TPTP Problem File: SLH0151^1.p
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- 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 : Median_Method/0000_Median/prob_00409_015447__14862714_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1389 ( 557 unt; 119 typ; 0 def)
% Number of atoms : 3788 (1222 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10275 ( 517 ~; 149 |; 199 &;7829 @)
% ( 0 <=>;1581 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 345 ( 345 >; 0 *; 0 +; 0 <<)
% Number of symbols : 103 ( 100 usr; 18 con; 0-3 aty)
% Number of variables : 3374 ( 259 ^;3092 !; 23 ?;3374 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:46:13.600
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $tType ).
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2417102609627094330l_num1: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
set_Extended_enat: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $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 ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (100)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Discrete_Olog,type,
log: nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
minus_838314146864362899l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
minus_minus_set_num: set_num > set_num > set_num ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__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__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__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nonnegative____Real__Oennreal,type,
numera4658534427948366547nnreal: num > extend8495563244428889912nnreal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2161328050825114965l_num1: num > numera2417102609627094330l_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_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Parity_Oadjust__mod,type,
adjust_mod: num > int > int ).
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_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Extended____Nat__Oenat,type,
set_or4374356025156299511d_enat: extended_enat > extended_enat > set_Extended_enat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Extended____Nonnegative____Real__Oennreal,type,
set_or2452457671691799911nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
set_or1222409239386451017an_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Extended____Nat__Oenat,type,
member_Extended_enat: extended_enat > set_Extended_enat > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
thf(sy_v_J____,type,
j: set_nat ).
thf(sy_v_v____,type,
v: nat ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1266)
thf(fact_0__092_060open_062card_AJ_A_092_060le_062_Acard_A_1230_O_O_060length_Axs_Adiv_A2_125_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_nat @ j ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% \<open>card J \<le> card {0..<length xs div 2}\<close>
thf(fact_1_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_2_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_3_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_4_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_5_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_6_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_7_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ S )
& ? [Y: nat] :
( ( member_nat @ Y @ S )
& ( X != Y )
& ! [Z: nat] :
( ( member_nat @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_8_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_9_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_10_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_11_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_12_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_13_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_14_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_15_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_16_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_17_v__1,axiom,
ord_less_nat @ v @ ( size_size_list_a @ xs ) ).
% v_1
thf(fact_18_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_19_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_20_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_21_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_22_order__refl,axiom,
! [X3: a] : ( ord_less_eq_a @ X3 @ X3 ) ).
% order_refl
thf(fact_23_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_24_order__refl,axiom,
! [X3: extended_enat] : ( ord_le2932123472753598470d_enat @ X3 @ X3 ) ).
% order_refl
thf(fact_25_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_26_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_31_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_32_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ M ) @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_33_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_34_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_35_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_36_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_37_ivl__subset,axiom,
! [I: num,J: num,M: num,N: num] :
( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M @ N ) )
= ( ( ord_less_eq_num @ J @ I )
| ( ( ord_less_eq_num @ M @ I )
& ( ord_less_eq_num @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_38_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_39_ivl__subset,axiom,
! [I: a,J: a,M: a,N: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ I @ J ) @ ( set_or5139330845457685135Than_a @ M @ N ) )
= ( ( ord_less_eq_a @ J @ I )
| ( ( ord_less_eq_a @ M @ I )
& ( ord_less_eq_a @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_40_ivl__subset,axiom,
! [I: real,J: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J @ I )
| ( ( ord_less_eq_real @ M @ I )
& ( ord_less_eq_real @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_41_ivl__subset,axiom,
! [I: extended_enat,J: extended_enat,M: extended_enat,N: extended_enat] :
( ( ord_le7203529160286727270d_enat @ ( set_or4374356025156299511d_enat @ I @ J ) @ ( set_or4374356025156299511d_enat @ M @ N ) )
= ( ( ord_le2932123472753598470d_enat @ J @ I )
| ( ( ord_le2932123472753598470d_enat @ M @ I )
& ( ord_le2932123472753598470d_enat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_42_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_43_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_44_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_45_atLeastLessThan__iff,axiom,
! [I: extend8495563244428889912nnreal,L: extend8495563244428889912nnreal,U: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I @ ( set_or2452457671691799911nnreal @ L @ U ) )
= ( ( ord_le3935885782089961368nnreal @ L @ I )
& ( ord_le7381754540660121996nnreal @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_46_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_47_atLeastLessThan__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_num @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_48_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_49_atLeastLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member_a @ I @ ( set_or5139330845457685135Than_a @ L @ U ) )
= ( ( ord_less_eq_a @ L @ I )
& ( ord_less_a @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_50_atLeastLessThan__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_real @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_51_atLeastLessThan__iff,axiom,
! [I: extended_enat,L: extended_enat,U: extended_enat] :
( ( member_Extended_enat @ I @ ( set_or4374356025156299511d_enat @ L @ U ) )
= ( ( ord_le2932123472753598470d_enat @ L @ I )
& ( ord_le72135733267957522d_enat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_52_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_53_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_54_assms_I3_J,axiom,
sorted_wrt_a @ ord_less_eq_a @ xs ).
% assms(3)
thf(fact_55_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_56_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_57_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_58_verit__comp__simplify1_I1_J,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_59_verit__comp__simplify1_I1_J,axiom,
! [A: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_60_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_61_atLeastLessThan__inj_I2_J,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_62_atLeastLessThan__inj_I2_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ( set_or4374356025156299511d_enat @ A @ B )
= ( set_or4374356025156299511d_enat @ C @ D ) )
=> ( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_63_atLeastLessThan__inj_I2_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ( set_or2452457671691799911nnreal @ A @ B )
= ( set_or2452457671691799911nnreal @ C @ D ) )
=> ( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_64_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_65_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_66_atLeastLessThan__inj_I1_J,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_67_atLeastLessThan__inj_I1_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ( set_or4374356025156299511d_enat @ A @ B )
= ( set_or4374356025156299511d_enat @ C @ D ) )
=> ( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_68_atLeastLessThan__inj_I1_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ( set_or2452457671691799911nnreal @ A @ B )
= ( set_or2452457671691799911nnreal @ C @ D ) )
=> ( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_69_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_70_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_71_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_72_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_73_less__numeral__extra_I3_J,axiom,
~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).
% less_numeral_extra(3)
thf(fact_74_less__numeral__extra_I3_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(3)
thf(fact_75_lt__ex,axiom,
! [X3: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X3 ) ).
% lt_ex
thf(fact_76_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_77_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_78_dense,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ? [Z2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z2 )
& ( ord_le7381754540660121996nnreal @ Z2 @ Y4 ) ) ) ).
% dense
thf(fact_79_less__imp__neq,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_80_less__imp__neq,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_81_less__imp__neq,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_82_less__imp__neq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_83_less__imp__neq,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% less_imp_neq
thf(fact_84_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_85_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_86_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_87_order_Oasym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).
% order.asym
thf(fact_88_order_Oasym,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ~ ( ord_le7381754540660121996nnreal @ B @ A ) ) ).
% order.asym
thf(fact_89_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_90_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_91_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_92_ord__eq__less__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_93_ord__eq__less__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A = B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_94_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_95_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_96_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_97_ord__less__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_98_ord__less__eq__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( B = C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_99_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_100_less__induct,axiom,
! [P: extended_enat > $o,A: extended_enat] :
( ! [X4: extended_enat] :
( ! [Y5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_101_antisym__conv3,axiom,
! [Y4: nat,X3: nat] :
( ~ ( ord_less_nat @ Y4 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_102_antisym__conv3,axiom,
! [Y4: int,X3: int] :
( ~ ( ord_less_int @ Y4 @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_103_antisym__conv3,axiom,
! [Y4: num,X3: num] :
( ~ ( ord_less_num @ Y4 @ X3 )
=> ( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_104_antisym__conv3,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_105_antisym__conv3,axiom,
! [Y4: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ Y4 @ X3 )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_106_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_107_Ico__eq__Ico,axiom,
! [L: num,H: num,L2: num,H2: num] :
( ( ( set_or1222409239386451017an_num @ L @ H )
= ( set_or1222409239386451017an_num @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_num @ L @ H )
& ~ ( ord_less_num @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_108_Ico__eq__Ico,axiom,
! [L: extended_enat,H: extended_enat,L2: extended_enat,H2: extended_enat] :
( ( ( set_or4374356025156299511d_enat @ L @ H )
= ( set_or4374356025156299511d_enat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le72135733267957522d_enat @ L @ H )
& ~ ( ord_le72135733267957522d_enat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_109_Ico__eq__Ico,axiom,
! [L: extend8495563244428889912nnreal,H: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal,H2: extend8495563244428889912nnreal] :
( ( ( set_or2452457671691799911nnreal @ L @ H )
= ( set_or2452457671691799911nnreal @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le7381754540660121996nnreal @ L @ H )
& ~ ( ord_le7381754540660121996nnreal @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_110_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_111_linorder__cases,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_112_linorder__cases,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_113_linorder__cases,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less_num @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_114_linorder__cases,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_115_linorder__cases,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ) ).
% linorder_cases
thf(fact_116_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_117_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_118_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_119_dual__order_Oasym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ).
% dual_order.asym
thf(fact_120_dual__order_Oasym,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ~ ( ord_le7381754540660121996nnreal @ A @ B ) ) ).
% dual_order.asym
thf(fact_121_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_122_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_123_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_124_dual__order_Oirrefl,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% dual_order.irrefl
thf(fact_125_dual__order_Oirrefl,axiom,
! [A: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A @ A ) ).
% dual_order.irrefl
thf(fact_126_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_127_exists__least__iff,axiom,
( ( ^ [P2: extended_enat > $o] :
? [X5: extended_enat] : ( P2 @ X5 ) )
= ( ^ [P3: extended_enat > $o] :
? [N2: extended_enat] :
( ( P3 @ N2 )
& ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_128_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_129_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int] : ( P @ A2 @ A2 )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_130_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num] : ( P @ A2 @ A2 )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_131_linorder__less__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A2: extended_enat,B2: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: extended_enat] : ( P @ A2 @ A2 )
=> ( ! [A2: extended_enat,B2: extended_enat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_132_linorder__less__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: extend8495563244428889912nnreal] : ( P @ A2 @ A2 )
=> ( ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_133_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_134_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_135_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_136_order_Ostrict__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_137_order_Ostrict__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_138_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_139_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y4: int] :
( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_140_not__less__iff__gr__or__eq,axiom,
! [X3: num,Y4: num] :
( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( ( ord_less_num @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_141_not__less__iff__gr__or__eq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( ( ord_le72135733267957522d_enat @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_142_not__less__iff__gr__or__eq,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 ) )
= ( ( ord_le7381754540660121996nnreal @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_143_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_144_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_145_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_146_dual__order_Ostrict__trans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_147_dual__order_Ostrict__trans,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ( ord_le7381754540660121996nnreal @ C @ B )
=> ( ord_le7381754540660121996nnreal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_148_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_149_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_150_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_151_order_Ostrict__implies__not__eq,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_152_order_Ostrict__implies__not__eq,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_153_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_154_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_155_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_156_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_157_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_160_linorder__neqE,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_161_linorder__neqE,axiom,
! [X3: int,Y4: int] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_162_linorder__neqE,axiom,
! [X3: num,Y4: num] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ord_less_num @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_163_linorder__neqE,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( X3 != Y4 )
=> ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_164_linorder__neqE,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( X3 != Y4 )
=> ( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_165_order__less__asym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_166_order__less__asym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_167_order__less__asym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_168_order__less__asym,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_169_order__less__asym,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ~ ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ).
% order_less_asym
thf(fact_170_linorder__neq__iff,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
= ( ( ord_less_nat @ X3 @ Y4 )
| ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_171_linorder__neq__iff,axiom,
! [X3: int,Y4: int] :
( ( X3 != Y4 )
= ( ( ord_less_int @ X3 @ Y4 )
| ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_172_linorder__neq__iff,axiom,
! [X3: num,Y4: num] :
( ( X3 != Y4 )
= ( ( ord_less_num @ X3 @ Y4 )
| ( ord_less_num @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_173_linorder__neq__iff,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( X3 != Y4 )
= ( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
| ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_174_linorder__neq__iff,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( X3 != Y4 )
= ( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
| ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_175_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_176_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_177_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_178_order__less__asym_H,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).
% order_less_asym'
thf(fact_179_order__less__asym_H,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ~ ( ord_le7381754540660121996nnreal @ B @ A ) ) ).
% order_less_asym'
thf(fact_180_order__less__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_181_order__less__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_182_order__less__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_183_order__less__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ Y4 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_184_order__less__trans,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ( ord_le7381754540660121996nnreal @ Y4 @ Z3 )
=> ( ord_le7381754540660121996nnreal @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_185_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_186_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_187_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_188_ord__eq__less__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_189_ord__eq__less__subst,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_190_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_191_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_192_ord__eq__less__subst,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_193_ord__eq__less__subst,axiom,
! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_194_ord__eq__less__subst,axiom,
! [A: extend8495563244428889912nnreal,F: int > extend8495563244428889912nnreal,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_195_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_196_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_197_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_198_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_199_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_200_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_201_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_202_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_203_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > extended_enat,C: extended_enat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_204_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_205_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_206_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_207_order__less__irrefl,axiom,
! [X3: num] :
~ ( ord_less_num @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_208_order__less__irrefl,axiom,
! [X3: extended_enat] :
~ ( ord_le72135733267957522d_enat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_209_order__less__irrefl,axiom,
! [X3: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_210_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_211_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_212_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_213_order__less__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_214_order__less__subst1,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_215_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_216_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_217_order__less__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_218_order__less__subst1,axiom,
! [A: int,F: extended_enat > int,B: extended_enat,C: extended_enat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_219_order__less__subst1,axiom,
! [A: int,F: extend8495563244428889912nnreal > int,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_220_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_221_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_222_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_223_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_224_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_225_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_226_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_227_order__less__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_228_order__less__subst2,axiom,
! [A: int,B: int,F: int > extended_enat,C: extended_enat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_229_order__less__subst2,axiom,
! [A: int,B: int,F: int > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_230_order__less__not__sym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_231_order__less__not__sym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_232_order__less__not__sym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_233_order__less__not__sym,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_234_order__less__not__sym,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ~ ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ).
% order_less_not_sym
thf(fact_235_order__less__imp__triv,axiom,
! [X3: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_236_order__less__imp__triv,axiom,
! [X3: int,Y4: int,P: $o] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_int @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_237_order__less__imp__triv,axiom,
! [X3: num,Y4: num,P: $o] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_num @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_238_order__less__imp__triv,axiom,
! [X3: extended_enat,Y4: extended_enat,P: $o] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_239_order__less__imp__triv,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,P: $o] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ( ord_le7381754540660121996nnreal @ Y4 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_240_linorder__less__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_nat @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_241_linorder__less__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_int @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_242_linorder__less__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less_num @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_243_linorder__less__linear,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_244_linorder__less__linear,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ).
% linorder_less_linear
thf(fact_245_order__less__imp__not__eq,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_246_order__less__imp__not__eq,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_247_order__less__imp__not__eq,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_248_order__less__imp__not__eq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_249_order__less__imp__not__eq,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_250_order__less__imp__not__eq2,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_251_order__less__imp__not__eq2,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_252_order__less__imp__not__eq2,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_253_order__less__imp__not__eq2,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_254_order__less__imp__not__eq2,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_255_order__less__imp__not__less,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_256_order__less__imp__not__less,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_257_order__less__imp__not__less,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_258_order__less__imp__not__less,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ~ ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_259_order__less__imp__not__less,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ~ ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_260_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_261_atLeastLessThan__eq__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D )
=> ( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_262_atLeastLessThan__eq__iff,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ C @ D )
=> ( ( ( set_or4374356025156299511d_enat @ A @ B )
= ( set_or4374356025156299511d_enat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_263_atLeastLessThan__eq__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ C @ D )
=> ( ( ( set_or2452457671691799911nnreal @ A @ B )
= ( set_or2452457671691799911nnreal @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_264_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_265_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le7381754540660121996nnreal @ ( numera4658534427948366547nnreal @ N ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_less_zero
thf(fact_266_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_less_zero
thf(fact_267_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_268_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_269_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_270_zero__less__numeral,axiom,
! [N: num] : ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N ) ) ).
% zero_less_numeral
thf(fact_271_zero__less__numeral,axiom,
! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_less_numeral
thf(fact_272_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_273_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_274_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_275_order__le__imp__less__or__eq,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_276_order__le__imp__less__or__eq,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_277_order__le__imp__less__or__eq,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_num @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_278_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_279_order__le__imp__less__or__eq,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_280_order__le__imp__less__or__eq,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_281_order__le__imp__less__or__eq,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_282_order__le__imp__less__or__eq,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_283_linorder__le__less__linear,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
| ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_284_linorder__le__less__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
| ( ord_less_num @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_285_linorder__le__less__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_nat @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_286_linorder__le__less__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
| ( ord_less_int @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_287_linorder__le__less__linear,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
| ( ord_less_a @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_288_linorder__le__less__linear,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
| ( ord_less_real @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_289_linorder__le__less__linear,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
| ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_290_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_291_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_292_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_num @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_293_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_294_order__less__le__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_295_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_296_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_297_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_298_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_299_order__less__le__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > num,C: num] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_300_order__less__le__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: num > extend8495563244428889912nnreal,B: num,C: num] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_301_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_302_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_303_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_304_order__less__le__subst1,axiom,
! [A: a,F: num > a,B: num,C: num] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_305_order__less__le__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_306_order__less__le__subst1,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_307_order__less__le__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_308_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_309_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_310_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_311_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_312_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_313_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_314_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_315_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_316_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_317_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_318_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_319_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_320_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_321_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: int > extend8495563244428889912nnreal,B: int,C: int] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_322_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: num > extend8495563244428889912nnreal,B: num,C: num] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_323_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extended_enat > extend8495563244428889912nnreal,B: extended_enat,C: extended_enat] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_324_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_325_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_326_order__le__less__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_327_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_328_order__le__less__subst1,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_329_order__le__less__subst1,axiom,
! [A: num,F: extend8495563244428889912nnreal > num,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_330_order__less__le__trans,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ( ord_le3935885782089961368nnreal @ Y4 @ Z3 )
=> ( ord_le7381754540660121996nnreal @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_331_order__less__le__trans,axiom,
! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z3 )
=> ( ord_less_set_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_332_order__less__le__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_333_order__less__le__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_334_order__less__le__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_335_order__less__le__trans,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ord_less_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ Y4 @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_336_order__less__le__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_337_order__less__le__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le2932123472753598470d_enat @ Y4 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_338_order__le__less__trans,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ( ord_le7381754540660121996nnreal @ Y4 @ Z3 )
=> ( ord_le7381754540660121996nnreal @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_339_order__le__less__trans,axiom,
! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_set_nat @ Y4 @ Z3 )
=> ( ord_less_set_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_340_order__le__less__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z3 )
=> ( ord_less_num @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_341_order__le__less__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_342_order__le__less__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_343_order__le__less__trans,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_a @ Y4 @ Z3 )
=> ( ord_less_a @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_344_order__le__less__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_345_order__le__less__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ord_le72135733267957522d_enat @ Y4 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_346_order__neq__le__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A != B )
=> ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le7381754540660121996nnreal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_347_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_348_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_349_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_350_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_351_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_352_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_353_order__neq__le__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A != B )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_354_order__le__neq__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( A != B )
=> ( ord_le7381754540660121996nnreal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_355_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_356_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_357_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_358_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_359_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_360_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_361_order__le__neq__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( A != B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_362_order__less__imp__le,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_363_order__less__imp__le,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_364_order__less__imp__le,axiom,
! [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_365_order__less__imp__le,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_366_order__less__imp__le,axiom,
! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_367_order__less__imp__le,axiom,
! [X3: a,Y4: a] :
( ( ord_less_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_368_order__less__imp__le,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_369_order__less__imp__le,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ X3 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_370_linorder__not__less,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 ) )
= ( ord_le3935885782089961368nnreal @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_371_linorder__not__less,axiom,
! [X3: num,Y4: num] :
( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_372_linorder__not__less,axiom,
! [X3: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_373_linorder__not__less,axiom,
! [X3: int,Y4: int] :
( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_374_linorder__not__less,axiom,
! [X3: a,Y4: a] :
( ( ~ ( ord_less_a @ X3 @ Y4 ) )
= ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_375_linorder__not__less,axiom,
! [X3: real,Y4: real] :
( ( ~ ( ord_less_real @ X3 @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_376_linorder__not__less,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) ) ).
% linorder_not_less
thf(fact_377_linorder__not__le,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ X3 @ Y4 ) )
= ( ord_le7381754540660121996nnreal @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_378_linorder__not__le,axiom,
! [X3: num,Y4: num] :
( ( ~ ( ord_less_eq_num @ X3 @ Y4 ) )
= ( ord_less_num @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_379_linorder__not__le,axiom,
! [X3: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y4 ) )
= ( ord_less_nat @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_380_linorder__not__le,axiom,
! [X3: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y4 ) )
= ( ord_less_int @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_381_linorder__not__le,axiom,
! [X3: a,Y4: a] :
( ( ~ ( ord_less_eq_a @ X3 @ Y4 ) )
= ( ord_less_a @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_382_linorder__not__le,axiom,
! [X3: real,Y4: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y4 ) )
= ( ord_less_real @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_383_linorder__not__le,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ X3 @ Y4 ) )
= ( ord_le72135733267957522d_enat @ Y4 @ X3 ) ) ).
% linorder_not_le
thf(fact_384_order__less__le,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_385_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_386_order__less__le,axiom,
( ord_less_num
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_387_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_388_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_389_order__less__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_390_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_391_order__less__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_392_order__le__less,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_393_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_394_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_395_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_396_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_397_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X: a,Y: a] :
( ( ord_less_a @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_398_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_399_order__le__less,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_400_dual__order_Ostrict__implies__order,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le3935885782089961368nnreal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_401_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_402_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_403_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_404_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_405_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_406_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_407_dual__order_Ostrict__implies__order,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_408_order_Ostrict__implies__order,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_409_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_410_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_411_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_412_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_413_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_414_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_415_order_Ostrict__implies__order,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_416_dual__order_Ostrict__iff__not,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B3: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A4 )
& ~ ( ord_le3935885782089961368nnreal @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_417_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_418_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_419_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_420_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_421_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ~ ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_422_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_423_dual__order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ~ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_424_dual__order_Ostrict__trans2,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le7381754540660121996nnreal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_425_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_426_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_427_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_428_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_429_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_430_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_431_dual__order_Ostrict__trans2,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_432_dual__order_Ostrict__trans1,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le7381754540660121996nnreal @ C @ B )
=> ( ord_le7381754540660121996nnreal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_433_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_434_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_435_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_436_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_437_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_438_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_439_dual__order_Ostrict__trans1,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_440_dual__order_Ostrict__iff__order,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B3: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_441_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_442_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_443_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_444_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_445_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_446_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_447_dual__order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_448_dual__order_Oorder__iff__strict,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B3: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_449_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_450_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_num @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_451_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_452_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_453_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B3: a,A4: a] :
( ( ord_less_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_454_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_455_dual__order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le72135733267957522d_enat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_456_dense__le__bounded,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ! [W: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ W )
=> ( ( ord_le7381754540660121996nnreal @ W @ Y4 )
=> ( ord_le3935885782089961368nnreal @ W @ Z3 ) ) )
=> ( ord_le3935885782089961368nnreal @ Y4 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_457_dense__le__bounded,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y4 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_458_dense__ge__bounded,axiom,
! [Z3: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z3 @ X3 )
=> ( ! [W: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z3 @ W )
=> ( ( ord_le7381754540660121996nnreal @ W @ X3 )
=> ( ord_le3935885782089961368nnreal @ Y4 @ W ) ) )
=> ( ord_le3935885782089961368nnreal @ Y4 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_459_dense__ge__bounded,axiom,
! [Z3: real,X3: real,Y4: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y4 @ W ) ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_460_order_Ostrict__iff__not,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B3 )
& ~ ( ord_le3935885782089961368nnreal @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_461_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_462_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_463_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_464_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_465_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ~ ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_466_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_467_order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ~ ( ord_le2932123472753598470d_enat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_468_order_Ostrict__trans2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_469_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_470_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_471_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_472_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_473_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_474_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_475_order_Ostrict__trans2,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_476_order_Ostrict__trans1,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_477_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_478_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_479_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_480_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_481_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_482_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_483_order_Ostrict__trans1,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_484_order_Ostrict__iff__order,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_485_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_486_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_487_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_488_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_489_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_490_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_491_order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_492_order_Oorder__iff__strict,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_493_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_494_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_495_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_496_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_497_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A4: a,B3: a] :
( ( ord_less_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_498_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_499_order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le72135733267957522d_enat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_500_not__le__imp__less,axiom,
! [Y4: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ Y4 @ X3 )
=> ( ord_le7381754540660121996nnreal @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_501_not__le__imp__less,axiom,
! [Y4: num,X3: num] :
( ~ ( ord_less_eq_num @ Y4 @ X3 )
=> ( ord_less_num @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_502_not__le__imp__less,axiom,
! [Y4: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ord_less_nat @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_503_not__le__imp__less,axiom,
! [Y4: int,X3: int] :
( ~ ( ord_less_eq_int @ Y4 @ X3 )
=> ( ord_less_int @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_504_not__le__imp__less,axiom,
! [Y4: a,X3: a] :
( ~ ( ord_less_eq_a @ Y4 @ X3 )
=> ( ord_less_a @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_505_not__le__imp__less,axiom,
! [Y4: real,X3: real] :
( ~ ( ord_less_eq_real @ Y4 @ X3 )
=> ( ord_less_real @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_506_not__le__imp__less,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ( ord_le72135733267957522d_enat @ X3 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_507_less__le__not__le,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
& ~ ( ord_le3935885782089961368nnreal @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_508_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_509_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_510_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_511_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_512_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ~ ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_513_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_514_less__le__not__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
& ~ ( ord_le2932123472753598470d_enat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_515_dense__le,axiom,
! [Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ! [X4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ X4 @ Z3 ) )
=> ( ord_le3935885782089961368nnreal @ Y4 @ Z3 ) ) ).
% dense_le
thf(fact_516_dense__le,axiom,
! [Y4: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ).
% dense_le
thf(fact_517_dense__ge,axiom,
! [Z3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ! [X4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z3 @ X4 )
=> ( ord_le3935885782089961368nnreal @ Y4 @ X4 ) )
=> ( ord_le3935885782089961368nnreal @ Y4 @ Z3 ) ) ).
% dense_ge
thf(fact_518_dense__ge,axiom,
! [Z3: real,Y4: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y4 @ X4 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ).
% dense_ge
thf(fact_519_antisym__conv2,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_520_antisym__conv2,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ~ ( ord_less_set_nat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_521_antisym__conv2,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ~ ( ord_less_num @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_522_antisym__conv2,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_523_antisym__conv2,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ~ ( ord_less_int @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_524_antisym__conv2,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ~ ( ord_less_a @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_525_antisym__conv2,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ~ ( ord_less_real @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_526_antisym__conv2,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_527_antisym__conv1,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_528_antisym__conv1,axiom,
! [X3: set_nat,Y4: set_nat] :
( ~ ( ord_less_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_529_antisym__conv1,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_530_antisym__conv1,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_531_antisym__conv1,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_532_antisym__conv1,axiom,
! [X3: a,Y4: a] :
( ~ ( ord_less_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_533_antisym__conv1,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_534_antisym__conv1,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_535_nless__le,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ A @ B ) )
= ( ~ ( ord_le3935885782089961368nnreal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_536_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_537_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_538_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_539_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_540_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_541_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_542_nless__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
= ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_543_leI,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ Y4 @ X3 ) ) ).
% leI
thf(fact_544_leI,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% leI
thf(fact_545_leI,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% leI
thf(fact_546_leI,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% leI
thf(fact_547_leI,axiom,
! [X3: a,Y4: a] :
( ~ ( ord_less_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% leI
thf(fact_548_leI,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% leI
thf(fact_549_leI,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) ) ).
% leI
thf(fact_550_leD,axiom,
! [Y4: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y4 @ X3 )
=> ~ ( ord_le7381754540660121996nnreal @ X3 @ Y4 ) ) ).
% leD
thf(fact_551_leD,axiom,
! [Y4: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ~ ( ord_less_set_nat @ X3 @ Y4 ) ) ).
% leD
thf(fact_552_leD,axiom,
! [Y4: num,X3: num] :
( ( ord_less_eq_num @ Y4 @ X3 )
=> ~ ( ord_less_num @ X3 @ Y4 ) ) ).
% leD
thf(fact_553_leD,axiom,
! [Y4: nat,X3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y4 ) ) ).
% leD
thf(fact_554_leD,axiom,
! [Y4: int,X3: int] :
( ( ord_less_eq_int @ Y4 @ X3 )
=> ~ ( ord_less_int @ X3 @ Y4 ) ) ).
% leD
thf(fact_555_leD,axiom,
! [Y4: a,X3: a] :
( ( ord_less_eq_a @ Y4 @ X3 )
=> ~ ( ord_less_a @ X3 @ Y4 ) ) ).
% leD
thf(fact_556_leD,axiom,
! [Y4: real,X3: real] :
( ( ord_less_eq_real @ Y4 @ X3 )
=> ~ ( ord_less_real @ X3 @ Y4 ) ) ).
% leD
thf(fact_557_leD,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ~ ( ord_le72135733267957522d_enat @ X3 @ Y4 ) ) ).
% leD
thf(fact_558_verit__comp__simplify1_I3_J,axiom,
! [B4: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ B4 @ A5 ) )
= ( ord_le7381754540660121996nnreal @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_559_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
= ( ord_less_num @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_560_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_561_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_562_verit__comp__simplify1_I3_J,axiom,
! [B4: a,A5: a] :
( ( ~ ( ord_less_eq_a @ B4 @ A5 ) )
= ( ord_less_a @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_563_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_564_verit__comp__simplify1_I3_J,axiom,
! [B4: extended_enat,A5: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ B4 @ A5 ) )
= ( ord_le72135733267957522d_enat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_565_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_566_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_567_atLeastLessThan__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D ) )
=> ( ( ord_less_eq_num @ B @ A )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_568_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_569_atLeastLessThan__subset__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ A @ B ) @ ( set_or5139330845457685135Than_a @ C @ D ) )
=> ( ( ord_less_eq_a @ B @ A )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_570_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_571_atLeastLessThan__subset__iff,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le7203529160286727270d_enat @ ( set_or4374356025156299511d_enat @ A @ B ) @ ( set_or4374356025156299511d_enat @ C @ D ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
| ( ( ord_le2932123472753598470d_enat @ C @ A )
& ( ord_le2932123472753598470d_enat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_572_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_573_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_574_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_575_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_576_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_577_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_578_zero__neq__numeral,axiom,
! [N: num] :
( zero_z7100319975126383169nnreal
!= ( numera4658534427948366547nnreal @ N ) ) ).
% zero_neq_numeral
thf(fact_579_zero__neq__numeral,axiom,
! [N: num] :
( zero_z5237406670263579293d_enat
!= ( numera1916890842035813515d_enat @ N ) ) ).
% zero_neq_numeral
thf(fact_580_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_581_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_582_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_583_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_584_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_585_order__antisym__conv,axiom,
! [Y4: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_586_order__antisym__conv,axiom,
! [Y4: num,X3: num] :
( ( ord_less_eq_num @ Y4 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_587_order__antisym__conv,axiom,
! [Y4: nat,X3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_588_order__antisym__conv,axiom,
! [Y4: int,X3: int] :
( ( ord_less_eq_int @ Y4 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_589_order__antisym__conv,axiom,
! [Y4: a,X3: a] :
( ( ord_less_eq_a @ Y4 @ X3 )
=> ( ( ord_less_eq_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_590_order__antisym__conv,axiom,
! [Y4: real,X3: real] :
( ( ord_less_eq_real @ Y4 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_591_order__antisym__conv,axiom,
! [Y4: extended_enat,X3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_592_linorder__le__cases,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_593_linorder__le__cases,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_594_linorder__le__cases,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_595_linorder__le__cases,axiom,
! [X3: a,Y4: a] :
( ~ ( ord_less_eq_a @ X3 @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_596_linorder__le__cases,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_597_linorder__le__cases,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_598_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_599_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_600_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_601_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_602_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_603_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_604_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_605_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_606_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_607_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_608_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_609_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_610_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_611_ord__eq__le__subst,axiom,
! [A: a,F: num > a,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_612_ord__eq__le__subst,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_613_ord__eq__le__subst,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_614_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_615_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_616_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_617_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_618_linorder__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
| ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_619_linorder__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_620_linorder__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_621_linorder__linear,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
| ( ord_less_eq_a @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_622_linorder__linear,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
| ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_623_linorder__linear,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
| ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_624_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_625_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_626_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_627_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_628_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_629_verit__la__disequality,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = B )
| ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ~ ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_630_order__eq__refl,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( X3 = Y4 )
=> ( ord_less_eq_set_nat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_631_order__eq__refl,axiom,
! [X3: num,Y4: num] :
( ( X3 = Y4 )
=> ( ord_less_eq_num @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_632_order__eq__refl,axiom,
! [X3: nat,Y4: nat] :
( ( X3 = Y4 )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_633_order__eq__refl,axiom,
! [X3: int,Y4: int] :
( ( X3 = Y4 )
=> ( ord_less_eq_int @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_634_order__eq__refl,axiom,
! [X3: a,Y4: a] :
( ( X3 = Y4 )
=> ( ord_less_eq_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_635_order__eq__refl,axiom,
! [X3: real,Y4: real] :
( ( X3 = Y4 )
=> ( ord_less_eq_real @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_636_order__eq__refl,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( X3 = Y4 )
=> ( ord_le2932123472753598470d_enat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_637_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_638_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_639_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_640_order__subst2,axiom,
! [A: num,B: num,F: num > a,C: a] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_641_order__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_642_order__subst2,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_643_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_644_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_645_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_646_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_647_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_648_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_649_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_650_order__subst1,axiom,
! [A: num,F: a > num,B: a,C: a] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_651_order__subst1,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_652_order__subst1,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X4: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_653_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_654_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_655_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_656_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_657_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_658_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_659_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_660_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_661_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: a,Z4: a] : ( Y6 = Z4 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
& ( ord_less_eq_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_662_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_663_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: extended_enat,Z4: extended_enat] : ( Y6 = Z4 ) )
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ( ord_le2932123472753598470d_enat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_664_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_665_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_666_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_667_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_668_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_669_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_670_antisym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_671_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_672_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_673_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_674_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_675_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_676_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_677_dual__order_Otrans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_678_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_679_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_680_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_681_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_682_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_683_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_684_dual__order_Oantisym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_685_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_686_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_687_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_688_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_689_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: a,Z4: a] : ( Y6 = Z4 ) )
= ( ^ [A4: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A4 )
& ( ord_less_eq_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_690_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_691_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: extended_enat,Z4: extended_enat] : ( Y6 = Z4 ) )
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ( ord_le2932123472753598470d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_692_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_693_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_694_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: int,B2: int] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_695_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: a,B2: a] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_696_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: real,B2: real] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_697_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: extended_enat,B2: extended_enat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_698_order__trans,axiom,
! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_699_order__trans,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z3 )
=> ( ord_less_eq_num @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_700_order__trans,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_701_order__trans,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_702_order__trans,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ Y4 @ Z3 )
=> ( ord_less_eq_a @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_703_order__trans,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_704_order__trans,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ord_le2932123472753598470d_enat @ Y4 @ Z3 )
=> ( ord_le2932123472753598470d_enat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_705_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_706_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_707_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_708_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_709_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_710_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_711_order_Otrans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_712_order__antisym,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_713_order__antisym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_714_order__antisym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_715_order__antisym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_716_order__antisym,axiom,
! [X3: a,Y4: a] :
( ( ord_less_eq_a @ X3 @ Y4 )
=> ( ( ord_less_eq_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_717_order__antisym,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_718_order__antisym,axiom,
! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_719_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_720_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_721_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_722_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_723_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_724_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_725_ord__le__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_726_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_727_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_728_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_729_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_730_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_731_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_732_ord__eq__le__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_733_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_734_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_735_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_736_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_737_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: a,Z4: a] : ( Y6 = Z4 ) )
= ( ^ [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
& ( ord_less_eq_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_738_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_739_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: extended_enat,Z4: extended_enat] : ( Y6 = Z4 ) )
= ( ^ [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
& ( ord_le2932123472753598470d_enat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_740_le__cases3,axiom,
! [X3: num,Y4: num,Z3: num] :
( ( ( ord_less_eq_num @ X3 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z3 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_741_le__cases3,axiom,
! [X3: nat,Y4: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_742_le__cases3,axiom,
! [X3: int,Y4: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_743_le__cases3,axiom,
! [X3: a,Y4: a,Z3: a] :
( ( ( ord_less_eq_a @ X3 @ Y4 )
=> ~ ( ord_less_eq_a @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_a @ Y4 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_a @ X3 @ Z3 )
=> ~ ( ord_less_eq_a @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_a @ Z3 @ Y4 )
=> ~ ( ord_less_eq_a @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_a @ Y4 @ Z3 )
=> ~ ( ord_less_eq_a @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_a @ Z3 @ X3 )
=> ~ ( ord_less_eq_a @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_744_le__cases3,axiom,
! [X3: real,Y4: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_745_le__cases3,axiom,
! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ~ ( ord_le2932123472753598470d_enat @ Y4 @ Z3 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
=> ~ ( ord_le2932123472753598470d_enat @ X3 @ Z3 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ X3 @ Z3 )
=> ~ ( ord_le2932123472753598470d_enat @ Z3 @ Y4 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Z3 @ Y4 )
=> ~ ( ord_le2932123472753598470d_enat @ Y4 @ X3 ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y4 @ Z3 )
=> ~ ( ord_le2932123472753598470d_enat @ Z3 @ X3 ) )
=> ~ ( ( ord_le2932123472753598470d_enat @ Z3 @ X3 )
=> ~ ( ord_le2932123472753598470d_enat @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_746_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_747_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_748_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_749_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_750_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_751_nle__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_752_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_753_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_754_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_755_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_756_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_757_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_758_verit__comp__simplify1_I2_J,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_759_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M3: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_760_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le3935885782089961368nnreal @ ( numera4658534427948366547nnreal @ N ) @ zero_z7100319975126383169nnreal ) ).
% not_numeral_le_zero
thf(fact_761_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_762_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_763_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_764_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_765_zero__le__numeral,axiom,
! [N: num] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( numera4658534427948366547nnreal @ N ) ) ).
% zero_le_numeral
thf(fact_766_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_767_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_768_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_769_zero__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_770_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_771_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_772_card__J__min,axiom,
ord_less_nat @ ( size_size_list_a @ xs ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ j ) ) ).
% card_J_min
thf(fact_773_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_774_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_775_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_776_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_777_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_778_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_779_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_780_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_781_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_782_zdiv__numeral__Bit0,axiom,
! [V: num,W2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_783_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z3: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ W2 ) @ Z3 ) )
= ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( times_times_num @ V @ W2 ) ) @ Z3 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_784_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z3: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z3 ) )
= ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W2 ) ) @ Z3 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_785_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z3: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z3 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Z3 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_786_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z3: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z3 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z3 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_787_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z3: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z3 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z3 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_788_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_789_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_790_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_791_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_792_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_803_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_804_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_805_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_806_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_807_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_808_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_809_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_810_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_left__diff__distrib__numeral,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,V: num] :
( ( times_8498157372700349887l_num1 @ ( minus_838314146864362899l_num1 @ A @ B ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( minus_838314146864362899l_num1 @ ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ V ) ) @ ( times_8498157372700349887l_num1 @ B @ ( numera2161328050825114965l_num1 @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_823_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_824_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_825_right__diff__distrib__numeral,axiom,
! [V: num,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( minus_838314146864362899l_num1 @ B @ C ) )
= ( minus_838314146864362899l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ B ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_826_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_827_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_828_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_829_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_830_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_831_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_832_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_833_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_834_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_835_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_836_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_837_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_838_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_839_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_840_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_841_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_842_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_843_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_844_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_845_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_846_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_847_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_848_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_849_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_850_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_851_times__divide__times__eq,axiom,
! [X3: real,Y4: real,Z3: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X3 @ Y4 ) @ ( divide_divide_real @ Z3 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X3 @ Z3 ) @ ( times_times_real @ Y4 @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_852_divide__divide__times__eq,axiom,
! [X3: real,Y4: real,Z3: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y4 ) @ ( divide_divide_real @ Z3 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X3 @ W2 ) @ ( times_times_real @ Y4 @ Z3 ) ) ) ).
% divide_divide_times_eq
thf(fact_853_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_854_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_855_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_856_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_857_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_858_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_859_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_860_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_861_divide__diff__eq__iff,axiom,
! [Z3: real,X3: real,Y4: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z3 ) @ Y4 )
= ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y4 @ Z3 ) ) @ Z3 ) ) ) ).
% divide_diff_eq_iff
thf(fact_862_diff__divide__eq__iff,axiom,
! [Z3: real,X3: real,Y4: real] :
( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y4 @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z3 ) @ Y4 ) @ Z3 ) ) ) ).
% diff_divide_eq_iff
thf(fact_863_diff__frac__eq,axiom,
! [Y4: real,Z3: real,X3: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y4 ) @ ( divide_divide_real @ W2 @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z3 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z3 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_864_add__divide__eq__if__simps_I4_J,axiom,
! [Z3: real,A: real,B: real] :
( ( ( Z3 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
= A ) )
& ( ( Z3 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_865_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_866_frac__eq__eq,axiom,
! [Y4: real,Z3: real,X3: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ( divide_divide_real @ X3 @ Y4 )
= ( divide_divide_real @ W2 @ Z3 ) )
= ( ( times_times_real @ X3 @ Z3 )
= ( times_times_real @ W2 @ Y4 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_876_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_877_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_878_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_879_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_880_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_881_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_882_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_883_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_884_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_885_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_886_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_887_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_888_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_889_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_890_frac__le__eq,axiom,
! [Y4: real,Z3: real,X3: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ ( divide_divide_real @ W2 @ Z3 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z3 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z3 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_891_frac__less__eq,axiom,
! [Y4: real,Z3: real,X3: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z3 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y4 ) @ ( divide_divide_real @ W2 @ Z3 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z3 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z3 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_892_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_893_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_894_mult__imp__less__div__pos,axiom,
! [Y4: real,Z3: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ ( times_times_real @ Z3 @ Y4 ) @ X3 )
=> ( ord_less_real @ Z3 @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_895_mult__imp__div__pos__less,axiom,
! [Y4: real,X3: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ X3 @ ( times_times_real @ Z3 @ Y4 ) )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y4 ) @ Z3 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_896_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_897_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_898_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_899_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_900_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_901_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_902_mult__numeral__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_903_mult__numeral__1,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_904_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_905_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_906_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_907_mult__numeral__1__right,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_908_mult__numeral__1__right,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_909_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_910_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_911_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_912_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_913_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_914_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_915_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_916_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_917_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_918_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_919_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_920_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_921_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_922_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_923_mult__imp__div__pos__le,axiom,
! [Y4: real,X3: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z3 @ Y4 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ Z3 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_924_mult__imp__le__div__pos,axiom,
! [Y4: real,Z3: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ Y4 ) @ X3 )
=> ( ord_less_eq_real @ Z3 @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_925_divide__left__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_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_926_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_927_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W2 ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_928_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W2 )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_929_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_930_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_931_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_932_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_933_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_934_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_935_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_936_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 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_937_linorder__neqE__nat,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_938_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_939_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_940_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_941_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_942_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_943_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_944_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_945_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y4: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_946_size__neq__size__imp__neq,axiom,
! [X3: num,Y4: num] :
( ( ( size_size_num @ X3 )
!= ( size_size_num @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_947_size__neq__size__imp__neq,axiom,
! [X3: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X3 )
!= ( size_size_list_nat @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_948_size__neq__size__imp__neq,axiom,
! [X3: char,Y4: char] :
( ( ( size_size_char @ X3 )
!= ( size_size_char @ Y4 ) )
=> ( X3 != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_949_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_950_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_951_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_952_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_953_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_954_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_955_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_956_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_957_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_958_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_959_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_960_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_961_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_962_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_963_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_964_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_965_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_966_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_967_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_968_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_969_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_970_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_971_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_972_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_973_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_974_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_975_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_976_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_977_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_978_divide__nonneg__nonneg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_979_divide__nonneg__nonpos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_980_divide__nonpos__nonneg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_981_divide__nonpos__nonpos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_982_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_983_divide__neg__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_neg_neg
thf(fact_984_divide__neg__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_985_divide__pos__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_986_divide__pos__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_pos_pos
thf(fact_987_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_988_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_989_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_990_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_991_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_992_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_993_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_994_frac__le,axiom,
! [Y4: real,X3: real,W2: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z3 ) @ ( divide_divide_real @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_995_frac__less,axiom,
! [X3: real,Y4: real,W2: real,Z3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z3 ) @ ( divide_divide_real @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_996_frac__less2,axiom,
! [X3: real,Y4: real,W2: real,Z3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z3 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z3 ) @ ( divide_divide_real @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_997_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_998_divide__nonneg__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_999_divide__nonneg__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1000_divide__nonpos__neg,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y4 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1001_divide__nonpos__pos,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_1002_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1003_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1011_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1012_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1013_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1014_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1015_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1016_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_1017_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1018_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1019_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1020_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1021_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1022_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1023_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1024_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1025_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1026_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1027_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_1028_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_1029_le__zero__eq,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N @ zero_z7100319975126383169nnreal )
= ( N = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_1030_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1031_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_1032_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1033_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_1034_not__gr__zero,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N ) )
= ( N = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_1035_mult__zero__left,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_left
thf(fact_1036_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1037_mult__zero__left,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat @ A )
= zero_z5237406670263579293d_enat ) ).
% mult_zero_left
thf(fact_1038_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1039_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1040_mult__zero__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ zero_z7100319975126383169nnreal )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_right
thf(fact_1041_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1042_mult__zero__right,axiom,
! [A: extended_enat] :
( ( times_7803423173614009249d_enat @ A @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% mult_zero_right
thf(fact_1043_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1044_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1045_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_1046_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_1047_mult__eq__0__iff,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ( times_7803423173614009249d_enat @ A @ B )
= zero_z5237406670263579293d_enat )
= ( ( A = zero_z5237406670263579293d_enat )
| ( B = zero_z5237406670263579293d_enat ) ) ) ).
% mult_eq_0_iff
thf(fact_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_1057_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1058_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_1059_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1060_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1061_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_1062_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1063_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1064_ivl__diff,axiom,
! [I: num,N: num,M: num] :
( ( ord_less_eq_num @ I @ N )
=> ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
= ( set_or1222409239386451017an_num @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1065_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1066_ivl__diff,axiom,
! [I: a,N: a,M: a] :
( ( ord_less_eq_a @ I @ N )
=> ( ( minus_minus_set_a @ ( set_or5139330845457685135Than_a @ I @ M ) @ ( set_or5139330845457685135Than_a @ I @ N ) )
= ( set_or5139330845457685135Than_a @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1067_ivl__diff,axiom,
! [I: real,N: real,M: real] :
( ( ord_less_eq_real @ I @ N )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I @ M ) @ ( set_or66887138388493659n_real @ I @ N ) )
= ( set_or66887138388493659n_real @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1068_ivl__diff,axiom,
! [I: extended_enat,N: extended_enat,M: extended_enat] :
( ( ord_le2932123472753598470d_enat @ I @ N )
=> ( ( minus_925952699566721837d_enat @ ( set_or4374356025156299511d_enat @ I @ M ) @ ( set_or4374356025156299511d_enat @ I @ N ) )
= ( set_or4374356025156299511d_enat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1069_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1070_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1071_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1072_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1073_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1074_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1075_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_1076_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_1077_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_1078_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_1079_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1080_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1081_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1082_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N4: extended_enat] :
( ! [M5: extended_enat] :
( ( ord_le72135733267957522d_enat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_1083_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_1084_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1085_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1086_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_1087_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1101_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1102_zero__reorient,axiom,
! [X3: extended_enat] :
( ( zero_z5237406670263579293d_enat = X3 )
= ( X3 = zero_z5237406670263579293d_enat ) ) ).
% zero_reorient
thf(fact_1103_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_1104_zero__reorient,axiom,
! [X3: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X3 )
= ( X3 = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_1105_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y4: int] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ Y4 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1106_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1107_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C )
= ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1108_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1109_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1110_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1111_mult_Oassoc,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C )
= ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1112_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1113_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1114_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1115_mult_Ocommute,axiom,
( times_7803423173614009249d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( times_7803423173614009249d_enat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1116_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1117_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1118_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1119_mult_Oleft__commute,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( times_7803423173614009249d_enat @ B @ ( times_7803423173614009249d_enat @ A @ C ) )
= ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1120_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1121_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1122_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_1123_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1124_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1125_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1126_zero__le,axiom,
! [X3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X3 ) ).
% zero_le
thf(fact_1127_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1128_zero__le,axiom,
! [X3: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X3 ) ).
% zero_le
thf(fact_1129_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1130_gr__zeroI,axiom,
! [N: extended_enat] :
( ( N != zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ).
% gr_zeroI
thf(fact_1131_gr__zeroI,axiom,
! [N: extend8495563244428889912nnreal] :
( ( N != zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N ) ) ).
% gr_zeroI
thf(fact_1132_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1133_not__less__zero,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_less_zero
thf(fact_1134_not__less__zero,axiom,
! [N: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ N @ zero_z7100319975126383169nnreal ) ).
% not_less_zero
thf(fact_1135_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1136_gr__implies__not__zero,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ M @ N )
=> ( N != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_1137_gr__implies__not__zero,axiom,
! [M: extend8495563244428889912nnreal,N: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ M @ N )
=> ( N != zero_z7100319975126383169nnreal ) ) ).
% gr_implies_not_zero
thf(fact_1138_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1139_zero__less__iff__neq__zero,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1140_zero__less__iff__neq__zero,axiom,
! [N: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N )
= ( N != zero_z7100319975126383169nnreal ) ) ).
% zero_less_iff_neq_zero
thf(fact_1141_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_1142_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_1143_mult__not__zero,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ( times_7803423173614009249d_enat @ A @ B )
!= zero_z5237406670263579293d_enat )
=> ( ( A != zero_z5237406670263579293d_enat )
& ( B != zero_z5237406670263579293d_enat ) ) ) ).
% mult_not_zero
thf(fact_1144_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_1145_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_1146_divisors__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% divisors_zero
thf(fact_1147_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_1148_divisors__zero,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ( times_7803423173614009249d_enat @ A @ B )
= zero_z5237406670263579293d_enat )
=> ( ( A = zero_z5237406670263579293d_enat )
| ( B = zero_z5237406670263579293d_enat ) ) ) ).
% divisors_zero
thf(fact_1149_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_1150_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_1151_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_1152_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_1153_no__zero__divisors,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A != zero_z5237406670263579293d_enat )
=> ( ( B != zero_z5237406670263579293d_enat )
=> ( ( times_7803423173614009249d_enat @ A @ B )
!= zero_z5237406670263579293d_enat ) ) ) ).
% no_zero_divisors
thf(fact_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1163_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1164_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1165_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1166_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1167_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1168_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1169_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1170_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [A4: real,B3: real] :
( ( minus_minus_real @ A4 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1171_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1172_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1173_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1174_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1175_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1176_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1177_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1178_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1179_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1180_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1181_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1182_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1183_c,axiom,
( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
& ( ord_less_nat @ v @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% c
thf(fact_1184_assms_I2_J,axiom,
interval_a @ i ).
% assms(2)
thf(fact_1185_v__2,axiom,
member_a @ ( nth_a @ xs @ v ) @ i ).
% v_2
thf(fact_1186__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_O_A_092_060lbrakk_062v_A_060_Alength_Axs_059_Axs_A_B_Av_A_092_060in_062_AI_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V2: nat] :
( ( ord_less_nat @ V2 @ ( size_size_list_a @ xs ) )
=> ~ ( member_a @ ( nth_a @ xs @ V2 ) @ i ) ) ).
% \<open>\<And>thesis. (\<And>v. \<lbrakk>v < length xs; xs ! v \<in> I\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1187_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1188_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_1189_p,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
=> ( ( ord_less_nat @ K @ ( size_size_list_a @ xs ) )
=> ~ ( member_a @ ( nth_a @ xs @ K ) @ i ) ) ) ).
% p
thf(fact_1190__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062xs_A_B_A_Ilength_Axs_Adiv_A2_J_A_092_060in_062_AI_A_092_060Longrightarrow_062_Athesis_059_Axs_A_B_A_Ilength_Axs_Adiv_A2_J_A_092_060notin_062_AI_A_092_060and_062_Alength_Axs_Adiv_A2_A_060_Av_A_092_060Longrightarrow_062_Athesis_059_Axs_A_B_A_Ilength_Axs_Adiv_A2_J_A_092_060notin_062_AI_A_092_060and_062_Av_A_060_Alength_Axs_Adiv_A2_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
=> ( ~ ( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
& ( ord_less_nat @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ v ) )
=> ( ~ ( member_a @ ( nth_a @ xs @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ i )
& ( ord_less_nat @ v @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>xs ! (length xs div 2) \<in> I \<Longrightarrow> thesis; xs ! (length xs div 2) \<notin> I \<and> length xs div 2 < v \<Longrightarrow> thesis; xs ! (length xs div 2) \<notin> I \<and> v < length xs div 2 \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_1191_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_1192_imp__le__cong,axiom,
! [X3: int,X7: int,P: $o,P4: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1193_conj__le__cong,axiom,
! [X3: int,X7: int,P: $o,P4: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1194_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1195_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K2: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1196_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1197_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_1198_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1199_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_1200_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1201_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1202_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1203_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1204_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1205_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1206_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1207_ediff__le__self,axiom,
! [X3: extended_enat,Y4: extended_enat] : ( ord_le2932123472753598470d_enat @ ( minus_3235023915231533773d_enat @ X3 @ Y4 ) @ X3 ) ).
% ediff_le_self
thf(fact_1208_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1209_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1210_Parity_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L3: num,R: int] : ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ ( numeral_numeral_int @ L3 ) @ R ) ) ) ) ).
% Parity.adjust_mod_def
thf(fact_1211_log__induct,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
=> ( ( P @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( P @ N4 ) ) )
=> ( P @ N ) ) ) ) ).
% log_induct
thf(fact_1212_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1213_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1214_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1215_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1216_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1217_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1218_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1219_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1220_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1221_log__half,axiom,
! [N: nat] :
( ( log @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ ( log @ N ) @ one_one_nat ) ) ).
% log_half
thf(fact_1222_one__less__numeral,axiom,
! [N: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral
thf(fact_1223_log__zero,axiom,
( ( log @ zero_zero_nat )
= zero_zero_nat ) ).
% log_zero
thf(fact_1224_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1225_Discrete_Olog__one,axiom,
( ( log @ one_one_nat )
= zero_zero_nat ) ).
% Discrete.log_one
thf(fact_1226_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_1227_Discrete_Olog__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( log @ M ) @ ( log @ N ) ) ) ).
% Discrete.log_le_iff
thf(fact_1228_exhaust__2,axiom,
! [X3: numera2417102609627094330l_num1] :
( ( X3 = one_on3868389512446148991l_num1 )
| ( X3
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_1229_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X5: numera2417102609627094330l_num1] : ( P2 @ X5 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_1230_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1231_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1232_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1233_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1234_int__div__less__self,axiom,
! [X3: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_1235_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 ) ) ) )
=> ? [L4: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L4 @ X6 @ I3 ) @ one_one_nat )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( X6 @ I3 )
= zero_zero_real ) )
=> ( ( L4 @ X6 @ I3 )
= zero_zero_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( X6 @ I3 )
= one_one_real ) )
=> ( ( L4 @ X6 @ I3 )
= one_one_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( L4 @ X6 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I3 ) @ ( F @ X6 @ I3 ) ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q2 @ I3 )
& ( ( L4 @ X6 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I3 ) @ ( X6 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1236_diff__gr0__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ).
% diff_gr0_ennreal
thf(fact_1237_half__bounded__equal,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ ( times_times_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real )
= ( X3
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% half_bounded_equal
thf(fact_1238_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1239_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_1240_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1241_zero__minus__ennreal,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% zero_minus_ennreal
thf(fact_1242_ennreal__minus__zero,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% ennreal_minus_zero
thf(fact_1243_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1244_zle__add1__eq__le,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1245_ennreal__minus__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ D @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ C @ D ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1246_ennreal__mono__minus,axiom,
! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).
% ennreal_mono_minus
thf(fact_1247_diff__le__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ A ) ).
% diff_le_self_ennreal
thf(fact_1248_ennreal__diff__le__mono__left,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1249_ennreal__minus__eq__0,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A @ B ) ) ).
% ennreal_minus_eq_0
thf(fact_1250_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1251_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1252_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1253_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1254_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1255_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1256_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1257_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1258_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1259_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1260_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1261_diff__diff__commute__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% diff_diff_commute_ennreal
thf(fact_1262_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1263_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_1264_zless__add1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z3 )
| ( W2 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1265_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
% Helper facts (3)
thf(help_If_3_1_If_001t__Int__Oint_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y4: int] :
( ( if_int @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y4: int] :
( ( if_int @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( finite_card_nat @ j ) @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
%------------------------------------------------------------------------------