_505_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_506_lift__Suc__mono__le,axiom,
! [F: nat > extended_enat,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_le2932123472753598470d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_507_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_508_lift__Suc__mono__le,axiom,
! [F: nat > num,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_509_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_510_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_511_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_512_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_513_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_514_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_515_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_516_power__commuting__commutes,axiom,
! [X: extended_ereal,Y: extended_ereal,N2: nat] :
( ( ( times_7703590493115627913_ereal @ X @ Y )
= ( times_7703590493115627913_ereal @ Y @ X ) )
=> ( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ X @ N2 ) @ Y )
= ( times_7703590493115627913_ereal @ Y @ ( power_1054015426188190660_ereal @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_517_power__commuting__commutes,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_518_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_519_power__commuting__commutes,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_520_power__commuting__commutes,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N2: nat] :
( ( ( times_1893300245718287421nnreal @ X @ Y )
= ( times_1893300245718287421nnreal @ Y @ X ) )
=> ( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ Y )
= ( times_1893300245718287421nnreal @ Y @ ( power_6007165696250533058nnreal @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_521_power__mult__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,N2: nat] :
( ( power_1054015426188190660_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ N2 )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ N2 ) @ ( power_1054015426188190660_ereal @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_522_power__mult__distrib,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_523_power__mult__distrib,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_524_power__mult__distrib,axiom,
! [A: int,B: int,N2: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_525_power__mult__distrib,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,N2: nat] :
( ( power_6007165696250533058nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ N2 )
= ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ N2 ) @ ( power_6007165696250533058nnreal @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_526_power__commutes,axiom,
! [A: extended_ereal,N2: nat] :
( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ N2 ) @ A )
= ( times_7703590493115627913_ereal @ A @ ( power_1054015426188190660_ereal @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_527_power__commutes,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_528_power__commutes,axiom,
! [A: nat,N2: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_529_power__commutes,axiom,
! [A: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_530_power__commutes,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ N2 ) @ A )
= ( times_1893300245718287421nnreal @ A @ ( power_6007165696250533058nnreal @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_531_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_532_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_533_mult__of__nat__commute,axiom,
! [X: nat,Y: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( semiri6283507881447550617nnreal @ X ) @ Y )
= ( times_1893300245718287421nnreal @ Y @ ( semiri6283507881447550617nnreal @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_534_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_535_mult__of__nat__commute,axiom,
! [X: nat,Y: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( semiri5667362542588693146l_num1 @ X ) @ Y )
= ( times_2938166955517408246l_num1 @ Y @ ( semiri5667362542588693146l_num1 @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_536_mult__of__nat__commute,axiom,
! [X: nat,Y: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( semiri1795386414920522267l_num1 @ X ) @ Y )
= ( times_8498157372700349887l_num1 @ Y @ ( semiri1795386414920522267l_num1 @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_537_power__divide,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
= ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_divide
thf(fact_538_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_539_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_540_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_541_one__le__numeral,axiom,
! [N2: num] : ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) ) ).
% one_le_numeral
thf(fact_542_one__le__numeral,axiom,
! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% one_le_numeral
thf(fact_543_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% one_le_numeral
thf(fact_544_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% one_le_numeral
thf(fact_545_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% one_le_numeral
thf(fact_546_power__increasing,axiom,
! [N2: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_547_power__increasing,axiom,
! [N2: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_548_power__increasing,axiom,
! [N2: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_549_one__le__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_550_one__le__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_551_one__le__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% one_le_power
thf(fact_552_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ X ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ X ) @ one_on7795324986448017462l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_553_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).
% one_plus_numeral_commute
thf(fact_554_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_555_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_556_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_557_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p1859984266308609217nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ X ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ X ) @ one_on2969667320475766781nnreal ) ) ).
% one_plus_numeral_commute
thf(fact_558_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p2313304076027620419l_num1 @ one_on3868389512446148991l_num1 @ ( numera2161328050825114965l_num1 @ X ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ X ) @ one_on3868389512446148991l_num1 ) ) ).
% one_plus_numeral_commute
thf(fact_559_mult__numeral__1__right,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( numera7754357348821619680l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_560_mult__numeral__1__right,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_561_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_562_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_563_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_564_mult__numeral__1__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ ( numera4658534427948366547nnreal @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_565_mult__numeral__1__right,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_566_mult__numeral__1,axiom,
! [A: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_567_mult__numeral__1,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_568_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_569_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_570_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_571_mult__numeral__1,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_572_mult__numeral__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_573_numeral__Bit0,axiom,
! [N2: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N2 ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N2 ) @ ( numera7754357348821619680l_num1 @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_574_numeral__Bit0,axiom,
! [N2: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_575_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_576_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_577_numeral__Bit0,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_578_numeral__Bit0,axiom,
! [N2: num] :
( ( numera4658534427948366547nnreal @ ( bit0 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ ( numera4658534427948366547nnreal @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_579_numeral__Bit0,axiom,
! [N2: num] :
( ( numera2161328050825114965l_num1 @ ( bit0 @ N2 ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N2 ) @ ( numera2161328050825114965l_num1 @ N2 ) ) ) ).
% numeral_Bit0
thf(fact_580_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_581_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_582_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_583_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_584_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_585_numeral__One,axiom,
( ( numera4658534427948366547nnreal @ one )
= one_on2969667320475766781nnreal ) ).
% numeral_One
thf(fact_586_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_587_left__right__inverse__power,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1,N2: nat] :
( ( ( times_8498157372700349887l_num1 @ X @ Y )
= one_on3868389512446148991l_num1 )
=> ( ( times_8498157372700349887l_num1 @ ( power_7402600760894073284l_num1 @ X @ N2 ) @ ( power_7402600760894073284l_num1 @ Y @ N2 ) )
= one_on3868389512446148991l_num1 ) ) ).
% left_right_inverse_power
thf(fact_588_left__right__inverse__power,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1,N2: nat] :
( ( ( times_2938166955517408246l_num1 @ X @ Y )
= one_on7795324986448017462l_num1 )
=> ( ( times_2938166955517408246l_num1 @ ( power_1002146276965246001l_num1 @ X @ N2 ) @ ( power_1002146276965246001l_num1 @ Y @ N2 ) )
= one_on7795324986448017462l_num1 ) ) ).
% left_right_inverse_power
thf(fact_589_left__right__inverse__power,axiom,
! [X: extended_enat,Y: extended_enat,N2: nat] :
( ( ( times_7803423173614009249d_enat @ X @ Y )
= one_on7984719198319812577d_enat )
=> ( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ X @ N2 ) @ ( power_8040749407984259932d_enat @ Y @ N2 ) )
= one_on7984719198319812577d_enat ) ) ).
% left_right_inverse_power
thf(fact_590_left__right__inverse__power,axiom,
! [X: extended_ereal,Y: extended_ereal,N2: nat] :
( ( ( times_7703590493115627913_ereal @ X @ Y )
= one_on4623092294121504201_ereal )
=> ( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ X @ N2 ) @ ( power_1054015426188190660_ereal @ Y @ N2 ) )
= one_on4623092294121504201_ereal ) ) ).
% left_right_inverse_power
thf(fact_591_left__right__inverse__power,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_592_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_593_left__right__inverse__power,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_594_left__right__inverse__power,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,N2: nat] :
( ( ( times_1893300245718287421nnreal @ X @ Y )
= one_on2969667320475766781nnreal )
=> ( ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ ( power_6007165696250533058nnreal @ Y @ N2 ) )
= one_on2969667320475766781nnreal ) ) ).
% left_right_inverse_power
thf(fact_595_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_596_power__one__over,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_597_power__Suc2,axiom,
! [A: extended_ereal,N2: nat] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ N2 ) )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_598_power__Suc2,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_599_power__Suc2,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_600_power__Suc2,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_601_power__Suc2,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( power_6007165696250533058nnreal @ A @ ( suc @ N2 ) )
= ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_602_power__Suc,axiom,
! [A: extended_ereal,N2: nat] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ N2 ) )
= ( times_7703590493115627913_ereal @ A @ ( power_1054015426188190660_ereal @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_603_power__Suc,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_604_power__Suc,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_605_power__Suc,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_606_power__Suc,axiom,
! [A: extend8495563244428889912nnreal,N2: nat] :
( ( power_6007165696250533058nnreal @ A @ ( suc @ N2 ) )
= ( times_1893300245718287421nnreal @ A @ ( power_6007165696250533058nnreal @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_607_power__add,axiom,
! [A: extended_ereal,M: nat,N2: nat] :
( ( power_1054015426188190660_ereal @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ M ) @ ( power_1054015426188190660_ereal @ A @ N2 ) ) ) ).
% power_add
thf(fact_608_power__add,axiom,
! [A: real,M: nat,N2: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% power_add
thf(fact_609_power__add,axiom,
! [A: nat,M: nat,N2: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_add
thf(fact_610_power__add,axiom,
! [A: int,M: nat,N2: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% power_add
thf(fact_611_power__add,axiom,
! [A: extend8495563244428889912nnreal,M: nat,N2: nat] :
( ( power_6007165696250533058nnreal @ A @ ( plus_plus_nat @ M @ N2 ) )
= ( times_1893300245718287421nnreal @ ( power_6007165696250533058nnreal @ A @ M ) @ ( power_6007165696250533058nnreal @ A @ N2 ) ) ) ).
% power_add
thf(fact_612_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera7754357348821619680l_num1 @ ( bit0 @ N2 ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N2 ) @ ( numera7754357348821619680l_num1 @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_613_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_614_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_real @ ( bit0 @ N2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_615_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_616_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numeral_numeral_int @ ( bit0 @ N2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_617_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera4658534427948366547nnreal @ ( bit0 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( numera4658534427948366547nnreal @ N2 ) @ ( numera4658534427948366547nnreal @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_618_numeral__code_I2_J,axiom,
! [N2: num] :
( ( numera2161328050825114965l_num1 @ ( bit0 @ N2 ) )
= ( plus_p2313304076027620419l_num1 @ ( numera2161328050825114965l_num1 @ N2 ) @ ( numera2161328050825114965l_num1 @ N2 ) ) ) ).
% numeral_code(2)
thf(fact_619_left__add__twice,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ A @ B ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_620_left__add__twice,axiom,
! [A: extended_enat,B: extended_enat] :
( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_621_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_622_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_623_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_624_left__add__twice,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ A @ ( plus_p1859984266308609217nnreal @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_625_left__add__twice,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1] :
( ( plus_p2313304076027620419l_num1 @ A @ ( plus_p2313304076027620419l_num1 @ A @ B ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_626_mult__2__right,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ Z @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2_right
thf(fact_627_mult__2__right,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_628_mult__2__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2_right
thf(fact_629_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_630_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_631_mult__2__right,axiom,
! [Z: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ Z @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) )
= ( plus_p1859984266308609217nnreal @ Z @ Z ) ) ).
% mult_2_right
thf(fact_632_mult__2__right,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ Z @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) )
= ( plus_p2313304076027620419l_num1 @ Z @ Z ) ) ).
% mult_2_right
thf(fact_633_mult__2,axiom,
! [Z: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p1441664204671982194l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_634_mult__2,axiom,
! [Z: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
= ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).
% mult_2
thf(fact_635_mult__2,axiom,
! [Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_real @ Z @ Z ) ) ).
% mult_2
thf(fact_636_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_637_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_638_mult__2,axiom,
! [Z: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ Z )
= ( plus_p1859984266308609217nnreal @ Z @ Z ) ) ).
% mult_2
thf(fact_639_mult__2,axiom,
! [Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ Z )
= ( plus_p2313304076027620419l_num1 @ Z @ Z ) ) ).
% mult_2
thf(fact_640_power2__sum,axiom,
! [X: numera4273646738625120315l_num1,Y: numera4273646738625120315l_num1] :
( ( power_1002146276965246001l_num1 @ ( plus_p1441664204671982194l_num1 @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p1441664204671982194l_num1 @ ( plus_p1441664204671982194l_num1 @ ( power_1002146276965246001l_num1 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_1002146276965246001l_num1 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_2938166955517408246l_num1 @ ( times_2938166955517408246l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_641_power2__sum,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_642_power2__sum,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_643_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_644_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_645_power2__sum,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( power_6007165696250533058nnreal @ ( plus_p1859984266308609217nnreal @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p1859984266308609217nnreal @ ( plus_p1859984266308609217nnreal @ ( power_6007165696250533058nnreal @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_6007165696250533058nnreal @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_646_power2__sum,axiom,
! [X: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( power_7402600760894073284l_num1 @ ( plus_p2313304076027620419l_num1 @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p2313304076027620419l_num1 @ ( plus_p2313304076027620419l_num1 @ ( power_7402600760894073284l_num1 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7402600760894073284l_num1 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_8498157372700349887l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_647_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_648_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_649_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_650_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_651_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_652_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_653_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_654_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_655_Suc__0__div__numeral_I2_J,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_656_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_657_arith__geo__mean,axiom,
! [U: real,X: real,Y: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_658_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_659_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_660_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_661_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_662_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_663_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_664_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_665_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_666_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_667_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_668_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_669_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_670_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_671_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_672_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_673_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_674_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_675_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_676_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_677_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_678_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_679_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_680_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_681_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_682_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_683_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_684_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_685_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_686_zdiv__int,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zdiv_int
thf(fact_687_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q ) ) ).
% div_mult2_eq
thf(fact_688_not__exp__less__eq__0__int,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_689_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_690_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_691_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z3: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_692_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_693_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_694_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_695_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_696_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_697_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_698_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_699_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_700_Suc__div__le__mono,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_701_div__mult2__eq_H,axiom,
! [A: int,M: nat,N2: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_702_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N2: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_703_real__of__nat__div4,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_704_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_705_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_706_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_707_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_708_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_709_numeral__Bit0__div__2,axiom,
! [N2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N2 ) ) ).
% numeral_Bit0_div_2
thf(fact_710_Suc__double__not__eq__double,axiom,
! [M: nat,N2: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% Suc_double_not_eq_double
thf(fact_711_double__not__eq__Suc__double,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_712_field__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_713_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_714_exp__add__not__zero__imp__left,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_715_exp__add__not__zero__imp__left,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_716_exp__add__not__zero__imp__right,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_717_exp__add__not__zero__imp__right,axiom,
! [M: nat,N2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_718_inverse__of__nat__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( N2 != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_719_div__exp__eq,axiom,
! [A: nat,M: nat,N2: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% div_exp_eq
thf(fact_720_div__exp__eq,axiom,
! [A: int,M: nat,N2: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% div_exp_eq
thf(fact_721_two__realpow__ge__one,axiom,
! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% two_realpow_ge_one
thf(fact_722_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_723_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_724_sum__squares__bound,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_725_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_726_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_727_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_728_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_729_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_730_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_731_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_732_semiring__norm_I87_J,axiom,
! [M: num,N2: num] :
( ( ( bit0 @ M )
= ( bit0 @ N2 ) )
= ( M = N2 ) ) ).
% semiring_norm(87)
thf(fact_733_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_734_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_735_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_736_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_737_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_738_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_739_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_740_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_741_semiring__norm_I83_J,axiom,
! [N2: num] :
( one
!= ( bit0 @ N2 ) ) ).
% semiring_norm(83)
thf(fact_742_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_743_semiring__norm_I6_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(6)
thf(fact_744_semiring__norm_I13_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).
% semiring_norm(13)
thf(fact_745_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_746_semiring__norm_I12_J,axiom,
! [N2: num] :
( ( times_times_num @ one @ N2 )
= N2 ) ).
% semiring_norm(12)
thf(fact_747_semiring__norm_I71_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(71)
thf(fact_748_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_749_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_750_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_751_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_752_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_753_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_754_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_755_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_756_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_757_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_758_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_759_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_760_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_761_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_762_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_763_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_764_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_765_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_766_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_767_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_768_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_769_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_770_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_771_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_772_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_773_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_774_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_775_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_776_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_777_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_778_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_779_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_780_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_781_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_782_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_783_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_784_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_785_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_786_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_787_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_788_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_789_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_790_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_791_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_792_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_793_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_794_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_795_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_796_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_797_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_798_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_799_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_800_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_801_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_802_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_803_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_804_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_805_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_806_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_807_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_808_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_809_mult__zero__left,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_left
thf(fact_810_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_811_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_812_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_813_mult__zero__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ zero_z7100319975126383169nnreal )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_right
thf(fact_814_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_815_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_816_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_817_mult__eq__0__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
= ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% mult_eq_0_iff
thf(fact_818_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_819_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_820_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_821_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_822_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_823_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_824_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_825_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_826_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_827_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_828_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_829_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_830_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_831_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_832_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_833_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_834_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_835_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_836_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_837_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_838_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_839_int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% int_int_eq
thf(fact_840_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_841_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_842_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_843_mult__not__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
!= zero_z7100319975126383169nnreal )
=> ( ( A != zero_z7100319975126383169nnreal )
& ( B != zero_z7100319975126383169nnreal ) ) ) ).
% mult_not_zero
thf(fact_844_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_845_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_846_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_847_divisors__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% divisors_zero
thf(fact_848_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_849_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_850_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_851_no__zero__divisors,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A != zero_z7100319975126383169nnreal )
=> ( ( B != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ A @ B )
!= zero_z7100319975126383169nnreal ) ) ) ).
% no_zero_divisors
thf(fact_852_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_853_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_854_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_855_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_856_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_857_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_858_zero__neq__one,axiom,
zero_z5982384998485459395l_num1 != one_on3868389512446148991l_num1 ).
% zero_neq_one
thf(fact_859_zero__neq__one,axiom,
zero_z2241845390563828978l_num1 != one_on7795324986448017462l_num1 ).
% zero_neq_one
thf(fact_860_zero__neq__one,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_neq_one
thf(fact_861_zero__neq__one,axiom,
zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).
% zero_neq_one
thf(fact_862_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_863_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_864_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_865_zero__neq__one,axiom,
zero_z2744965634713055877_ereal != one_on4623092294121504201_ereal ).
% zero_neq_one
thf(fact_866_ring__class_Oring__distribs_I2_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ C ) @ ( times_2938166955517408246l_num1 @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_867_ring__class_Oring__distribs_I2_J,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ C )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ C ) @ ( times_8498157372700349887l_num1 @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_868_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_869_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_870_ring__class_Oring__distribs_I1_J,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ B @ C ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_2938166955517408246l_num1 @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_871_ring__class_Oring__distribs_I1_J,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( plus_p2313304076027620419l_num1 @ B @ C ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ B ) @ ( times_8498157372700349887l_num1 @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_872_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_873_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_874_comm__semiring__class_Odistrib,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_875_comm__semiring__class_Odistrib,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ C ) @ ( times_2938166955517408246l_num1 @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_876_comm__semiring__class_Odistrib,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ C )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ C ) @ ( times_8498157372700349887l_num1 @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_877_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_878_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_879_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_880_comm__semiring__class_Odistrib,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_881_distrib__left,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ ( times_7803423173614009249d_enat @ A @ C ) ) ) ).
% distrib_left
thf(fact_882_distrib__left,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ A @ ( plus_p1441664204671982194l_num1 @ B @ C ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ B ) @ ( times_2938166955517408246l_num1 @ A @ C ) ) ) ).
% distrib_left
thf(fact_883_distrib__left,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( plus_p2313304076027620419l_num1 @ B @ C ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ B ) @ ( times_8498157372700349887l_num1 @ A @ C ) ) ) ).
% distrib_left
thf(fact_884_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_885_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_886_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_887_distrib__left,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ ( plus_p1859984266308609217nnreal @ B @ C ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ ( times_1893300245718287421nnreal @ A @ C ) ) ) ).
% distrib_left
thf(fact_888_distrib__right,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% distrib_right
thf(fact_889_distrib__right,axiom,
! [A: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ C )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ C ) @ ( times_2938166955517408246l_num1 @ B @ C ) ) ) ).
% distrib_right
thf(fact_890_distrib__right,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ C )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ C ) @ ( times_8498157372700349887l_num1 @ B @ C ) ) ) ).
% distrib_right
thf(fact_891_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_892_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_893_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_894_distrib__right,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% distrib_right
thf(fact_895_combine__common__factor,axiom,
! [A: extended_enat,E: extended_enat,B: extended_enat,C: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ E ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ B @ E ) @ C ) )
= ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_896_combine__common__factor,axiom,
! [A: numera4273646738625120315l_num1,E: numera4273646738625120315l_num1,B: numera4273646738625120315l_num1,C: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ A @ E ) @ ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ B @ E ) @ C ) )
= ( plus_p1441664204671982194l_num1 @ ( times_2938166955517408246l_num1 @ ( plus_p1441664204671982194l_num1 @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_897_combine__common__factor,axiom,
! [A: numera2417102609627094330l_num1,E: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ A @ E ) @ ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ B @ E ) @ C ) )
= ( plus_p2313304076027620419l_num1 @ ( times_8498157372700349887l_num1 @ ( plus_p2313304076027620419l_num1 @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_898_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_899_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_900_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_901_combine__common__factor,axiom,
! [A: extend8495563244428889912nnreal,E: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ A @ E ) @ ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ B @ E ) @ C ) )
= ( plus_p1859984266308609217nnreal @ ( times_1893300245718287421nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_902_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_903_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_904_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_905_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_906_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_907_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_908_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_909_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_910_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_911_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_912_lambda__zero,axiom,
( ( ^ [H: extend8495563244428889912nnreal] : zero_z7100319975126383169nnreal )
= ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal ) ) ).
% lambda_zero
thf(fact_913_lambda__one,axiom,
( ( ^ [X3: numera2417102609627094330l_num1] : X3 )
= ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% lambda_one
thf(fact_914_lambda__one,axiom,
( ( ^ [X3: numera4273646738625120315l_num1] : X3 )
= ( times_2938166955517408246l_num1 @ one_on7795324986448017462l_num1 ) ) ).
% lambda_one
thf(fact_915_lambda__one,axiom,
( ( ^ [X3: extended_enat] : X3 )
= ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat ) ) ).
% lambda_one
thf(fact_916_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_917_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_918_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_919_lambda__one,axiom,
( ( ^ [X3: extend8495563244428889912nnreal] : X3 )
= ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal ) ) ).
% lambda_one
thf(fact_920_mult__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ C @ D )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_921_mult__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ C @ D )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_922_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_923_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_924_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_925_mult__mono_H,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ C @ D )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_926_mult__mono_H,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ C @ D )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_927_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_928_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_929_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_930_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_931_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_932_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_933_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_934_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_935_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_936_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_937_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_938_mult__left__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_939_mult__left__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ C @ A ) @ ( times_7803423173614009249d_enat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_940_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_941_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_942_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_943_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_944_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_945_mult__right__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_946_mult__right__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_947_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_948_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_949_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_950_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_951_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_952_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_953_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_954_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_955_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_956_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_957_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_958_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_959_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_960_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_961_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_962_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_963_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_964_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_965_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_966_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_967_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_968_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_969_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_970_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ C @ A ) @ ( times_7803423173614009249d_enat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_971_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_972_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_973_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_974_zero__less__one__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% zero_less_one_class.zero_le_one
thf(fact_975_zero__less__one__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one_class.zero_le_one
thf(fact_976_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_977_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_978_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_979_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_980_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_981_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_982_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_983_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_984_not__one__le__zero,axiom,
~ ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% not_one_le_zero
thf(fact_985_not__one__le__zero,axiom,
~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_le_zero
thf(fact_986_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_987_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_988_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_989_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_990_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_991_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_992_zadd__int__left,axiom,
! [M: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_993_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_994_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_995_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_996_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_997_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_998_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_999_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1000_mult__le__one,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ one_on7984719198319812577d_enat )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ one_on7984719198319812577d_enat )
=> ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ one_on7984719198319812577d_enat ) ) ) ) ).
% mult_le_one
thf(fact_1001_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1002_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1003_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1004_half__bounded__equal,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( times_times_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real )
= ( X
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% half_bounded_equal
thf(fact_1005_linear__plus__1__le__power,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).
% linear_plus_1_le_power
thf(fact_1006_triangle__lemma,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ Z ) ) ) ) ) ) ).
% triangle_lemma
thf(fact_1007_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_1008_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I3 ) @ one_one_nat )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I3 )
= zero_zero_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( X5 @ I3 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I3 )
= one_one_nat ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L2 @ X5 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I3 ) @ ( F @ X5 @ I3 ) ) )
& ! [X5: nat > real,I3: nat] :
( ( ( P @ X5 )
& ( Q2 @ I3 )
& ( ( L2 @ X5 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I3 ) @ ( X5 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1009_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1010_enat__ord__number_I1_J,axiom,
! [M: num,N2: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% enat_ord_number(1)
thf(fact_1011_set__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1012_set__bit__greater__eq,axiom,
! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).
% set_bit_greater_eq
thf(fact_1013_nat__add__1__add__1,axiom,
! [N2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_1014_power__le__one__iff,axiom,
! [A: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real )
= ( ( N2 = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1015_numeral__eq__of__nat,axiom,
( numera4658534427948366547nnreal
= ( ^ [A3: num] : ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ A3 ) ) ) ) ).
% numeral_eq_of_nat
thf(fact_1016_exhaust__2,axiom,
! [X: numera2417102609627094330l_num1] :
( ( X = one_on3868389512446148991l_num1 )
| ( X
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_1017_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X6: numera2417102609627094330l_num1] : ( P2 @ X6 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_1018_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_1019_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1020_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1021_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_1022_unset__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1023_unset__bit__less__eq,axiom,
! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_1024_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1025_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1026_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_1027_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_1028_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1029_int__ops_I3_J,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% int_ops(3)
thf(fact_1030_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_1031_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1032_int__plus,axiom,
! [N2: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1033_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_1034_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_1035_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_1036_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1037_real__inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ X ) @ one_one_real )
= ( ( X = one_one_real )
| ( X = zero_zero_real ) ) ) ) ) ).
% real_inverse_le_1_iff
thf(fact_1038_semiring__norm_I90_J,axiom,
! [M: num,N2: num] :
( ( ( bit1 @ M )
= ( bit1 @ N2 ) )
= ( M = N2 ) ) ).
% semiring_norm(90)
thf(fact_1039_semiring__norm_I88_J,axiom,
! [M: num,N2: num] :
( ( bit0 @ M )
!= ( bit1 @ N2 ) ) ).
% semiring_norm(88)
thf(fact_1040_semiring__norm_I89_J,axiom,
! [M: num,N2: num] :
( ( bit1 @ M )
!= ( bit0 @ N2 ) ) ).
% semiring_norm(89)
thf(fact_1041_semiring__norm_I84_J,axiom,
! [N2: num] :
( one
!= ( bit1 @ N2 ) ) ).
% semiring_norm(84)
thf(fact_1042_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_1043_semiring__norm_I73_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(73)
thf(fact_1044_semiring__norm_I9_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(9)
thf(fact_1045_semiring__norm_I7_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% semiring_norm(7)
thf(fact_1046_semiring__norm_I15_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).
% semiring_norm(15)
thf(fact_1047_semiring__norm_I14_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).
% semiring_norm(14)
thf(fact_1048_semiring__norm_I72_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(72)
thf(fact_1049_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1050_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_1051_semiring__norm_I3_J,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
= ( bit1 @ N2 ) ) ).
% semiring_norm(3)
thf(fact_1052_semiring__norm_I4_J,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).
% semiring_norm(4)
thf(fact_1053_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_1054_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_1055_semiring__norm_I10_J,axiom,
! [M: num,N2: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_1056_semiring__norm_I16_J,axiom,
! [M: num,N2: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_1057_Suc__0__div__numeral_I3_J,axiom,
! [N2: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(3)
thf(fact_1058_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1059_div__Suc__eq__div__add3,axiom,
! [M: nat,N2: nat] :
( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_1060_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1061_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1062_exhaust__4,axiom,
! [X: numera4273646738625120315l_num1] :
( ( X = one_on7795324986448017462l_num1 )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_1063_forall__4,axiom,
( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
! [X6: numera4273646738625120315l_num1] : ( P2 @ X6 ) )
= ( ^ [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_1064_exhaust__3,axiom,
! [X: numera6367994245245682809l_num1] :
( ( X = one_on7819281148064737470l_num1 )
| ( X
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_1065_forall__3,axiom,
( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
! [X6: numera6367994245245682809l_num1] : ( P2 @ X6 ) )
= ( ^ [P3: numera6367994245245682809l_num1 > $o] :
( ( P3 @ one_on7819281148064737470l_num1 )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_1066_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1067_eval__nat__numeral_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1068_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1069_Suc3__eq__add__3,axiom,
! [N2: nat] :
( ( suc @ ( suc @ ( suc @ N2 ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).
% Suc3_eq_add_3
thf(fact_1070_Suc__div__eq__add3__div,axiom,
! [M: nat,N2: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% Suc_div_eq_add3_div
thf(fact_1071_neq__4k1__k43,axiom,
! [M: nat,N2: nat,M3: 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 ) ) @ N2 ) ) )
!= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( 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_1072_suminf__half__series__ereal,axiom,
( ( suminf4411151127299490740_ereal
@ ^ [N: nat] : ( power_1054015426188190660_ereal @ ( divide8893690120176169980_ereal @ one_on4623092294121504201_ereal @ ( numera1204434989813589363_ereal @ ( bit0 @ one ) ) ) @ ( suc @ N ) ) )
= one_on4623092294121504201_ereal ) ).
% suminf_half_series_ereal
thf(fact_1073_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_1074_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1075_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1076_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_1077_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1078_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1079_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1080_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1081_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1082_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1083_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1084_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1085_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1086_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1087_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1088_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1089_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1090_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1091_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1092_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1093_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1094_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1095_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1096_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1097_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1098_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1099_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1100_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1101_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1102_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1103_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1104_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1105_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1106_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1107_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1108_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1109_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1110_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_1111_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1112_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1113_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1114_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1115_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1116_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1117_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1118_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1119_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1120_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1121_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1122_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1123_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1124_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1125_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_1126_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_1127_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1128_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1129_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_1130_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_1131_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_1132_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1133_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( M5 != N ) ) ) ) ).
% nat_less_le
thf(fact_1134_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1135_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1136_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1137_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1138_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_1139_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1140_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1141_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1142_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1143_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1144_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1145_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1146_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1147_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1148_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1149_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1150_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1151_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1152_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_1153_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1154_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_1155_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1156_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1157_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1158_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1159_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1160_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_1161_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
thf(fact_1162_mult__less__mono1,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 @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1163_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1164_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1165_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1166_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1167_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1168_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1169_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1170_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1171_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1172_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1173_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1174_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1175_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_1176_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1177_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1178_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N2 )
= M )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1179_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1180_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N2 )
= ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1181_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1182_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1183_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_1184_kuhn__lemma,axiom,
! [P4: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P4 )
=> ( ! [X4: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I3 ) @ P4 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( Label @ X4 @ I2 )
= zero_zero_nat )
| ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I3 ) @ P4 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X4 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I3 ) @ P4 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X4 @ I2 )
= P4 )
=> ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_nat @ ( Q3 @ I3 ) @ P4 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ? [R3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( R3 @ J4 ) )
& ( ord_less_eq_nat @ ( R3 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( S3 @ J4 ) )
& ( ord_less_eq_nat @ ( S3 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I3 )
!= ( Label @ S3 @ I3 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1185_less__exp,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% less_exp
thf(fact_1186_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1187_div__nat__eqI,axiom,
! [N2: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N2 )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1188_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1189_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1190_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N2 )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_div
thf(fact_1191_suminf__ereal__offset__le,axiom,
! [F: nat > extended_ereal,K: nat] :
( ! [I2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ I2 ) )
=> ( ord_le1083603963089353582_ereal
@ ( suminf4411151127299490740_ereal
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ K ) ) )
@ ( suminf4411151127299490740_ereal @ F ) ) ) ).
% suminf_ereal_offset_le
thf(fact_1192_less__2__cases,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N2 = zero_zero_nat )
| ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1193_less__2__cases__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N2 = zero_zero_nat )
| ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1194_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1195_split__div_H,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1196_nat__bit__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_1197_div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_1198_Suc__n__div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_1199_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_1200_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_1201_log__induct,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
=> ( ( P @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( P @ N3 ) ) )
=> ( P @ N2 ) ) ) ) ).
% log_induct
thf(fact_1202_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1203_semiring__norm_I78_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(78)
thf(fact_1204_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1205_semiring__norm_I80_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(80)
thf(fact_1206_unset__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_1207_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1208_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1209_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1210_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1211_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1212_semiring__norm_I81_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(81)
thf(fact_1213_semiring__norm_I77_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).
% semiring_norm(77)
thf(fact_1214_enat__ord__number_I2_J,axiom,
! [M: num,N2: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% enat_ord_number(2)
thf(fact_1215_one__less__numeral,axiom,
! [N2: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N2 ) )
= ( ord_less_num @ one @ N2 ) ) ).
% one_less_numeral
thf(fact_1216_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_1217_numeral__less__real__of__nat__iff,axiom,
! [W: num,N2: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_1218_semiring__norm_I74_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(74)
thf(fact_1219_semiring__norm_I79_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(79)
thf(fact_1220_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1221_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1222_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_real @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_1223_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_1224_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1225_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1226_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_1227_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1228_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1229_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1230_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1231_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1232_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1233_realpow__pos__nth2,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N2 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1234_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1235_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1236_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1237_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( M = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1238_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1239_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1240_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1241_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1242_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1243_zdiv__mono2,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1244_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1245_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1246_zdiv__mono2__neg,axiom,
! [A: int,B3: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1247_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1248_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1249_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1250_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1251_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1252_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1253_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1254_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1255_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1256_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N2 )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N2 )
= A ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1257_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1258_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1259_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1260_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1261_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1262_int__div__neg__eq,axiom,
! [A: int,B: int,Q: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_neg_eq
thf(fact_1263_int__div__pos__eq,axiom,
! [A: int,B: int,Q: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q ) ) ) ) ).
% int_div_pos_eq
thf(fact_1264_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K3 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1265_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1266_r01__binary__expression__correct,axiom,
! [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( ord_less_real @ R2 @ one_one_real )
=> ( R2
= ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( r01_binary_expansion @ R2 @ N ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) ) ) ) ) ) ) ).
% r01_binary_expression_correct
thf(fact_1267_ennreal__suminf__lessD,axiom,
! [F: nat > extend8495563244428889912nnreal,X: extend8495563244428889912nnreal,I: nat] :
( ( ord_le7381754540660121996nnreal @ ( suminf7725996343205245138nnreal @ F ) @ X )
=> ( ord_le7381754540660121996nnreal @ ( F @ I ) @ X ) ) ).
% ennreal_suminf_lessD
% 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,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_real
@ ( suminf_real
@ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( a @ ( plus_plus_nat @ N @ ( suc @ i ) ) ) ) @ ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ ( suc @ N ) @ ( suc @ i ) ) ) ) )
@ ( suminf_real
@ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ ( suc @ N ) @ ( suc @ i ) ) ) ) ) ).
%------------------------------------------------------------------------------