TPTP Problem File: SLH0657^1.p
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%------------------------------------------------------------------------------
% 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 : Dedekind_Real/0000_Dedekind_Real/prob_00511_015252__5644456_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1351 ( 489 unt; 83 typ; 0 def)
% Number of atoms : 3909 (1051 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10323 ( 450 ~; 151 |; 194 &;7782 @)
% ( 0 <=>;1746 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 327 ( 327 >; 0 *; 0 +; 0 <<)
% Number of symbols : 73 ( 72 usr; 12 con; 0-2 aty)
% Number of variables : 3303 ( 200 ^;2976 !; 127 ?;3303 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:24:40.223
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $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__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (72)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Dedekind__Real_Oadd__set,type,
dedekind_add_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Ocut,type,
dedekind_cut: set_rat > $o ).
thf(sy_c_Dedekind__Real_Oinverse__set,type,
dedekind_inverse_set: set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Omult__set,type,
dedekind_mult_set: set_rat > set_rat > set_rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
inverse_inverse_rat: rat > rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
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__Rat__Orat,type,
one_one_rat: rat ).
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_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__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__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
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__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
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__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__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Rat__Orat_M_Eo_J,type,
bot_bot_rat_o: rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
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__Rat__Orat,type,
ord_less_rat: rat > rat > $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__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $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__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $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__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__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $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_Rat_Ofield__char__0__class_Oof__rat_001t__Rat__Orat,type,
field_2639924705303425560at_rat: rat > rat ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
field_7254667332652039916t_real: rat > real ).
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__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_Ois__empty_001t__Rat__Orat,type,
is_empty_rat: set_rat > $o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
set_or1207661135979820486an_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Num__Onum,type,
set_or6990855429499425204an_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6174011595382531368et_rat: set_rat > set_set_rat ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
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__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_v_A,type,
a: set_rat ).
% Relevant facts (1267)
thf(fact_0__092_060open_062_092_060exists_062x_Ay_O_A0_A_060_Ax_A_092_060and_062_Ax_A_060_Ay_A_092_060and_062_Ainverse_Ay_A_092_060notin_062_AA_092_060close_062,axiom,
? [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
& ( ord_less_rat @ X @ Y )
& ~ ( member_rat @ ( inverse_inverse_rat @ Y ) @ a ) ) ).
% \<open>\<exists>x y. 0 < x \<and> x < y \<and> inverse y \<notin> A\<close>
thf(fact_1_assms,axiom,
dedekind_cut @ a ).
% assms
thf(fact_2_empty__iff,axiom,
! [C: rat] :
~ ( member_rat @ C @ bot_bot_set_rat ) ).
% empty_iff
thf(fact_3_all__not__in__conv,axiom,
! [A: set_rat] :
( ( ! [X2: rat] :
~ ( member_rat @ X2 @ A ) )
= ( A = bot_bot_set_rat ) ) ).
% all_not_in_conv
thf(fact_4_Collect__empty__eq,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( ! [X2: rat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_5_empty__Collect__eq,axiom,
! [P: rat > $o] :
( ( bot_bot_set_rat
= ( collect_rat @ P ) )
= ( ! [X2: rat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_6_not__psubset__empty,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ bot_bot_set_rat ) ).
% not_psubset_empty
thf(fact_7_bot_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ bot_bot_set_rat ) ).
% bot.extremum_strict
thf(fact_8_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_9_bot_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != bot_bot_set_rat )
= ( ord_less_set_rat @ bot_bot_set_rat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_10_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_11_psubsetD,axiom,
! [A: set_rat,B: set_rat,C: rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% psubsetD
thf(fact_12_psubset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_13_emptyE,axiom,
! [A2: rat] :
~ ( member_rat @ A2 @ bot_bot_set_rat ) ).
% emptyE
thf(fact_14_equals0D,axiom,
! [A: set_rat,A2: rat] :
( ( A = bot_bot_set_rat )
=> ~ ( member_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_15_preal__Ex__mem,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] : ( member_rat @ X @ A ) ) ).
% preal_Ex_mem
thf(fact_16_preal__imp__pos,axiom,
! [A: set_rat,R: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ R @ A )
=> ( ord_less_rat @ zero_zero_rat @ R ) ) ) ).
% preal_imp_pos
thf(fact_17_preal__nonempty,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ zero_zero_rat @ X ) ) ) ).
% preal_nonempty
thf(fact_18_not__in__preal__ub,axiom,
! [A: set_rat,X3: rat,Y2: rat] :
( ( dedekind_cut @ A )
=> ( ~ ( member_rat @ X3 @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ( ord_less_rat @ Y2 @ X3 ) ) ) ) ) ).
% not_in_preal_ub
thf(fact_19_preal__exists__bound,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
& ~ ( member_rat @ X @ A ) ) ) ).
% preal_exists_bound
thf(fact_20_preal__exists__greater,axiom,
! [A: set_rat,Y2: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y2 @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ Y2 @ X ) ) ) ) ).
% preal_exists_greater
thf(fact_21_preal__downwards__closed,axiom,
! [A: set_rat,Y2: rat,Z: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ Z @ Y2 )
=> ( member_rat @ Z @ A ) ) ) ) ) ).
% preal_downwards_closed
thf(fact_22_bot__set__def,axiom,
( bot_bot_set_rat
= ( collect_rat @ bot_bot_rat_o ) ) ).
% bot_set_def
thf(fact_23_order__less__imp__not__less,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_24_order__less__imp__not__less,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_25_order__less__imp__not__less,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_26_order__less__imp__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_27_order__less__imp__not__less,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_28_order__less__imp__not__eq2,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_29_order__less__imp__not__eq2,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_30_order__less__imp__not__eq2,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_31_order__less__imp__not__eq2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_32_order__less__imp__not__eq2,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_33_order__less__imp__not__eq,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_34_order__less__imp__not__eq,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_35_order__less__imp__not__eq,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_36_order__less__imp__not__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_37_order__less__imp__not__eq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_38_linorder__less__linear,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_rat @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_39_linorder__less__linear,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_num @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_40_linorder__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_41_linorder__less__linear,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_42_order__less__imp__triv,axiom,
! [X3: set_rat,Y2: set_rat,P: $o] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_43_order__less__imp__triv,axiom,
! [X3: rat,Y2: rat,P: $o] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_44_order__less__imp__triv,axiom,
! [X3: num,Y2: num,P: $o] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ( ord_less_num @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_45_order__less__imp__triv,axiom,
! [X3: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_46_order__less__imp__triv,axiom,
! [X3: int,Y2: int,P: $o] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_47_order__less__not__sym,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_48_order__less__not__sym,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_49_order__less__not__sym,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_50_order__less__not__sym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_51_order__less__not__sym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_52_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_53_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_54_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_55_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_56_order__less__subst2,axiom,
! [A2: num,B2: num,F: num > rat,C: rat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_57_order__less__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_58_order__less__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_59_order__less__subst2,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_60_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_61_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_62_order__less__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_63_order__less__subst1,axiom,
! [A2: rat,F: num > rat,B2: num,C: num] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_64_order__less__subst1,axiom,
! [A2: rat,F: nat > rat,B2: nat,C: nat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_65_order__less__subst1,axiom,
! [A2: rat,F: int > rat,B2: int,C: int] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_66_order__less__subst1,axiom,
! [A2: num,F: rat > num,B2: rat,C: rat] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_67_order__less__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_68_order__less__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_69_order__less__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_70_order__less__subst1,axiom,
! [A2: nat,F: rat > nat,B2: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_71_order__less__subst1,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_72_order__less__irrefl,axiom,
! [X3: set_rat] :
~ ( ord_less_set_rat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_73_order__less__irrefl,axiom,
! [X3: rat] :
~ ( ord_less_rat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_74_order__less__irrefl,axiom,
! [X3: num] :
~ ( ord_less_num @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_75_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_76_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_77_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_78_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_79_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_80_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_81_ord__less__eq__subst,axiom,
! [A2: num,B2: num,F: num > rat,C: rat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_82_ord__less__eq__subst,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_83_ord__less__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_84_ord__less__eq__subst,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_85_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_86_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_87_ord__eq__less__subst,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_88_ord__eq__less__subst,axiom,
! [A2: num,F: rat > num,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_89_ord__eq__less__subst,axiom,
! [A2: nat,F: rat > nat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_90_ord__eq__less__subst,axiom,
! [A2: int,F: rat > int,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_91_ord__eq__less__subst,axiom,
! [A2: rat,F: num > rat,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_92_ord__eq__less__subst,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_93_ord__eq__less__subst,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_94_ord__eq__less__subst,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_95_ord__eq__less__subst,axiom,
! [A2: rat,F: nat > rat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_96_ord__eq__less__subst,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_order__less__trans,axiom,
! [X3: set_rat,Y2: set_rat,Z: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ Z )
=> ( ord_less_set_rat @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_98_order__less__trans,axiom,
! [X3: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_99_order__less__trans,axiom,
! [X3: num,Y2: num,Z: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z )
=> ( ord_less_num @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_100_order__less__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_101_order__less__trans,axiom,
! [X3: int,Y2: int,Z: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_102_order__less__asym_H,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_rat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_103_order__less__asym_H,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ~ ( ord_less_rat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_104_order__less__asym_H,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ~ ( ord_less_num @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_105_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_106_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_107_linorder__neq__iff,axiom,
! [X3: rat,Y2: rat] :
( ( X3 != Y2 )
= ( ( ord_less_rat @ X3 @ Y2 )
| ( ord_less_rat @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_108_linorder__neq__iff,axiom,
! [X3: num,Y2: num] :
( ( X3 != Y2 )
= ( ( ord_less_num @ X3 @ Y2 )
| ( ord_less_num @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_109_linorder__neq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
= ( ( ord_less_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_110_linorder__neq__iff,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
= ( ( ord_less_int @ X3 @ Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_111_order__less__asym,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_112_order__less__asym,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_113_order__less__asym,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_114_order__less__asym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_115_order__less__asym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_116_linorder__neqE,axiom,
! [X3: rat,Y2: rat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_rat @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_117_linorder__neqE,axiom,
! [X3: num,Y2: num] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_num @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_118_linorder__neqE,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_119_linorder__neqE,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_120_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_121_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_122_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_123_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_124_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_125_order_Ostrict__implies__not__eq,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_126_order_Ostrict__implies__not__eq,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_127_order_Ostrict__implies__not__eq,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_128_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_129_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_130_dual__order_Ostrict__trans,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( ord_less_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_131_dual__order_Ostrict__trans,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ B2 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_132_dual__order_Ostrict__trans,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_133_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_134_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_135_not__less__iff__gr__or__eq,axiom,
! [X3: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X3 @ Y2 ) )
= ( ( ord_less_rat @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_136_not__less__iff__gr__or__eq,axiom,
! [X3: num,Y2: num] :
( ( ~ ( ord_less_num @ X3 @ Y2 ) )
= ( ( ord_less_num @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_137_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_138_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_139_order_Ostrict__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_140_order_Ostrict__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_141_order_Ostrict__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_142_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_143_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_144_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B2: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: rat] : ( P @ A3 @ A3 )
=> ( ! [A3: rat,B3: rat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_145_linorder__less__wlog,axiom,
! [P: num > num > $o,A2: num,B2: num] :
( ! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: num] : ( P @ A3 @ A3 )
=> ( ! [A3: num,B3: num] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_146_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_147_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_148_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_149_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X2: rat] : ( member_rat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_Collect__cong,axiom,
! [P: rat > $o,Q: rat > $o] :
( ! [X: rat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_rat @ P )
= ( collect_rat @ Q ) ) ) ).
% Collect_cong
thf(fact_152_dual__order_Oirrefl,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_153_dual__order_Oirrefl,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_154_dual__order_Oirrefl,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_155_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_156_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_157_dual__order_Oasym,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ~ ( ord_less_set_rat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_158_dual__order_Oasym,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ~ ( ord_less_rat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_159_dual__order_Oasym,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ~ ( ord_less_num @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_160_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_161_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_162_linorder__cases,axiom,
! [X3: rat,Y2: rat] :
( ~ ( ord_less_rat @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_rat @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_163_linorder__cases,axiom,
! [X3: num,Y2: num] :
( ~ ( ord_less_num @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_num @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_164_linorder__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_165_linorder__cases,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_166_antisym__conv3,axiom,
! [Y2: rat,X3: rat] :
( ~ ( ord_less_rat @ Y2 @ X3 )
=> ( ( ~ ( ord_less_rat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_167_antisym__conv3,axiom,
! [Y2: num,X3: num] :
( ~ ( ord_less_num @ Y2 @ X3 )
=> ( ( ~ ( ord_less_num @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_168_antisym__conv3,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_nat @ Y2 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_169_antisym__conv3,axiom,
! [Y2: int,X3: int] :
( ~ ( ord_less_int @ Y2 @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_170_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_171_ord__less__eq__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_172_ord__less__eq__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_173_ord__less__eq__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_174_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_175_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_176_ord__eq__less__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( A2 = B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_177_ord__eq__less__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( A2 = B2 )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_178_ord__eq__less__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( A2 = B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_179_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_180_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_181_order_Oasym,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_rat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_182_order_Oasym,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ~ ( ord_less_rat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_183_order_Oasym,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ~ ( ord_less_num @ B2 @ A2 ) ) ).
% order.asym
thf(fact_184_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_185_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_186_less__imp__neq,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_187_less__imp__neq,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_188_less__imp__neq,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_189_less__imp__neq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_190_less__imp__neq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_191_dense,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ? [Z2: rat] :
( ( ord_less_rat @ X3 @ Z2 )
& ( ord_less_rat @ Z2 @ Y2 ) ) ) ).
% dense
thf(fact_192_gt__ex,axiom,
! [X3: rat] :
? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_193_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_194_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_195_lt__ex,axiom,
! [X3: rat] :
? [Y: rat] : ( ord_less_rat @ Y @ X3 ) ).
% lt_ex
thf(fact_196_lt__ex,axiom,
! [X3: int] :
? [Y: int] : ( ord_less_int @ Y @ X3 ) ).
% lt_ex
thf(fact_197_ex__in__conv,axiom,
! [A: set_rat] :
( ( ? [X2: rat] : ( member_rat @ X2 @ A ) )
= ( A != bot_bot_set_rat ) ) ).
% ex_in_conv
thf(fact_198_equals0I,axiom,
! [A: set_rat] :
( ! [Y: rat] :
~ ( member_rat @ Y @ A )
=> ( A = bot_bot_set_rat ) ) ).
% equals0I
thf(fact_199_inverse__positive__iff__positive,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% inverse_positive_iff_positive
thf(fact_200_inverse__positive__iff__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% inverse_positive_iff_positive
thf(fact_201_inverse__negative__iff__negative,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% inverse_negative_iff_negative
thf(fact_202_inverse__negative__iff__negative,axiom,
! [A2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_203_inverse__less__iff__less__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_204_inverse__less__iff__less__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_205_inverse__less__iff__less,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_206_inverse__less__iff__less,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_207_inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% inverse_zero
thf(fact_208_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_209_inverse__nonzero__iff__nonzero,axiom,
! [A2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_210_inverse__nonzero__iff__nonzero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_211_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_212_positive__imp__inverse__positive,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_213_positive__imp__inverse__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_214_negative__imp__inverse__negative,axiom,
! [A2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat ) ) ).
% negative_imp_inverse_negative
thf(fact_215_negative__imp__inverse__negative,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_216_inverse__positive__imp__positive,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) )
=> ( ( A2 != zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_217_inverse__positive__imp__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
=> ( ( A2 != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A2 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_218_inverse__negative__imp__negative,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat )
=> ( ( A2 != zero_zero_rat )
=> ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).
% inverse_negative_imp_negative
thf(fact_219_inverse__negative__imp__negative,axiom,
! [A2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
=> ( ( A2 != zero_zero_real )
=> ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_220_less__imp__inverse__less__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_221_less__imp__inverse__less__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_222_inverse__inverse__eq,axiom,
! [A2: rat] :
( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A2 ) )
= A2 ) ).
% inverse_inverse_eq
thf(fact_223_inverse__inverse__eq,axiom,
! [A2: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A2 ) )
= A2 ) ).
% inverse_inverse_eq
thf(fact_224_inverse__eq__iff__eq,axiom,
! [A2: rat,B2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= ( inverse_inverse_rat @ B2 ) )
= ( A2 = B2 ) ) ).
% inverse_eq_iff_eq
thf(fact_225_inverse__eq__iff__eq,axiom,
! [A2: real,B2: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B2 ) )
= ( A2 = B2 ) ) ).
% inverse_eq_iff_eq
thf(fact_226_zero__reorient,axiom,
! [X3: rat] :
( ( zero_zero_rat = X3 )
= ( X3 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_227_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_228_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_229_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_230_linordered__field__no__lb,axiom,
! [X5: rat] :
? [Y: rat] : ( ord_less_rat @ Y @ X5 ) ).
% linordered_field_no_lb
thf(fact_231_linordered__field__no__ub,axiom,
! [X5: rat] :
? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_232_inverse__eq__imp__eq,axiom,
! [A2: rat,B2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= ( inverse_inverse_rat @ B2 ) )
=> ( A2 = B2 ) ) ).
% inverse_eq_imp_eq
thf(fact_233_inverse__eq__imp__eq,axiom,
! [A2: real,B2: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B2 ) )
=> ( A2 = B2 ) ) ).
% inverse_eq_imp_eq
thf(fact_234_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_235_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_236_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_237_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_238_nonzero__imp__inverse__nonzero,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A2 )
!= zero_zero_rat ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_239_nonzero__imp__inverse__nonzero,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ A2 )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_240_nonzero__inverse__inverse__eq,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A2 ) )
= A2 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_241_nonzero__inverse__inverse__eq,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A2 ) )
= A2 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_242_nonzero__inverse__eq__imp__eq,axiom,
! [A2: rat,B2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= ( inverse_inverse_rat @ B2 ) )
=> ( ( A2 != zero_zero_rat )
=> ( ( B2 != zero_zero_rat )
=> ( A2 = B2 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_243_nonzero__inverse__eq__imp__eq,axiom,
! [A2: real,B2: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B2 ) )
=> ( ( A2 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( A2 = B2 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_244_inverse__zero__imp__zero,axiom,
! [A2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= zero_zero_rat )
=> ( A2 = zero_zero_rat ) ) ).
% inverse_zero_imp_zero
thf(fact_245_inverse__zero__imp__zero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_246_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% field_class.field_inverse_zero
thf(fact_247_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_248_inverse__less__imp__less,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ).
% inverse_less_imp_less
thf(fact_249_inverse__less__imp__less,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ B2 @ A2 ) ) ) ).
% inverse_less_imp_less
thf(fact_250_less__imp__inverse__less,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_251_less__imp__inverse__less,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_252_inverse__less__imp__less__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_253_inverse__less__imp__less__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_254_bot__empty__eq,axiom,
( bot_bot_rat_o
= ( ^ [X2: rat] : ( member_rat @ X2 @ bot_bot_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_255_Collect__empty__eq__bot,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( P = bot_bot_rat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_256_Dedekind__Real_Ocut__def,axiom,
( dedekind_cut
= ( ^ [A4: set_rat] :
( ( ord_less_set_rat @ bot_bot_set_rat @ A4 )
& ( ord_less_set_rat @ A4 @ ( set_or575021546402375026an_rat @ zero_zero_rat ) )
& ! [X2: rat] :
( ( member_rat @ X2 @ A4 )
=> ( ! [Z3: rat] :
( ( ( ord_less_rat @ zero_zero_rat @ Z3 )
& ( ord_less_rat @ Z3 @ X2 ) )
=> ( member_rat @ Z3 @ A4 ) )
& ? [Y4: rat] :
( ( member_rat @ Y4 @ A4 )
& ( ord_less_rat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% Dedekind_Real.cut_def
thf(fact_257_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_258_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_259_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_260_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_261_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_262_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_263_Set_Ois__empty__def,axiom,
( is_empty_rat
= ( ^ [A4: set_rat] : ( A4 = bot_bot_set_rat ) ) ) ).
% Set.is_empty_def
thf(fact_264_mem__mult__set,axiom,
! [A: set_rat,B: set_rat] :
( ( dedekind_cut @ A )
=> ( ( dedekind_cut @ B )
=> ( dedekind_cut @ ( dedekind_mult_set @ A @ B ) ) ) ) ).
% mem_mult_set
thf(fact_265_mem__add__set,axiom,
! [A: set_rat,B: set_rat] :
( ( dedekind_cut @ A )
=> ( ( dedekind_cut @ B )
=> ( dedekind_cut @ ( dedekind_add_set @ A @ B ) ) ) ) ).
% mem_add_set
thf(fact_266_inverse__le__iff__le,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% inverse_le_iff_le
thf(fact_267_inverse__le__iff__le,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% inverse_le_iff_le
thf(fact_268_inverse__le__iff__le__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_269_inverse__le__iff__le__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_270_dual__order_Orefl,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_271_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_272_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_273_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_274_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_275_order__refl,axiom,
! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).
% order_refl
thf(fact_276_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_277_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_278_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_279_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_280_empty__subsetI,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A ) ).
% empty_subsetI
thf(fact_281_subset__empty,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ A @ bot_bot_set_rat )
= ( A = bot_bot_set_rat ) ) ).
% subset_empty
thf(fact_282_psubsetI,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_rat @ A @ B ) ) ) ).
% psubsetI
thf(fact_283_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_284_inverse__nonnegative__iff__nonnegative,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_285_inverse__nonnegative__iff__nonnegative,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_286_inverse__nonpositive__iff__nonpositive,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_287_inverse__nonpositive__iff__nonpositive,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_288_order__antisym__conv,axiom,
! [Y2: rat,X3: rat] :
( ( ord_less_eq_rat @ Y2 @ X3 )
=> ( ( ord_less_eq_rat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_289_order__antisym__conv,axiom,
! [Y2: num,X3: num] :
( ( ord_less_eq_num @ Y2 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_290_order__antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_291_order__antisym__conv,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_292_order__antisym__conv,axiom,
! [Y2: int,X3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_293_linorder__le__cases,axiom,
! [X3: rat,Y2: rat] :
( ~ ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_294_linorder__le__cases,axiom,
! [X3: num,Y2: num] :
( ~ ( ord_less_eq_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_295_linorder__le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_296_linorder__le__cases,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_297_linorder__le__cases,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_298_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_299_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_300_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_301_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_302_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_303_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_304_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_305_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_306_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_307_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_308_ord__eq__le__subst,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_309_ord__eq__le__subst,axiom,
! [A2: num,F: rat > num,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_310_ord__eq__le__subst,axiom,
! [A2: nat,F: rat > nat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_311_ord__eq__le__subst,axiom,
! [A2: real,F: rat > real,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_312_ord__eq__le__subst,axiom,
! [A2: int,F: rat > int,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_313_ord__eq__le__subst,axiom,
! [A2: rat,F: num > rat,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_314_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_315_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_316_ord__eq__le__subst,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_317_ord__eq__le__subst,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_318_linorder__linear,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
| ( ord_less_eq_rat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_319_linorder__linear,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
| ( ord_less_eq_num @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_320_linorder__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_321_linorder__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_322_linorder__linear,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_323_order__eq__refl,axiom,
! [X3: rat,Y2: rat] :
( ( X3 = Y2 )
=> ( ord_less_eq_rat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_324_order__eq__refl,axiom,
! [X3: num,Y2: num] :
( ( X3 = Y2 )
=> ( ord_less_eq_num @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_325_order__eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_326_order__eq__refl,axiom,
! [X3: real,Y2: real] :
( ( X3 = Y2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_327_order__eq__refl,axiom,
! [X3: int,Y2: int] :
( ( X3 = Y2 )
=> ( ord_less_eq_int @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_328_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_329_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_330_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_331_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_332_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_333_order__subst2,axiom,
! [A2: num,B2: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_334_order__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_335_order__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_336_order__subst2,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_337_order__subst2,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_338_order__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_339_order__subst1,axiom,
! [A2: rat,F: num > rat,B2: num,C: num] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_340_order__subst1,axiom,
! [A2: rat,F: nat > rat,B2: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_341_order__subst1,axiom,
! [A2: rat,F: real > rat,B2: real,C: real] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_342_order__subst1,axiom,
! [A2: rat,F: int > rat,B2: int,C: int] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_343_order__subst1,axiom,
! [A2: num,F: rat > num,B2: rat,C: rat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_344_order__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_345_order__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_346_order__subst1,axiom,
! [A2: num,F: real > num,B2: real,C: real] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_347_order__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_348_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_349_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_350_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_351_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_352_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_353_antisym,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_354_antisym,axiom,
! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_355_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_356_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_357_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_358_dual__order_Otrans,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ B2 )
=> ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_359_dual__order_Otrans,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_360_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_361_dual__order_Otrans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_362_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_363_dual__order_Oantisym,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_364_dual__order_Oantisym,axiom,
! [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_365_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_366_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_367_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_368_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_369_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_370_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_371_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_372_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_373_linorder__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B2: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: rat,B3: rat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_374_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B2: num] :
( ! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: num,B3: num] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_375_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_376_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_377_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_378_order__trans,axiom,
! [X3: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_eq_rat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_379_order__trans,axiom,
! [X3: num,Y2: num,Z: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z )
=> ( ord_less_eq_num @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_380_order__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_381_order__trans,axiom,
! [X3: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z )
=> ( ord_less_eq_real @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_382_order__trans,axiom,
! [X3: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_383_order_Otrans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_384_order_Otrans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_385_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_386_order_Otrans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_387_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_388_order__antisym,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_389_order__antisym,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_390_order__antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_391_order__antisym,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_392_order__antisym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_393_ord__le__eq__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_394_ord__le__eq__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_395_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_396_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_397_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_398_ord__eq__le__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_399_ord__eq__le__trans,axiom,
! [A2: num,B2: num,C: num] :
( ( A2 = B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_400_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_401_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_402_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_403_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 ) )
= ( ^ [X2: rat,Y4: rat] :
( ( ord_less_eq_rat @ X2 @ Y4 )
& ( ord_less_eq_rat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_404_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_405_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_406_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z4: real] : ( Y5 = Z4 ) )
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_407_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_408_le__cases3,axiom,
! [X3: rat,Y2: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X3 @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_rat @ X3 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_409_le__cases3,axiom,
! [X3: num,Y2: num,Z: num] :
( ( ( ord_less_eq_num @ X3 @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_num @ Y2 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_num @ Z @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y2 @ Z )
=> ~ ( ord_less_eq_num @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_410_le__cases3,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_411_le__cases3,axiom,
! [X3: real,Y2: real,Z: real] :
( ( ( ord_less_eq_real @ X3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_412_le__cases3,axiom,
! [X3: int,Y2: int,Z: int] :
( ( ( ord_less_eq_int @ X3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_413_nle__le,axiom,
! [A2: rat,B2: rat] :
( ( ~ ( ord_less_eq_rat @ A2 @ B2 ) )
= ( ( ord_less_eq_rat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_414_nle__le,axiom,
! [A2: num,B2: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B2 ) )
= ( ( ord_less_eq_num @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_415_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_416_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_417_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_418_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_419_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_420_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_421_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_422_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_423_order__le__imp__less__or__eq,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X3 @ Y2 )
=> ( ( ord_less_set_rat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_424_order__le__imp__less__or__eq,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ( ord_less_rat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_425_order__le__imp__less__or__eq,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ( ord_less_num @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_426_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_427_order__le__imp__less__or__eq,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_428_order__le__imp__less__or__eq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_429_linorder__le__less__linear,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
| ( ord_less_rat @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_430_linorder__le__less__linear,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
| ( ord_less_num @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_431_linorder__le__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_432_linorder__le__less__linear,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
| ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_433_linorder__le__less__linear,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_434_order__less__le__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_435_order__less__le__subst2,axiom,
! [A2: num,B2: num,F: num > rat,C: rat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_436_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_437_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > rat,C: rat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_438_order__less__le__subst2,axiom,
! [A2: rat,B2: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_439_order__less__le__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_440_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_441_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > num,C: num] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_442_order__less__le__subst2,axiom,
! [A2: rat,B2: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_443_order__less__le__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_444_order__less__le__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_445_order__less__le__subst1,axiom,
! [A2: num,F: rat > num,B2: rat,C: rat] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_446_order__less__le__subst1,axiom,
! [A2: nat,F: rat > nat,B2: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_447_order__less__le__subst1,axiom,
! [A2: real,F: rat > real,B2: rat,C: rat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_448_order__less__le__subst1,axiom,
! [A2: int,F: rat > int,B2: rat,C: rat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_449_order__less__le__subst1,axiom,
! [A2: rat,F: num > rat,B2: num,C: num] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_450_order__less__le__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_451_order__less__le__subst1,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_452_order__less__le__subst1,axiom,
! [A2: real,F: num > real,B2: num,C: num] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_453_order__less__le__subst1,axiom,
! [A2: int,F: num > int,B2: num,C: num] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_454_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_455_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_456_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_457_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_458_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_459_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_460_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_461_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_462_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_463_order__le__less__subst2,axiom,
! [A2: num,B2: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_464_order__le__less__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_465_order__le__less__subst1,axiom,
! [A2: rat,F: num > rat,B2: num,C: num] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_466_order__le__less__subst1,axiom,
! [A2: rat,F: nat > rat,B2: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_467_order__le__less__subst1,axiom,
! [A2: rat,F: int > rat,B2: int,C: int] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_468_order__le__less__subst1,axiom,
! [A2: num,F: rat > num,B2: rat,C: rat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_469_order__le__less__subst1,axiom,
! [A2: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_470_order__le__less__subst1,axiom,
! [A2: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_471_order__le__less__subst1,axiom,
! [A2: num,F: int > num,B2: int,C: int] :
( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_472_order__le__less__subst1,axiom,
! [A2: nat,F: rat > nat,B2: rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_473_order__le__less__subst1,axiom,
! [A2: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_474_order__less__le__trans,axiom,
! [X3: set_rat,Y2: set_rat,Z: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ Z )
=> ( ord_less_set_rat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_475_order__less__le__trans,axiom,
! [X3: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_rat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_476_order__less__le__trans,axiom,
! [X3: num,Y2: num,Z: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z )
=> ( ord_less_num @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_477_order__less__le__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_478_order__less__le__trans,axiom,
! [X3: real,Y2: real,Z: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z )
=> ( ord_less_real @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_479_order__less__le__trans,axiom,
! [X3: int,Y2: int,Z: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_int @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_480_order__le__less__trans,axiom,
! [X3: set_rat,Y2: set_rat,Z: set_rat] :
( ( ord_less_eq_set_rat @ X3 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ Z )
=> ( ord_less_set_rat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_481_order__le__less__trans,axiom,
! [X3: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_482_order__le__less__trans,axiom,
! [X3: num,Y2: num,Z: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z )
=> ( ord_less_num @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_483_order__le__less__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_484_order__le__less__trans,axiom,
! [X3: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z )
=> ( ord_less_real @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_485_order__le__less__trans,axiom,
! [X3: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_486_order__neq__le__trans,axiom,
! [A2: set_rat,B2: set_rat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ord_less_set_rat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_487_order__neq__le__trans,axiom,
! [A2: rat,B2: rat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_488_order__neq__le__trans,axiom,
! [A2: num,B2: num] :
( ( A2 != B2 )
=> ( ( ord_less_eq_num @ A2 @ B2 )
=> ( ord_less_num @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_489_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_490_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_491_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_492_order__le__neq__trans,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_rat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_493_order__le__neq__trans,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_494_order__le__neq__trans,axiom,
! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_num @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_495_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_496_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_497_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_498_order__less__imp__le,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X3 @ Y2 )
=> ( ord_less_eq_set_rat @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_499_order__less__imp__le,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_500_order__less__imp__le,axiom,
! [X3: num,Y2: num] :
( ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_501_order__less__imp__le,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_502_order__less__imp__le,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_503_order__less__imp__le,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_504_linorder__not__less,axiom,
! [X3: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X3 @ Y2 ) )
= ( ord_less_eq_rat @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_505_linorder__not__less,axiom,
! [X3: num,Y2: num] :
( ( ~ ( ord_less_num @ X3 @ Y2 ) )
= ( ord_less_eq_num @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_506_linorder__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_507_linorder__not__less,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_508_linorder__not__less,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_509_linorder__not__le,axiom,
! [X3: rat,Y2: rat] :
( ( ~ ( ord_less_eq_rat @ X3 @ Y2 ) )
= ( ord_less_rat @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_510_linorder__not__le,axiom,
! [X3: num,Y2: num] :
( ( ~ ( ord_less_eq_num @ X3 @ Y2 ) )
= ( ord_less_num @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_511_linorder__not__le,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_512_linorder__not__le,axiom,
! [X3: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y2 ) )
= ( ord_less_real @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_513_linorder__not__le,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y2 ) )
= ( ord_less_int @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_514_order__less__le,axiom,
( ord_less_set_rat
= ( ^ [X2: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_515_order__less__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y4: rat] :
( ( ord_less_eq_rat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_516_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_517_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_518_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_519_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_520_order__le__less,axiom,
( ord_less_eq_set_rat
= ( ^ [X2: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_521_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X2: rat,Y4: rat] :
( ( ord_less_rat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_522_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_num @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_523_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_524_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_525_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_526_dual__order_Ostrict__implies__order,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ord_less_eq_set_rat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_527_dual__order_Ostrict__implies__order,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_528_dual__order_Ostrict__implies__order,axiom,
! [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ord_less_eq_num @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_529_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_530_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_531_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_532_order_Ostrict__implies__order,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ord_less_eq_set_rat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_533_order_Ostrict__implies__order,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_534_order_Ostrict__implies__order,axiom,
! [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ord_less_eq_num @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_535_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_536_order_Ostrict__implies__order,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_537_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_538_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [B4: set_rat,A5: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A5 )
& ~ ( ord_less_eq_set_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_539_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_540_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_541_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_542_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_543_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_544_dual__order_Ostrict__trans2,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_545_dual__order_Ostrict__trans2,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ B2 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_546_dual__order_Ostrict__trans2,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_num @ B2 @ A2 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_547_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_548_dual__order_Ostrict__trans2,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_549_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_550_dual__order_Ostrict__trans1,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A2 )
=> ( ( ord_less_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_551_dual__order_Ostrict__trans1,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ B2 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_552_dual__order_Ostrict__trans1,axiom,
! [B2: num,A2: num,C: num] :
( ( ord_less_eq_num @ B2 @ A2 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_553_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_554_dual__order_Ostrict__trans1,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_555_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_556_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [B4: set_rat,A5: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_557_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_558_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_559_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_560_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_561_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_562_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [B4: set_rat,A5: set_rat] :
( ( ord_less_set_rat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_563_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_rat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_564_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_num @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_565_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_566_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_567_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_568_dense__le__bounded,axiom,
! [X3: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ! [W: rat] :
( ( ord_less_rat @ X3 @ W )
=> ( ( ord_less_rat @ W @ Y2 )
=> ( ord_less_eq_rat @ W @ Z ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_569_dense__le__bounded,axiom,
! [X3: real,Y2: real,Z: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_570_dense__ge__bounded,axiom,
! [Z: rat,X3: rat,Y2: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z @ W )
=> ( ( ord_less_rat @ W @ X3 )
=> ( ord_less_eq_rat @ Y2 @ W ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_571_dense__ge__bounded,axiom,
! [Z: real,X3: real,Y2: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_572_order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B4 )
& ~ ( ord_less_eq_set_rat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_573_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_574_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_575_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_576_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_577_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_578_order_Ostrict__trans2,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_579_order_Ostrict__trans2,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_580_order_Ostrict__trans2,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_num @ A2 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_581_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_582_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_583_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_584_order_Ostrict__trans1,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_585_order_Ostrict__trans1,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_586_order_Ostrict__trans1,axiom,
! [A2: num,B2: num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_587_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_588_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_589_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_590_order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_591_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_592_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_593_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_594_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_595_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_596_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B4: set_rat] :
( ( ord_less_set_rat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_597_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_rat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_598_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_599_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_600_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_601_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_602_not__le__imp__less,axiom,
! [Y2: rat,X3: rat] :
( ~ ( ord_less_eq_rat @ Y2 @ X3 )
=> ( ord_less_rat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_603_not__le__imp__less,axiom,
! [Y2: num,X3: num] :
( ~ ( ord_less_eq_num @ Y2 @ X3 )
=> ( ord_less_num @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_604_not__le__imp__less,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ord_less_nat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_605_not__le__imp__less,axiom,
! [Y2: real,X3: real] :
( ~ ( ord_less_eq_real @ Y2 @ X3 )
=> ( ord_less_real @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_606_not__le__imp__less,axiom,
! [Y2: int,X3: int] :
( ~ ( ord_less_eq_int @ Y2 @ X3 )
=> ( ord_less_int @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_607_less__le__not__le,axiom,
( ord_less_set_rat
= ( ^ [X2: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y4 )
& ~ ( ord_less_eq_set_rat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_608_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y4: rat] :
( ( ord_less_eq_rat @ X2 @ Y4 )
& ~ ( ord_less_eq_rat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_609_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y4: num] :
( ( ord_less_eq_num @ X2 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_610_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_611_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_612_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_613_dense__le,axiom,
! [Y2: rat,Z: rat] :
( ! [X: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ X @ Z ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ).
% dense_le
thf(fact_614_dense__le,axiom,
! [Y2: real,Z: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ).
% dense_le
thf(fact_615_dense__ge,axiom,
! [Z: rat,Y2: rat] :
( ! [X: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ord_less_eq_rat @ Y2 @ X ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_616_dense__ge,axiom,
! [Z: real,Y2: real] :
( ! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_617_antisym__conv2,axiom,
! [X3: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_rat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_618_antisym__conv2,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_rat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_619_antisym__conv2,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_num @ X3 @ Y2 )
=> ( ( ~ ( ord_less_num @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_620_antisym__conv2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_621_antisym__conv2,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ( ~ ( ord_less_real @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_622_antisym__conv2,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_623_antisym__conv1,axiom,
! [X3: set_rat,Y2: set_rat] :
( ~ ( ord_less_set_rat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_rat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_624_antisym__conv1,axiom,
! [X3: rat,Y2: rat] :
( ~ ( ord_less_rat @ X3 @ Y2 )
=> ( ( ord_less_eq_rat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_625_antisym__conv1,axiom,
! [X3: num,Y2: num] :
( ~ ( ord_less_num @ X3 @ Y2 )
=> ( ( ord_less_eq_num @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_626_antisym__conv1,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_627_antisym__conv1,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ( ord_less_eq_real @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_628_antisym__conv1,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_629_nless__le,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ~ ( ord_less_set_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_630_nless__le,axiom,
! [A2: rat,B2: rat] :
( ( ~ ( ord_less_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_631_nless__le,axiom,
! [A2: num,B2: num] :
( ( ~ ( ord_less_num @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_632_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_633_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_634_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_635_leI,axiom,
! [X3: rat,Y2: rat] :
( ~ ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X3 ) ) ).
% leI
thf(fact_636_leI,axiom,
! [X3: num,Y2: num] :
( ~ ( ord_less_num @ X3 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X3 ) ) ).
% leI
thf(fact_637_leI,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% leI
thf(fact_638_leI,axiom,
! [X3: real,Y2: real] :
( ~ ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% leI
thf(fact_639_leI,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% leI
thf(fact_640_leD,axiom,
! [Y2: set_rat,X3: set_rat] :
( ( ord_less_eq_set_rat @ Y2 @ X3 )
=> ~ ( ord_less_set_rat @ X3 @ Y2 ) ) ).
% leD
thf(fact_641_leD,axiom,
! [Y2: rat,X3: rat] :
( ( ord_less_eq_rat @ Y2 @ X3 )
=> ~ ( ord_less_rat @ X3 @ Y2 ) ) ).
% leD
thf(fact_642_leD,axiom,
! [Y2: num,X3: num] :
( ( ord_less_eq_num @ Y2 @ X3 )
=> ~ ( ord_less_num @ X3 @ Y2 ) ) ).
% leD
thf(fact_643_leD,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_644_leD,axiom,
! [Y2: real,X3: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ~ ( ord_less_real @ X3 @ Y2 ) ) ).
% leD
thf(fact_645_leD,axiom,
! [Y2: int,X3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ~ ( ord_less_int @ X3 @ Y2 ) ) ).
% leD
thf(fact_646_bot_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
=> ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_uniqueI
thf(fact_647_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_648_bot_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
= ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_unique
thf(fact_649_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_650_bot_Oextremum,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A2 ) ).
% bot.extremum
thf(fact_651_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_652_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B5: set_rat] :
( ( ord_less_set_rat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_653_subset__psubset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_654_subset__not__subset__eq,axiom,
( ord_less_set_rat
= ( ^ [A4: set_rat,B5: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B5 )
& ~ ( ord_less_eq_set_rat @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_655_psubset__subset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_656_psubset__imp__subset,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_657_psubset__eq,axiom,
( ord_less_set_rat
= ( ^ [A4: set_rat,B5: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_658_psubsetE,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ( ord_less_eq_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ B @ A ) ) ) ).
% psubsetE
thf(fact_659_inverse__le__imp__le,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% inverse_le_imp_le
thf(fact_660_inverse__le__imp__le,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% inverse_le_imp_le
thf(fact_661_le__imp__inverse__le,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% le_imp_inverse_le
thf(fact_662_le__imp__inverse__le,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% le_imp_inverse_le
thf(fact_663_inverse__le__imp__le__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_664_inverse__le__imp__le__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_665_le__imp__inverse__le__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_666_le__imp__inverse__le__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_667_preal__downwards__closed_H,axiom,
! [A: set_rat,Y2: rat,Z: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ Z @ Y2 )
=> ( member_rat @ Z @ A ) ) ) ) ) ).
% preal_downwards_closed'
thf(fact_668_greaterThan__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or6174011595382531368et_rat @ K ) )
= ( ord_less_set_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_669_greaterThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_or6990855429499425204an_num @ K ) )
= ( ord_less_num @ K @ I ) ) ).
% greaterThan_iff
thf(fact_670_greaterThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_or1210151606488870762an_nat @ K ) )
= ( ord_less_nat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_671_greaterThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_or1207661135979820486an_int @ K ) )
= ( ord_less_int @ K @ I ) ) ).
% greaterThan_iff
thf(fact_672_greaterThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_or575021546402375026an_rat @ K ) )
= ( ord_less_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_673_greaterThan__subset__iff,axiom,
! [X3: num,Y2: num] :
( ( ord_less_eq_set_num @ ( set_or6990855429499425204an_num @ X3 ) @ ( set_or6990855429499425204an_num @ Y2 ) )
= ( ord_less_eq_num @ Y2 @ X3 ) ) ).
% greaterThan_subset_iff
thf(fact_674_greaterThan__subset__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1210151606488870762an_nat @ X3 ) @ ( set_or1210151606488870762an_nat @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% greaterThan_subset_iff
thf(fact_675_greaterThan__subset__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_eq_set_real @ ( set_or5849166863359141190n_real @ X3 ) @ ( set_or5849166863359141190n_real @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X3 ) ) ).
% greaterThan_subset_iff
thf(fact_676_greaterThan__subset__iff,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_or1207661135979820486an_int @ X3 ) @ ( set_or1207661135979820486an_int @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% greaterThan_subset_iff
thf(fact_677_greaterThan__subset__iff,axiom,
! [X3: rat,Y2: rat] :
( ( ord_less_eq_set_rat @ ( set_or575021546402375026an_rat @ X3 ) @ ( set_or575021546402375026an_rat @ Y2 ) )
= ( ord_less_eq_rat @ Y2 @ X3 ) ) ).
% greaterThan_subset_iff
thf(fact_678_greaterThan__eq__iff,axiom,
! [X3: rat,Y2: rat] :
( ( ( set_or575021546402375026an_rat @ X3 )
= ( set_or575021546402375026an_rat @ Y2 ) )
= ( X3 = Y2 ) ) ).
% greaterThan_eq_iff
thf(fact_679_greaterThan__non__empty,axiom,
! [X3: rat] :
( ( set_or575021546402375026an_rat @ X3 )
!= bot_bot_set_rat ) ).
% greaterThan_non_empty
thf(fact_680_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B2 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_681_complete__interval,axiom,
! [A2: real,B2: real,P: real > $o] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A2 @ C3 )
& ( ord_less_eq_real @ C3 @ B2 )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A2 @ X5 )
& ( ord_less_real @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: real] :
( ! [X: real] :
( ( ( ord_less_eq_real @ A2 @ X )
& ( ord_less_real @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_real @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_682_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B2 )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A2 @ X )
& ( ord_less_int @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_683_verit__comp__simplify1_I3_J,axiom,
! [B6: rat,A6: rat] :
( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
= ( ord_less_rat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_684_verit__comp__simplify1_I3_J,axiom,
! [B6: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
= ( ord_less_num @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_685_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_686_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_687_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_688_pinf_I6_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_689_pinf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_690_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_691_pinf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_692_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_693_pinf_I8_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ( ord_less_eq_rat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_694_pinf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_695_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_696_pinf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_697_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_698_minf_I6_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ( ord_less_eq_rat @ X5 @ T ) ) ).
% minf(6)
thf(fact_699_minf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_700_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_701_minf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_702_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_703_minf_I8_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% minf(8)
thf(fact_704_minf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_705_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_706_minf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_707_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_708_subsetI,axiom,
! [A: set_rat,B: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ X @ B ) )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% subsetI
thf(fact_709_Collect__mono__iff,axiom,
! [P: rat > $o,Q: rat > $o] :
( ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) )
= ( ! [X2: rat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_710_Collect__mono,axiom,
! [P: rat > $o,Q: rat > $o] :
( ! [X: rat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) ) ) ).
% Collect_mono
thf(fact_711_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B5: set_rat] :
! [T2: rat] :
( ( member_rat @ T2 @ A4 )
=> ( member_rat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_712_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B5: set_rat] :
! [X2: rat] :
( ( member_rat @ X2 @ A4 )
=> ( member_rat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_713_subsetD,axiom,
! [A: set_rat,B: set_rat,C: rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% subsetD
thf(fact_714_in__mono,axiom,
! [A: set_rat,B: set_rat,X3: rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( member_rat @ X3 @ A )
=> ( member_rat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_715_verit__la__disequality,axiom,
! [A2: rat,B2: rat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_rat @ A2 @ B2 )
| ~ ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_716_verit__la__disequality,axiom,
! [A2: num,B2: num] :
( ( A2 = B2 )
| ~ ( ord_less_eq_num @ A2 @ B2 )
| ~ ( ord_less_eq_num @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_717_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_718_verit__la__disequality,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
| ~ ( ord_less_eq_real @ A2 @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_719_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_720_verit__comp__simplify1_I2_J,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_721_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_722_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_723_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_724_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_725_minf_I7_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ~ ( ord_less_rat @ T @ X5 ) ) ).
% minf(7)
thf(fact_726_minf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ~ ( ord_less_num @ T @ X5 ) ) ).
% minf(7)
thf(fact_727_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_728_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_729_minf_I5_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ( ord_less_rat @ X5 @ T ) ) ).
% minf(5)
thf(fact_730_minf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ( ord_less_num @ X5 @ T ) ) ).
% minf(5)
thf(fact_731_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_732_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_733_minf_I4_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_734_minf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_735_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_736_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_737_minf_I3_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_738_minf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_739_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_740_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_741_minf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_742_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_743_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_744_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_745_minf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_746_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_747_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_748_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_749_pinf_I7_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ( ord_less_rat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_750_pinf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ( ord_less_num @ T @ X5 ) ) ).
% pinf(7)
thf(fact_751_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_752_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_753_pinf_I5_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ~ ( ord_less_rat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_754_pinf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ~ ( ord_less_num @ X5 @ T ) ) ).
% pinf(5)
thf(fact_755_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_756_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_757_pinf_I4_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_758_pinf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_759_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_760_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_761_pinf_I3_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_762_pinf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_763_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_764_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_765_pinf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_766_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_767_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_768_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_769_pinf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_770_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: num] :
! [X: num] :
( ( ord_less_num @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X5: num] :
( ( ord_less_num @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_771_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_772_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less_int @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_773_verit__comp__simplify1_I1_J,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_774_verit__comp__simplify1_I1_J,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_775_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_776_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_777_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_778_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X2: rat,Y4: rat] :
( ( ord_less_rat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_rat_def
thf(fact_779_subset__emptyI,axiom,
! [A: set_rat] :
( ! [X: rat] :
~ ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ A @ bot_bot_set_rat ) ) ).
% subset_emptyI
thf(fact_780_one__le__inverse__iff,axiom,
! [X3: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X3 ) )
= ( ( ord_less_rat @ zero_zero_rat @ X3 )
& ( ord_less_eq_rat @ X3 @ one_one_rat ) ) ) ).
% one_le_inverse_iff
thf(fact_781_one__le__inverse__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X3 ) )
= ( ( ord_less_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_782_inverse__less__1__iff,axiom,
! [X3: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ X3 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
| ( ord_less_rat @ one_one_rat @ X3 ) ) ) ).
% inverse_less_1_iff
thf(fact_783_inverse__less__1__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X3 ) @ one_one_real )
= ( ( ord_less_eq_real @ X3 @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% inverse_less_1_iff
thf(fact_784_one__le__inverse,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ one_one_rat )
=> ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% one_le_inverse
thf(fact_785_one__le__inverse,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% one_le_inverse
thf(fact_786_inverse__less__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
=> ( ord_less_rat @ B2 @ A2 ) )
& ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ) ).
% inverse_less_iff
thf(fact_787_inverse__less__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
=> ( ord_less_real @ B2 @ A2 ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
=> ( ord_less_real @ A2 @ B2 ) ) ) ) ).
% inverse_less_iff
thf(fact_788_inverse__le__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
=> ( ord_less_eq_rat @ B2 @ A2 ) )
& ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).
% inverse_le_iff
thf(fact_789_inverse__le__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
=> ( ord_less_eq_real @ B2 @ A2 ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% inverse_le_iff
thf(fact_790_of__rat__le__0__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ zero_zero_rat )
= ( ord_less_eq_rat @ R @ zero_zero_rat ) ) ).
% of_rat_le_0_iff
thf(fact_791_of__rat__le__0__iff,axiom,
! [R: rat] :
( ( ord_less_eq_real @ ( field_7254667332652039916t_real @ R ) @ zero_zero_real )
= ( ord_less_eq_rat @ R @ zero_zero_rat ) ) ).
% of_rat_le_0_iff
thf(fact_792_mult__1,axiom,
! [A2: rat] :
( ( times_times_rat @ one_one_rat @ A2 )
= A2 ) ).
% mult_1
thf(fact_793_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_794_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_795_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_796_mult_Oright__neutral,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ one_one_rat )
= A2 ) ).
% mult.right_neutral
thf(fact_797_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_798_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_799_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_800_inverse__mult__distrib,axiom,
! [A2: rat,B2: rat] :
( ( inverse_inverse_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) ) ) ).
% inverse_mult_distrib
thf(fact_801_inverse__mult__distrib,axiom,
! [A2: real,B2: real] :
( ( inverse_inverse_real @ ( times_times_real @ A2 @ B2 ) )
= ( times_times_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B2 ) ) ) ).
% inverse_mult_distrib
thf(fact_802_inverse__eq__1__iff,axiom,
! [X3: rat] :
( ( ( inverse_inverse_rat @ X3 )
= one_one_rat )
= ( X3 = one_one_rat ) ) ).
% inverse_eq_1_iff
thf(fact_803_inverse__eq__1__iff,axiom,
! [X3: real] :
( ( ( inverse_inverse_real @ X3 )
= one_one_real )
= ( X3 = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_804_inverse__1,axiom,
( ( inverse_inverse_rat @ one_one_rat )
= one_one_rat ) ).
% inverse_1
thf(fact_805_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_806_of__rat__1,axiom,
( ( field_7254667332652039916t_real @ one_one_rat )
= one_one_real ) ).
% of_rat_1
thf(fact_807_of__rat__eq__1__iff,axiom,
! [A2: rat] :
( ( ( field_7254667332652039916t_real @ A2 )
= one_one_real )
= ( A2 = one_one_rat ) ) ).
% of_rat_eq_1_iff
thf(fact_808_one__eq__of__rat__iff,axiom,
! [A2: rat] :
( ( one_one_real
= ( field_7254667332652039916t_real @ A2 ) )
= ( one_one_rat = A2 ) ) ).
% one_eq_of_rat_iff
thf(fact_809_of__rat__0,axiom,
( ( field_2639924705303425560at_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% of_rat_0
thf(fact_810_of__rat__0,axiom,
( ( field_7254667332652039916t_real @ zero_zero_rat )
= zero_zero_real ) ).
% of_rat_0
thf(fact_811_of__rat__eq__0__iff,axiom,
! [A2: rat] :
( ( ( field_2639924705303425560at_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% of_rat_eq_0_iff
thf(fact_812_of__rat__eq__0__iff,axiom,
! [A2: rat] :
( ( ( field_7254667332652039916t_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_rat ) ) ).
% of_rat_eq_0_iff
thf(fact_813_zero__eq__of__rat__iff,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( field_2639924705303425560at_rat @ A2 ) )
= ( zero_zero_rat = A2 ) ) ).
% zero_eq_of_rat_iff
thf(fact_814_zero__eq__of__rat__iff,axiom,
! [A2: rat] :
( ( zero_zero_real
= ( field_7254667332652039916t_real @ A2 ) )
= ( zero_zero_rat = A2 ) ) ).
% zero_eq_of_rat_iff
thf(fact_815_left__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ A2 )
= one_one_rat ) ) ).
% left_inverse
thf(fact_816_left__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A2 ) @ A2 )
= one_one_real ) ) ).
% left_inverse
thf(fact_817_right__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( times_times_rat @ A2 @ ( inverse_inverse_rat @ A2 ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_818_right__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ ( inverse_inverse_real @ A2 ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_819_one__less__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_real @ one_one_real @ ( field_7254667332652039916t_real @ R ) )
= ( ord_less_rat @ one_one_rat @ R ) ) ).
% one_less_of_rat_iff
thf(fact_820_one__less__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_rat @ one_one_rat @ R ) ) ).
% one_less_of_rat_iff
thf(fact_821_of__rat__less__1__iff,axiom,
! [R: rat] :
( ( ord_less_real @ ( field_7254667332652039916t_real @ R ) @ one_one_real )
= ( ord_less_rat @ R @ one_one_rat ) ) ).
% of_rat_less_1_iff
thf(fact_822_of__rat__less__1__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ one_one_rat )
= ( ord_less_rat @ R @ one_one_rat ) ) ).
% of_rat_less_1_iff
thf(fact_823_of__rat__le__1__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ one_one_rat )
= ( ord_less_eq_rat @ R @ one_one_rat ) ) ).
% of_rat_le_1_iff
thf(fact_824_of__rat__le__1__iff,axiom,
! [R: rat] :
( ( ord_less_eq_real @ ( field_7254667332652039916t_real @ R ) @ one_one_real )
= ( ord_less_eq_rat @ R @ one_one_rat ) ) ).
% of_rat_le_1_iff
thf(fact_825_one__le__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_eq_rat @ one_one_rat @ R ) ) ).
% one_le_of_rat_iff
thf(fact_826_one__le__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_eq_real @ one_one_real @ ( field_7254667332652039916t_real @ R ) )
= ( ord_less_eq_rat @ one_one_rat @ R ) ) ).
% one_le_of_rat_iff
thf(fact_827_zero__less__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_real @ zero_zero_real @ ( field_7254667332652039916t_real @ R ) )
= ( ord_less_rat @ zero_zero_rat @ R ) ) ).
% zero_less_of_rat_iff
thf(fact_828_zero__less__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_rat @ zero_zero_rat @ R ) ) ).
% zero_less_of_rat_iff
thf(fact_829_of__rat__less__0__iff,axiom,
! [R: rat] :
( ( ord_less_real @ ( field_7254667332652039916t_real @ R ) @ zero_zero_real )
= ( ord_less_rat @ R @ zero_zero_rat ) ) ).
% of_rat_less_0_iff
thf(fact_830_of__rat__less__0__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ zero_zero_rat )
= ( ord_less_rat @ R @ zero_zero_rat ) ) ).
% of_rat_less_0_iff
thf(fact_831_zero__le__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_eq_rat @ zero_zero_rat @ R ) ) ).
% zero_le_of_rat_iff
thf(fact_832_zero__le__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_eq_real @ zero_zero_real @ ( field_7254667332652039916t_real @ R ) )
= ( ord_less_eq_rat @ zero_zero_rat @ R ) ) ).
% zero_le_of_rat_iff
thf(fact_833_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_834_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_835_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_836_mult_Oleft__commute,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( times_times_rat @ B2 @ ( times_times_rat @ A2 @ C ) )
= ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_837_mult_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_838_mult_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_839_mult_Oleft__commute,axiom,
! [B2: real,A2: real,C: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A2 @ C ) )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_840_mult_Ocomm__neutral,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ one_one_rat )
= A2 ) ).
% mult.comm_neutral
thf(fact_841_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_842_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_843_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_844_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_845_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_846_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_847_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_848_mult_Oassoc,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
= ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_849_mult_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_850_mult_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_851_mult_Oassoc,axiom,
! [A2: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_852_comm__monoid__mult__class_Omult__1,axiom,
! [A2: rat] :
( ( times_times_rat @ one_one_rat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_853_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_854_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_855_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_856_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
= ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_857_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_858_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_859_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_860_of__rat__mult,axiom,
! [A2: rat,B2: rat] :
( ( field_2639924705303425560at_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_rat @ ( field_2639924705303425560at_rat @ A2 ) @ ( field_2639924705303425560at_rat @ B2 ) ) ) ).
% of_rat_mult
thf(fact_861_of__rat__mult,axiom,
! [A2: rat,B2: rat] :
( ( field_7254667332652039916t_real @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_real @ ( field_7254667332652039916t_real @ A2 ) @ ( field_7254667332652039916t_real @ B2 ) ) ) ).
% of_rat_mult
thf(fact_862_inverse__unique,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
= one_one_rat )
=> ( ( inverse_inverse_rat @ A2 )
= B2 ) ) ).
% inverse_unique
thf(fact_863_inverse__unique,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= one_one_real )
=> ( ( inverse_inverse_real @ A2 )
= B2 ) ) ).
% inverse_unique
thf(fact_864_field__class_Ofield__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ A2 )
= one_one_rat ) ) ).
% field_class.field_inverse
thf(fact_865_field__class_Ofield__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A2 ) @ A2 )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_866_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_867_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_868_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_869_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_870_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_871_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_872_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_873_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_874_mult__commute__imp__mult__inverse__commute,axiom,
! [Y2: rat,X3: rat] :
( ( ( times_times_rat @ Y2 @ X3 )
= ( times_times_rat @ X3 @ Y2 ) )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ Y2 ) @ X3 )
= ( times_times_rat @ X3 @ ( inverse_inverse_rat @ Y2 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_875_mult__commute__imp__mult__inverse__commute,axiom,
! [Y2: real,X3: real] :
( ( ( times_times_real @ Y2 @ X3 )
= ( times_times_real @ X3 @ Y2 ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y2 ) @ X3 )
= ( times_times_real @ X3 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_876_of__rat__less,axiom,
! [R: rat,S: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S ) )
= ( ord_less_rat @ R @ S ) ) ).
% of_rat_less
thf(fact_877_field__le__mult__one__interval,axiom,
! [X3: rat,Y2: rat] :
( ! [Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ Z2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X3 ) @ Y2 ) ) )
=> ( ord_less_eq_rat @ X3 @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_878_field__le__mult__one__interval,axiom,
! [X3: real,Y2: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X3 @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_879_of__rat__less__eq,axiom,
! [R: rat,S: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S ) )
= ( ord_less_eq_rat @ R @ S ) ) ).
% of_rat_less_eq
thf(fact_880_of__rat__less__eq,axiom,
! [R: rat,S: rat] :
( ( ord_less_eq_real @ ( field_7254667332652039916t_real @ R ) @ ( field_7254667332652039916t_real @ S ) )
= ( ord_less_eq_rat @ R @ S ) ) ).
% of_rat_less_eq
thf(fact_881_of__rat__inverse,axiom,
! [A2: rat] :
( ( field_2639924705303425560at_rat @ ( inverse_inverse_rat @ A2 ) )
= ( inverse_inverse_rat @ ( field_2639924705303425560at_rat @ A2 ) ) ) ).
% of_rat_inverse
thf(fact_882_of__rat__inverse,axiom,
! [A2: rat] :
( ( field_7254667332652039916t_real @ ( inverse_inverse_rat @ A2 ) )
= ( inverse_inverse_real @ ( field_7254667332652039916t_real @ A2 ) ) ) ).
% of_rat_inverse
thf(fact_883_nonzero__inverse__mult__distrib,axiom,
! [A2: rat,B2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( B2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_884_nonzero__inverse__mult__distrib,axiom,
! [A2: real,B2: real] :
( ( A2 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A2 @ B2 ) )
= ( times_times_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A2 ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_885_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_886_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_887_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_888_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_889_nonzero__of__rat__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( field_2639924705303425560at_rat @ ( inverse_inverse_rat @ A2 ) )
= ( inverse_inverse_rat @ ( field_2639924705303425560at_rat @ A2 ) ) ) ) ).
% nonzero_of_rat_inverse
thf(fact_890_nonzero__of__rat__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( field_7254667332652039916t_real @ ( inverse_inverse_rat @ A2 ) )
= ( inverse_inverse_real @ ( field_7254667332652039916t_real @ A2 ) ) ) ) ).
% nonzero_of_rat_inverse
thf(fact_891_inverse__le__1__iff,axiom,
! [X3: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X3 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
| ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).
% inverse_le_1_iff
thf(fact_892_inverse__le__1__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X3 ) @ one_one_real )
= ( ( ord_less_eq_real @ X3 @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% inverse_le_1_iff
thf(fact_893_one__less__inverse,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ A2 @ one_one_rat )
=> ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% one_less_inverse
thf(fact_894_one__less__inverse,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ A2 @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% one_less_inverse
thf(fact_895_one__less__inverse__iff,axiom,
! [X3: rat] :
( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X3 ) )
= ( ( ord_less_rat @ zero_zero_rat @ X3 )
& ( ord_less_rat @ X3 @ one_one_rat ) ) ) ).
% one_less_inverse_iff
thf(fact_896_one__less__inverse__iff,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X3 ) )
= ( ( ord_less_real @ zero_zero_real @ X3 )
& ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_897_mult__cancel__left1,axiom,
! [C: rat,B2: rat] :
( ( C
= ( times_times_rat @ C @ B2 ) )
= ( ( C = zero_zero_rat )
| ( B2 = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_898_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_899_mult__cancel__left1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_900_mult__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ( times_times_rat @ C @ A2 )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_901_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_902_mult__cancel__left2,axiom,
! [C: real,A2: real] :
( ( ( times_times_real @ C @ A2 )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_903_mult__cancel__right1,axiom,
! [C: rat,B2: rat] :
( ( C
= ( times_times_rat @ B2 @ C ) )
= ( ( C = zero_zero_rat )
| ( B2 = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_904_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_905_mult__cancel__right1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_906_mult__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ( times_times_rat @ A2 @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_907_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_908_mult__cancel__right2,axiom,
! [A2: real,C: real] :
( ( ( times_times_real @ A2 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_909_mult__less__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_910_mult__less__cancel__right2,axiom,
! [A2: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_911_mult__less__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_912_mult__less__cancel__right1,axiom,
! [C: rat,B2: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B2 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_913_mult__less__cancel__right1,axiom,
! [C: real,B2: real] :
( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_914_mult__less__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_915_mult__less__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_916_mult__less__cancel__left2,axiom,
! [C: real,A2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_917_mult__less__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_918_mult__zero__left,axiom,
! [A2: rat] :
( ( times_times_rat @ zero_zero_rat @ A2 )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_919_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_920_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_921_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_922_mult__zero__right,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_923_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_924_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_925_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_926_mult__eq__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
= zero_zero_rat )
= ( ( A2 = zero_zero_rat )
| ( B2 = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_927_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_928_mult__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_929_mult__eq__0__iff,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_930_mult__cancel__left,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B2 ) )
= ( ( C = zero_zero_rat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_931_mult__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_932_mult__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_933_mult__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_934_mult__cancel__right,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B2 @ C ) )
= ( ( C = zero_zero_rat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_935_mult__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_936_mult__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_937_mult__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_938_linorder__neqE__linordered__idom,axiom,
! [X3: rat,Y2: rat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_rat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_939_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_940_mult__not__zero,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
!= zero_zero_rat )
=> ( ( A2 != zero_zero_rat )
& ( B2 != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_941_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_942_mult__not__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_943_mult__not__zero,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_944_divisors__zero,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
= zero_zero_rat )
=> ( ( A2 = zero_zero_rat )
| ( B2 = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_945_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_946_divisors__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_947_divisors__zero,axiom,
! [A2: real,B2: real] :
( ( ( times_times_real @ A2 @ B2 )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_948_no__zero__divisors,axiom,
! [A2: rat,B2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( B2 != zero_zero_rat )
=> ( ( times_times_rat @ A2 @ B2 )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_949_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_950_no__zero__divisors,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_951_no__zero__divisors,axiom,
! [A2: real,B2: real] :
( ( A2 != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_952_mult__left__cancel,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_953_mult__left__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_954_mult__left__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_955_mult__left__cancel,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_956_mult__right__cancel,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_957_mult__right__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_958_mult__right__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_959_mult__right__cancel,axiom,
! [C: real,A2: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_960_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_961_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_962_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_963_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_964_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_965_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_966_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_967_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_968_zero__le__mult__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_969_zero__le__mult__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_970_zero__le__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_971_mult__nonneg__nonpos2,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_972_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_973_mult__nonneg__nonpos2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_974_mult__nonneg__nonpos2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_975_mult__nonpos__nonneg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_976_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_977_mult__nonpos__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_978_mult__nonpos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_979_mult__nonneg__nonpos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_980_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_981_mult__nonneg__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_982_mult__nonneg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_983_mult__nonneg__nonneg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_984_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_985_mult__nonneg__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_986_mult__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_987_split__mult__neg__le,axiom,
! [A2: rat,B2: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_988_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_989_split__mult__neg__le,axiom,
! [A2: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_990_split__mult__neg__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_991_mult__le__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_992_mult__le__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_993_mult__le__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_994_mult__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_995_mult__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_996_mult__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_997_mult__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_998_mult__right__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_999_mult__right__mono__neg,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1000_mult__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1001_mult__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1002_mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1003_mult__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1004_mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1005_mult__nonpos__nonpos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1006_mult__nonpos__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1007_mult__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1008_mult__left__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1009_mult__left__mono__neg,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1010_mult__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1011_split__mult__pos__le,axiom,
! [A2: rat,B2: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1012_split__mult__pos__le,axiom,
! [A2: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1013_split__mult__pos__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1014_zero__le__square,axiom,
! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1015_zero__le__square,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1016_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1017_mult__mono_H,axiom,
! [A2: rat,B2: rat,C: rat,D3: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ D3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1018_mult__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1019_mult__mono_H,axiom,
! [A2: real,B2: real,C: real,D3: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1020_mult__mono_H,axiom,
! [A2: int,B2: int,C: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1021_mult__mono,axiom,
! [A2: rat,B2: rat,C: rat,D3: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ D3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1022_mult__mono,axiom,
! [A2: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1023_mult__mono,axiom,
! [A2: real,B2: real,C: real,D3: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1024_mult__mono,axiom,
! [A2: int,B2: int,C: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1025_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_1026_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1027_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1028_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1029_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1030_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1031_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1032_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1033_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_1034_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1035_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1036_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1037_mult__neg__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_1038_mult__neg__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_1039_mult__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_1040_not__square__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_1041_not__square__less__zero,axiom,
! [A2: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_1042_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1043_mult__less__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_1044_mult__less__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_1045_mult__less__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_1046_mult__neg__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_1047_mult__neg__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_1048_mult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1049_mult__neg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1050_mult__pos__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_1051_mult__pos__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_1052_mult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1053_mult__pos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1054_mult__pos__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1055_mult__pos__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1056_mult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1057_mult__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1058_mult__pos__neg2,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_1059_mult__pos__neg2,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_1060_mult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1061_mult__pos__neg2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1062_zero__less__mult__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1063_zero__less__mult__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1064_zero__less__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1065_zero__less__mult__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1066_zero__less__mult__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1067_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1068_zero__less__mult__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1069_zero__less__mult__pos2,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1070_zero__less__mult__pos2,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1071_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1072_zero__less__mult__pos2,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1073_mult__less__cancel__left__neg,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_real @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1074_mult__less__cancel__left__neg,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1075_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1076_mult__less__cancel__left__pos,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1077_mult__less__cancel__left__pos,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_rat @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1078_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1079_mult__strict__left__mono__neg,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1080_mult__strict__left__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1081_mult__strict__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1082_mult__strict__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1083_mult__strict__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1084_mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1085_mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1086_mult__less__cancel__left__disj,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1087_mult__less__cancel__left__disj,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A2 @ B2 ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1088_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1089_mult__strict__right__mono__neg,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1090_mult__strict__right__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1091_mult__strict__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1092_mult__strict__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1093_mult__strict__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1094_mult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1095_mult__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1096_mult__less__cancel__right__disj,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B2 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1097_mult__less__cancel__right__disj,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A2 @ B2 ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1098_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1099_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1100_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1101_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1102_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1103_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1104_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_1105_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1106_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1107_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1108_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_1109_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1110_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1111_less__1__mult,axiom,
! [M2: real,N2: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_1112_less__1__mult,axiom,
! [M2: rat,N2: rat] :
( ( ord_less_rat @ one_one_rat @ M2 )
=> ( ( ord_less_rat @ one_one_rat @ N2 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_1113_less__1__mult,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_1114_less__1__mult,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_1115_mult__less__le__imp__less,axiom,
! [A2: rat,B2: rat,C: rat,D3: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ D3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1116_mult__less__le__imp__less,axiom,
! [A2: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1117_mult__less__le__imp__less,axiom,
! [A2: real,B2: real,C: real,D3: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1118_mult__less__le__imp__less,axiom,
! [A2: int,B2: int,C: int,D3: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1119_mult__le__less__imp__less,axiom,
! [A2: rat,B2: rat,C: rat,D3: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D3 )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1120_mult__le__less__imp__less,axiom,
! [A2: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1121_mult__le__less__imp__less,axiom,
! [A2: real,B2: real,C: real,D3: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1122_mult__le__less__imp__less,axiom,
! [A2: int,B2: int,C: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1123_mult__right__le__imp__le,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1124_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1125_mult__right__le__imp__le,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1126_mult__right__le__imp__le,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1127_mult__left__le__imp__le,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1128_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1129_mult__left__le__imp__le,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1130_mult__left__le__imp__le,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1131_mult__le__cancel__left__pos,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1132_mult__le__cancel__left__pos,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1133_mult__le__cancel__left__pos,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1134_mult__le__cancel__left__neg,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1135_mult__le__cancel__left__neg,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1136_mult__le__cancel__left__neg,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1137_mult__less__cancel__right,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B2 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1138_mult__less__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1139_mult__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1140_mult__strict__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1141_mult__strict__mono_H,axiom,
! [A2: real,B2: real,C: real,D3: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1142_mult__strict__mono_H,axiom,
! [A2: int,B2: int,C: int,D3: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1143_obtain__pos__sum,axiom,
! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ~ ! [S2: rat] :
( ( ord_less_rat @ zero_zero_rat @ S2 )
=> ! [T3: rat] :
( ( ord_less_rat @ zero_zero_rat @ T3 )
=> ( R
!= ( plus_plus_rat @ S2 @ T3 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_1144_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q3: rat,R2: rat] : ( times_times_rat @ Q3 @ ( inverse_inverse_rat @ R2 ) ) ) ) ).
% divide_rat_def
thf(fact_1145_add__inc,axiom,
! [X3: num,Y2: num] :
( ( plus_plus_num @ X3 @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus_num @ X3 @ Y2 ) ) ) ).
% add_inc
thf(fact_1146_mult__inc,axiom,
! [X3: num,Y2: num] :
( ( times_times_num @ X3 @ ( inc @ Y2 ) )
= ( plus_plus_num @ ( times_times_num @ X3 @ Y2 ) @ X3 ) ) ).
% mult_inc
thf(fact_1147_semiring__norm_I12_J,axiom,
! [N2: num] :
( ( times_times_num @ one @ N2 )
= N2 ) ).
% semiring_norm(12)
thf(fact_1148_semiring__norm_I11_J,axiom,
! [M2: num] :
( ( times_times_num @ M2 @ one )
= M2 ) ).
% semiring_norm(11)
thf(fact_1149_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1150_num__induct,axiom,
! [P: num > $o,X3: num] :
( ( P @ one )
=> ( ! [X: num] :
( ( P @ X )
=> ( P @ ( inc @ X ) ) )
=> ( P @ X3 ) ) ) ).
% num_induct
thf(fact_1151_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1152_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_1153_add__One__commute,axiom,
! [N2: num] :
( ( plus_plus_num @ one @ N2 )
= ( plus_plus_num @ N2 @ one ) ) ).
% add_One_commute
thf(fact_1154_add__One,axiom,
! [X3: num] :
( ( plus_plus_num @ X3 @ one )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_1155_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1156_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1157_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1158_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1159_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1160_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1161_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1162_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1163_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1164_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1165_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1166_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1167_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1168_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1169_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1170_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1171_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1172_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1173_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1174_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1175_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1176_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1177_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1178_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1179_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1180_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1181_div__mult2__eq,axiom,
! [M2: nat,N2: nat,Q4: nat] :
( ( divide_divide_nat @ M2 @ ( times_times_nat @ N2 @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N2 ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_1182_div__less__iff__less__mult,axiom,
! [Q4: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q4 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1183_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1184_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1185_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q4 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1186_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1187_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1188_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I2: nat,J: nat] :
( ( ( ord_less_nat @ J @ N2 )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1189_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1190_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1191_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1192_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1193_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1194_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1195_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_1196_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1197_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1198_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1199_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1200_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1201_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1202_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1203_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1204_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1205_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1206_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1207_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1208_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_1209_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1210_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1211_mult__less__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1212_mult__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1213_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1214_mult__le__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1215_mult__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1216_mult__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1217_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1218_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1219_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1220_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1221_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1222_trans__less__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1223_trans__less__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1224_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1225_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1226_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1227_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1228_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1229_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1230_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1231_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1232_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1233_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1234_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1235_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1236_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1237_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1238_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1239_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1240_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1241_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1242_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1243_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_1244_semiring__norm_I13_J,axiom,
! [M2: num,N2: num] :
( ( times_times_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M2 @ N2 ) ) ) ) ).
% semiring_norm(13)
thf(fact_1245_semiring__norm_I78_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M2 @ N2 ) ) ).
% semiring_norm(78)
thf(fact_1246_num__double,axiom,
! [N2: num] :
( ( times_times_num @ ( bit0 @ one ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_1247_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1248_plus__inverse__ge__2,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_1249_neg__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1250_pos__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ B2 @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1251_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1252_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1253_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1254_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1255_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_1256_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1257_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= zero_zero_real )
= ( X3 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1258_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_1259_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1260_ln__inj__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ( ln_ln_real @ X3 )
= ( ln_ln_real @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% ln_inj_iff
thf(fact_1261_ln__less__cancel__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_real @ X3 @ Y2 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1262_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
= ( X3 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1263_real__add__minus__iff,axiom,
! [X3: real,A2: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X3 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1264_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_1265_ln__le__cancel__iff,axiom,
! [X3: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_eq_real @ X3 @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1266_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% ln_le_zero_iff
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_set_rat @ bot_bot_set_rat @ ( dedekind_inverse_set @ a ) ).
%------------------------------------------------------------------------------