TPTP Problem File: SLH0299^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_01461_047827__5773440_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1386 ( 506 unt; 116 typ; 0 def)
% Number of atoms : 3869 ( 964 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10222 ( 405 ~; 126 |; 181 &;7816 @)
% ( 0 <=>;1694 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 552 ( 552 >; 0 *; 0 +; 0 <<)
% Number of symbols : 112 ( 109 usr; 24 con; 0-3 aty)
% Number of variables : 3472 ( 327 ^;2997 !; 148 ?;3472 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:35.671
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
set_Dedekind_preal: $tType ).
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
set_Dedekind_real: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Oreal,type,
dedekind_real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (109)
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Dedekind____Real__Opreal,type,
condit4019698779847608237_preal: ( dedekind_preal > dedekind_preal > $o ) > ( dedekind_preal > dedekind_preal > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Dedekind____Real__Oreal,type,
condit6212436594666024831d_real: ( dedekind_real > dedekind_real > $o ) > ( dedekind_real > dedekind_real > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Int__Oint,type,
condit7933062003635074389dd_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Nat__Onat,type,
condit7935552474144124665dd_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Num__Onum,type,
condit4492884260299903299dd_num: ( num > num > $o ) > ( num > num > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Dedekind____Real__Opreal,type,
condit1308845025010339211_preal: ( dedekind_preal > dedekind_preal > $o ) > set_Dedekind_preal > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Dedekind____Real__Oreal,type,
condit3870944721565864481d_real: ( dedekind_real > dedekind_real > $o ) > set_Dedekind_real > $o ).
thf(sy_c_Dedekind__Real_Opsup,type,
dedekind_psup: set_Dedekind_preal > dedekind_preal ).
thf(sy_c_Dedekind__Real_Oreal__of__preal,type,
dedeki829075424511776921_preal: dedekind_preal > dedekind_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Dedekind____Real__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Dedekind____Real__Oreal,type,
one_on6069100329679821595d_real: dedekind_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_Oplus__class_Oplus_001t__Dedekind____Real__Opreal,type,
plus_p3173629198307831117_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Dedekind____Real__Oreal,type,
plus_p4060926892116697567d_real: dedekind_real > dedekind_real > dedekind_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_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Oreal,type,
times_2157731159493324635d_real: dedekind_real > dedekind_real > dedekind_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_Ouminus__class_Ouminus_001t__Dedekind____Real__Oreal,type,
uminus7714077491378687647d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Dedekind____Real__Oreal,type,
zero_z580800474297136991d_real: dedekind_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_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Dedekind____Real__Oreal,type,
neg_nu8665065731627423525d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Dedekind____Real__Oreal,type,
neg_nu2849763295382964967d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__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_Orderings_Obot__class_Obot_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
bot_bo4815710026557797371real_o: dedekind_preal > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Dedekind____Real__Oreal_M_Eo_J,type,
bot_bo7891357765246109669real_o: dedekind_real > $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__Dedekind____Real__Opreal_J,type,
bot_bo4848840305443107682_preal: set_Dedekind_preal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
bot_bo2548302898904406688d_real: set_Dedekind_real ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
ord_le1914443497901267899real_o: ( dedekind_preal > $o ) > ( dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Dedekind____Real__Oreal_M_Eo_J,type,
ord_le2006262546704330789real_o: ( dedekind_real > $o ) > ( dedekind_real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Oreal,type,
ord_le2991122432403439658d_real: dedekind_real > dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
ord_le1802228187270208418_preal: set_Dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
ord_le1729950913864802912d_real: set_Dedekind_real > set_Dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Dedekind____Real__Opreal_J,type,
ord_le6023059474077774659_preal: ( $o > dedekind_preal ) > ( $o > dedekind_preal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Dedekind____Real__Oreal_J,type,
ord_le267469853237711487d_real: ( $o > dedekind_real ) > ( $o > dedekind_real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Int__Oint_J,type,
ord_less_eq_o_int: ( $o > int ) > ( $o > int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Num__Onum_J,type,
ord_less_eq_o_num: ( $o > num ) > ( $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
ord_le5990576202697070023real_o: ( dedekind_preal > $o ) > ( dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Dedekind____Real__Oreal_M_Eo_J,type,
ord_le60669685330047769real_o: ( dedekind_real > $o ) > ( dedekind_real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Oreal,type,
ord_le2716243287969276982d_real: dedekind_real > dedekind_real > $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__Set__Oset_It__Dedekind____Real__Opreal_J,type,
ord_le7349499860212017814_preal: set_Dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
ord_le5071027967042688620d_real: set_Dedekind_real > set_Dedekind_real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Dedekind____Real__Opreal,type,
order_958373252487505263_preal: ( dedekind_preal > $o ) > dedekind_preal ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Dedekind____Real__Oreal,type,
order_1105151008113787197d_real: ( dedekind_real > $o ) > dedekind_real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Num__Onum,type,
order_Greatest_num: ( num > $o ) > num ).
thf(sy_c_Orderings_Oordering_001t__Dedekind____Real__Opreal,type,
orderi4673934450047582894_preal: ( dedekind_preal > dedekind_preal > $o ) > ( dedekind_preal > dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Dedekind____Real__Oreal,type,
orderi8957706999020659774d_real: ( dedekind_real > dedekind_real > $o ) > ( dedekind_real > dedekind_real > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Int__Oint,type,
ordering_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Num__Onum,type,
ordering_num: ( num > num > $o ) > ( num > num > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
orderi8630438233043249959_preal: ( set_Dedekind_preal > set_Dedekind_preal > $o ) > ( set_Dedekind_preal > set_Dedekind_preal > $o ) > set_Dedekind_preal > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Dedekind____Real__Opreal,type,
partia2807573505362346828_preal: ( dedekind_preal > dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Dedekind____Real__Oreal,type,
partia4124895370616003616d_real: ( dedekind_real > dedekind_real > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Int__Oint,type,
partia6820327588127286646ng_int: ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Nat__Onat,type,
partia6822818058636336922ng_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Num__Onum,type,
partia3380149844792115556ng_num: ( num > num > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Dedekind____Real__Opreal,type,
preord1889057019909701233_preal: ( dedekind_preal > dedekind_preal > $o ) > ( dedekind_preal > dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Dedekind____Real__Oreal,type,
preord5216658436359260091d_real: ( dedekind_real > dedekind_real > $o ) > ( dedekind_real > dedekind_real > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Int__Oint,type,
preordering_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Nat__Onat,type,
preordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Num__Onum,type,
preordering_num: ( num > num > $o ) > ( num > num > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
top_to6295660221356622871real_o: dedekind_preal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Dedekind____Real__Oreal_M_Eo_J,type,
top_to4979224932776997065real_o: dedekind_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
top_to2164892472777521222_preal: set_Dedekind_preal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
top_to4551142129947008188d_real: set_Dedekind_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Dedekind____Real__Oreal,type,
divide9119111104558704680d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Opreal,type,
collec1132657498972982273_preal: ( dedekind_preal > $o ) > set_Dedekind_preal ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Oreal,type,
collec1873404778653292075d_real: ( dedekind_real > $o ) > set_Dedekind_real ).
thf(sy_c_Set_Ois__empty_001t__Dedekind____Real__Opreal,type,
is_emp2142545888348893928_preal: set_Dedekind_preal > $o ).
thf(sy_c_member_001t__Dedekind____Real__Opreal,type,
member6871284927547481791_preal: dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_c_member_001t__Dedekind____Real__Oreal,type,
member_Dedekind_real: dedekind_real > set_Dedekind_real > $o ).
thf(sy_v_P,type,
p: set_Dedekind_real ).
thf(sy_v_a____,type,
a: dedekind_real ).
thf(sy_v_pa____,type,
pa: dedekind_preal ).
thf(sy_v_possup____,type,
possup: dedekind_preal ).
thf(sy_v_px____,type,
px: dedekind_preal ).
thf(sy_v_py____,type,
py: dedekind_preal ).
thf(sy_v_sup____,type,
sup: dedekind_real ).
thf(sy_v_x____,type,
x: dedekind_real ).
thf(sy_v_y____,type,
y: dedekind_real ).
% Relevant facts (1269)
thf(fact_0_not__empty__P,axiom,
? [X: dedekind_real] : ( member_Dedekind_real @ X @ p ) ).
% not_empty_P
thf(fact_1_upper__bound__Ex,axiom,
? [Y: dedekind_real] :
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
=> ( ord_le2991122432403439658d_real @ X2 @ Y ) ) ).
% upper_bound_Ex
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_O_Aa_A_092_060in_062_AP_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [A: dedekind_real] :
~ ( member_Dedekind_real @ A @ p ) ).
% \<open>\<And>thesis. (\<And>a. a \<in> P \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_y__is__py,axiom,
( y
= ( dedeki829075424511776921_preal @ py ) ) ).
% y_is_py
thf(fact_4__092_060open_062a_A_092_060in_062_AP_092_060close_062,axiom,
member_Dedekind_real @ a @ p ).
% \<open>a \<in> P\<close>
thf(fact_5_x__in__P,axiom,
member_Dedekind_real @ x @ p ).
% x_in_P
thf(fact_6__092_060open_062py_A_060_Apsup_A_123w_O_Areal__of__preal_Aw_A_092_060in_062_AP_125_092_060close_062,axiom,
( ord_le5708704896291381698_preal @ py
@ ( dedekind_psup
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) ) ) ).
% \<open>py < psup {w. real_of_preal w \<in> P}\<close>
thf(fact_7__092_060open_062_092_060exists_062x_092_060in_062P_O_Ay_A_060_Ax_092_060close_062,axiom,
? [X: dedekind_real] :
( ( member_Dedekind_real @ X @ p )
& ( ord_le2991122432403439658d_real @ y @ X ) ) ).
% \<open>\<exists>x\<in>P. y < x\<close>
thf(fact_8_x__is__px,axiom,
( x
= ( dedeki829075424511776921_preal @ px ) ) ).
% x_is_px
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sup_O_A_092_060forall_062x_092_060in_062P_O_Ax_A_060_Asup_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Sup: dedekind_real] :
~ ! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
=> ( ord_le2991122432403439658d_real @ X2 @ Sup ) ) ).
% \<open>\<And>thesis. (\<And>sup. \<forall>x\<in>P. x < sup \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062py_O_Ay_A_061_Areal__of__preal_Apy_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Py: dedekind_preal] :
( y
!= ( dedeki829075424511776921_preal @ Py ) ) ).
% \<open>\<And>thesis. (\<And>py. y = real_of_preal py \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_11_sup,axiom,
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
=> ( ord_le2991122432403439658d_real @ X2 @ sup ) ) ).
% sup
thf(fact_12_y__less__x,axiom,
ord_le2991122432403439658d_real @ y @ x ).
% y_less_x
thf(fact_13__092_060open_062a_A_061_Areal__of__preal_Apa_092_060close_062,axiom,
( a
= ( dedeki829075424511776921_preal @ pa ) ) ).
% \<open>a = real_of_preal pa\<close>
thf(fact_14__092_060open_062sup_A_061_Areal__of__preal_Apossup_092_060close_062,axiom,
( sup
= ( dedeki829075424511776921_preal @ possup ) ) ).
% \<open>sup = real_of_preal possup\<close>
thf(fact_15__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_062x_A_092_060in_062_AP_059_Ay_A_060_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: dedekind_real] :
( ( member_Dedekind_real @ X @ p )
=> ~ ( ord_le2991122432403439658d_real @ y @ X ) ) ).
% \<open>\<And>thesis. (\<And>x. \<lbrakk>x \<in> P; y < x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_16_psup,axiom,
! [Z: dedekind_preal] :
( ( ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) )
& ( ord_le5708704896291381698_preal @ Z @ X3 ) ) )
= ( ord_le5708704896291381698_preal @ Z
@ ( dedekind_psup
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) ) ) ) ).
% psup
thf(fact_17_positive__P,axiom,
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ p )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X2 ) ) ).
% positive_P
thf(fact_18__092_060open_062pa_A_092_060in_062_A_123w_O_Areal__of__preal_Aw_A_092_060in_062_AP_125_092_060close_062,axiom,
( member6871284927547481791_preal @ pa
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) ) ).
% \<open>pa \<in> {w. real_of_preal w \<in> P}\<close>
thf(fact_19_pos__y,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ y ).
% pos_y
thf(fact_20_py__less__X,axiom,
? [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) )
& ( ord_le5708704896291381698_preal @ py @ X ) ) ).
% py_less_X
thf(fact_21__092_060open_062_092_060exists_062X_092_060in_062_123w_O_Areal__of__preal_Aw_A_092_060in_062_AP_125_O_Apy_A_060_AX_A_092_060Longrightarrow_062_Apy_A_060_Apsup_A_123w_O_Areal__of__preal_Aw_A_092_060in_062_AP_125_092_060close_062,axiom,
( ? [X2: dedekind_preal] :
( ( member6871284927547481791_preal @ X2
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) )
& ( ord_le5708704896291381698_preal @ py @ X2 ) )
=> ( ord_le5708704896291381698_preal @ py
@ ( dedekind_psup
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) ) ) ) ).
% \<open>\<exists>X\<in>{w. real_of_preal w \<in> P}. py < X \<Longrightarrow> py < psup {w. real_of_preal w \<in> P}\<close>
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pa_O_Aa_A_061_Areal__of__preal_Apa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pa: dedekind_preal] :
( a
!= ( dedeki829075424511776921_preal @ Pa ) ) ).
% \<open>\<And>thesis. (\<And>pa. a = real_of_preal pa \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062possup_O_Asup_A_061_Areal__of__preal_Apossup_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Possup: dedekind_preal] :
( sup
!= ( dedeki829075424511776921_preal @ Possup ) ) ).
% \<open>\<And>thesis. (\<And>possup. sup = real_of_preal possup \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_24__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062px_O_Ax_A_061_Areal__of__preal_Apx_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Px: dedekind_preal] :
( x
!= ( dedeki829075424511776921_preal @ Px ) ) ).
% \<open>\<And>thesis. (\<And>px. x = real_of_preal px \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_25_real__of__preal__less__iff,axiom,
! [M1: dedekind_preal,M2: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ ( dedeki829075424511776921_preal @ M1 ) @ ( dedeki829075424511776921_preal @ M2 ) )
= ( ord_le5708704896291381698_preal @ M1 @ M2 ) ) ).
% real_of_preal_less_iff
thf(fact_26_pP__not__empty,axiom,
( ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) )
!= bot_bo4848840305443107682_preal ) ).
% pP_not_empty
thf(fact_27__092_060open_062_092_060forall_062X_092_060in_062_123w_O_Areal__of__preal_Aw_A_092_060in_062_AP_125_O_AX_A_092_060le_062_Apossup_092_060close_062,axiom,
! [X2: dedekind_preal] :
( ( member6871284927547481791_preal @ X2
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) )
=> ( ord_le5604041210740703414_preal @ X2 @ possup ) ) ).
% \<open>\<forall>X\<in>{w. real_of_preal w \<in> P}. X \<le> possup\<close>
thf(fact_28_minf_I7_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ~ ( ord_le2991122432403439658d_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_29_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ~ ( ord_le5708704896291381698_preal @ T @ X2 ) ) ).
% minf(7)
thf(fact_30_minf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ~ ( ord_less_num @ T @ X2 ) ) ).
% minf(7)
thf(fact_31_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_32_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_33_minf_I5_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ord_le2991122432403439658d_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_34_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ( ord_le5708704896291381698_preal @ X2 @ T ) ) ).
% minf(5)
thf(fact_35_minf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ord_less_num @ X2 @ T ) ) ).
% minf(5)
thf(fact_36_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_37_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_38_minf_I4_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_39_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_40_minf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_41_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_42_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_43_minf_I3_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_44_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_45_minf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_46_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_47_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_48_minf_I2_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_49_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_50_minf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_51_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_52_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_53_a__pos,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ a ).
% a_pos
thf(fact_54__092_060open_0620_A_060_Ax_092_060close_062,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ x ).
% \<open>0 < x\<close>
thf(fact_55__092_060open_0620_A_060_Asup_092_060close_062,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ sup ).
% \<open>0 < sup\<close>
thf(fact_56__092_060open_062a_A_060_Asup_092_060close_062,axiom,
ord_le2991122432403439658d_real @ a @ sup ).
% \<open>a < sup\<close>
thf(fact_57_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ T ) ) ).
% pinf(6)
thf(fact_58_pinf_I6_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ~ ( ord_le2716243287969276982d_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_59_pinf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% pinf(6)
thf(fact_60_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_61_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_62_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ( ord_le5604041210740703414_preal @ T @ X2 ) ) ).
% pinf(8)
thf(fact_63_pinf_I8_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ord_le2716243287969276982d_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_64_pinf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ord_less_eq_num @ T @ X2 ) ) ).
% pinf(8)
thf(fact_65_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_66_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_67_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ( ord_le5604041210740703414_preal @ X2 @ T ) ) ).
% minf(6)
thf(fact_68_minf_I6_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ord_le2716243287969276982d_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_69_minf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ord_less_eq_num @ X2 @ T ) ) ).
% minf(6)
thf(fact_70_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_71_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_72_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ~ ( ord_le5604041210740703414_preal @ T @ X2 ) ) ).
% minf(8)
thf(fact_73_minf_I8_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ~ ( ord_le2716243287969276982d_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_74_minf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% minf(8)
thf(fact_75_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_76_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_77_preal__complete,axiom,
! [P: set_Dedekind_preal,Y2: dedekind_preal,Z: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y2 ) )
=> ( ( P != bot_bo4848840305443107682_preal )
=> ( ( ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ P )
& ( ord_le5708704896291381698_preal @ Z @ X3 ) ) )
= ( ord_le5708704896291381698_preal @ Z @ ( dedekind_psup @ P ) ) ) ) ) ).
% preal_complete
thf(fact_78_psup__le__ub,axiom,
! [P: set_Dedekind_preal,Y2: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y2 ) )
=> ( ( P != bot_bo4848840305443107682_preal )
=> ( ord_le5604041210740703414_preal @ ( dedekind_psup @ P ) @ Y2 ) ) ) ).
% psup_le_ub
thf(fact_79_preal__psup__le,axiom,
! [P: set_Dedekind_preal,Y2: dedekind_preal,X5: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y2 ) )
=> ( ( member6871284927547481791_preal @ X5 @ P )
=> ( ord_le5604041210740703414_preal @ X5 @ ( dedekind_psup @ P ) ) ) ) ).
% preal_psup_le
thf(fact_80_mem__Collect__eq,axiom,
! [A2: dedekind_real,P: dedekind_real > $o] :
( ( member_Dedekind_real @ A2 @ ( collec1873404778653292075d_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A2: dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ A2 @ ( collec1132657498972982273_preal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A3: set_Dedekind_real] :
( ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A3: set_Dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_84_Collect__cong,axiom,
! [P: dedekind_preal > $o,Q: dedekind_preal > $o] :
( ! [X: dedekind_preal] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec1132657498972982273_preal @ P )
= ( collec1132657498972982273_preal @ Q ) ) ) ).
% Collect_cong
thf(fact_85_real__of__preal__zero__less,axiom,
! [M: dedekind_preal] : ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( dedeki829075424511776921_preal @ M ) ) ).
% real_of_preal_zero_less
thf(fact_86_real__gt__zero__preal__Ex,axiom,
! [X5: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X5 )
= ( ? [Y3: dedekind_preal] :
( X5
= ( dedeki829075424511776921_preal @ Y3 ) ) ) ) ).
% real_gt_zero_preal_Ex
thf(fact_87_pinf_I1_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_88_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_89_pinf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_90_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_91_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_92_pinf_I2_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_93_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_94_pinf_I2_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_95_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_96_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_97_pinf_I3_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_98_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_99_pinf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_100_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_101_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_102_pinf_I4_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_103_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_104_pinf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_105_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_106_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_107_pinf_I5_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ~ ( ord_le2991122432403439658d_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_108_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ~ ( ord_le5708704896291381698_preal @ X2 @ T ) ) ).
% pinf(5)
thf(fact_109_pinf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ~ ( ord_less_num @ X2 @ T ) ) ).
% pinf(5)
thf(fact_110_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_111_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_112_pinf_I7_J,axiom,
! [T: dedekind_real] :
? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z2 @ X2 )
=> ( ord_le2991122432403439658d_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_113_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X2 )
=> ( ord_le5708704896291381698_preal @ T @ X2 ) ) ).
% pinf(7)
thf(fact_114_pinf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ Z2 @ X2 )
=> ( ord_less_num @ T @ X2 ) ) ).
% pinf(7)
thf(fact_115_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_116_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_117_minf_I1_J,axiom,
! [P: dedekind_real > $o,P2: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_118_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_119_minf_I1_J,axiom,
! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: num] :
! [X: num] :
( ( ord_less_num @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: num] :
! [X2: num] :
( ( ord_less_num @ X2 @ Z2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_120_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_121_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_122_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_123_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_124_empty__iff,axiom,
! [C: dedekind_real] :
~ ( member_Dedekind_real @ C @ bot_bo2548302898904406688d_real ) ).
% empty_iff
thf(fact_125_empty__iff,axiom,
! [C: dedekind_preal] :
~ ( member6871284927547481791_preal @ C @ bot_bo4848840305443107682_preal ) ).
% empty_iff
thf(fact_126_all__not__in__conv,axiom,
! [A3: set_Dedekind_real] :
( ( ! [X3: dedekind_real] :
~ ( member_Dedekind_real @ X3 @ A3 ) )
= ( A3 = bot_bo2548302898904406688d_real ) ) ).
% all_not_in_conv
thf(fact_127_all__not__in__conv,axiom,
! [A3: set_Dedekind_preal] :
( ( ! [X3: dedekind_preal] :
~ ( member6871284927547481791_preal @ X3 @ A3 ) )
= ( A3 = bot_bo4848840305443107682_preal ) ) ).
% all_not_in_conv
thf(fact_128_Collect__empty__eq,axiom,
! [P: dedekind_preal > $o] :
( ( ( collec1132657498972982273_preal @ P )
= bot_bo4848840305443107682_preal )
= ( ! [X3: dedekind_preal] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_129_empty__Collect__eq,axiom,
! [P: dedekind_preal > $o] :
( ( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal @ P ) )
= ( ! [X3: dedekind_preal] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_130_order__refl,axiom,
! [X5: dedekind_preal] : ( ord_le5604041210740703414_preal @ X5 @ X5 ) ).
% order_refl
thf(fact_131_order__refl,axiom,
! [X5: dedekind_real] : ( ord_le2716243287969276982d_real @ X5 @ X5 ) ).
% order_refl
thf(fact_132_order__refl,axiom,
! [X5: num] : ( ord_less_eq_num @ X5 @ X5 ) ).
% order_refl
thf(fact_133_order__refl,axiom,
! [X5: int] : ( ord_less_eq_int @ X5 @ X5 ) ).
% order_refl
thf(fact_134_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_135_dual__order_Orefl,axiom,
! [A2: dedekind_preal] : ( ord_le5604041210740703414_preal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_136_dual__order_Orefl,axiom,
! [A2: dedekind_real] : ( ord_le2716243287969276982d_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_137_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_138_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_139_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_140_bot_Onot__eq__extremum,axiom,
! [A2: set_Dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
= ( ord_le1802228187270208418_preal @ bot_bo4848840305443107682_preal @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_141_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_142_empty__subsetI,axiom,
! [A3: set_Dedekind_preal] : ( ord_le7349499860212017814_preal @ bot_bo4848840305443107682_preal @ A3 ) ).
% empty_subsetI
thf(fact_143_subset__empty,axiom,
! [A3: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A3 @ bot_bo4848840305443107682_preal )
= ( A3 = bot_bo4848840305443107682_preal ) ) ).
% subset_empty
thf(fact_144_real__of__preal__le__iff,axiom,
! [M1: dedekind_preal,M2: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ ( dedeki829075424511776921_preal @ M1 ) @ ( dedeki829075424511776921_preal @ M2 ) )
= ( ord_le5604041210740703414_preal @ M1 @ M2 ) ) ).
% real_of_preal_le_iff
thf(fact_145_bot__set__def,axiom,
( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal @ bot_bo4815710026557797371real_o ) ) ).
% bot_set_def
thf(fact_146_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% real_less_def
thf(fact_147_order__antisym__conv,axiom,
! [Y4: dedekind_preal,X5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y4 @ X5 )
=> ( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_148_order__antisym__conv,axiom,
! [Y4: dedekind_real,X5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_149_order__antisym__conv,axiom,
! [Y4: num,X5: num] :
( ( ord_less_eq_num @ Y4 @ X5 )
=> ( ( ord_less_eq_num @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_150_order__antisym__conv,axiom,
! [Y4: int,X5: int] :
( ( ord_less_eq_int @ Y4 @ X5 )
=> ( ( ord_less_eq_int @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_151_order__antisym__conv,axiom,
! [Y4: nat,X5: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ( ord_less_eq_nat @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_152_linorder__le__cases,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ( ord_le5604041210740703414_preal @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_153_linorder__le__cases,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ord_le2716243287969276982d_real @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_154_linorder__le__cases,axiom,
! [X5: num,Y4: num] :
( ~ ( ord_less_eq_num @ X5 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_155_linorder__le__cases,axiom,
! [X5: int,Y4: int] :
( ~ ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_156_linorder__le__cases,axiom,
! [X5: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_157_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_158_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_159_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_161_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_162_ord__le__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_163_ord__le__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_164_ord__le__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_165_ord__le__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_166_ord__le__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_167_ord__eq__le__subst,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_168_ord__eq__le__subst,axiom,
! [A2: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_169_ord__eq__le__subst,axiom,
! [A2: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_170_ord__eq__le__subst,axiom,
! [A2: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_171_ord__eq__le__subst,axiom,
! [A2: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_172_ord__eq__le__subst,axiom,
! [A2: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A2: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A2: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_ord__eq__le__subst,axiom,
! [A2: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_176_ord__eq__le__subst,axiom,
! [A2: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_177_linorder__linear,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
| ( ord_le5604041210740703414_preal @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_178_linorder__linear,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
| ( ord_le2716243287969276982d_real @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_179_linorder__linear,axiom,
! [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
| ( ord_less_eq_num @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_180_linorder__linear,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_181_linorder__linear,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_182_order__eq__refl,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( X5 = Y4 )
=> ( ord_le5604041210740703414_preal @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_183_order__eq__refl,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( X5 = Y4 )
=> ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_184_order__eq__refl,axiom,
! [X5: num,Y4: num] :
( ( X5 = Y4 )
=> ( ord_less_eq_num @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_185_order__eq__refl,axiom,
! [X5: int,Y4: int] :
( ( X5 = Y4 )
=> ( ord_less_eq_int @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_186_order__eq__refl,axiom,
! [X5: nat,Y4: nat] :
( ( X5 = Y4 )
=> ( ord_less_eq_nat @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_187_order__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_188_order__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_189_order__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_190_order__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_192_order__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_193_order__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_194_order__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_195_order__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_196_order__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_197_order__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_198_order__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_199_order__subst1,axiom,
! [A2: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_200_order__subst1,axiom,
! [A2: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_201_order__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_202_order__subst1,axiom,
! [A2: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_203_order__subst1,axiom,
! [A2: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_204_order__subst1,axiom,
! [A2: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_205_order__subst1,axiom,
! [A2: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_206_order__subst1,axiom,
! [A2: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_207_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z4: dedekind_preal] : ( Y5 = Z4 ) )
= ( ^ [A4: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B2 )
& ( ord_le5604041210740703414_preal @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_208_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z4: dedekind_real] : ( Y5 = Z4 ) )
= ( ^ [A4: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B2 )
& ( ord_le2716243287969276982d_real @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_209_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A4: num,B2: num] :
( ( ord_less_eq_num @ A4 @ B2 )
& ( ord_less_eq_num @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_210_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ( ord_less_eq_int @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_211_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_212_antisym,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_213_antisym,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_214_antisym,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_215_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_216_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_217_dual__order_Otrans,axiom,
! [B: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A2 )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5604041210740703414_preal @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_218_dual__order_Otrans,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A2 )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2716243287969276982d_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_219_dual__order_Otrans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_220_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_221_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_222_dual__order_Oantisym,axiom,
! [B: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A2 )
=> ( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_223_dual__order_Oantisym,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A2 )
=> ( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_224_dual__order_Oantisym,axiom,
! [B: num,A2: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_225_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_226_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_227_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z4: dedekind_preal] : ( Y5 = Z4 ) )
= ( ^ [A4: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A4 )
& ( ord_le5604041210740703414_preal @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_228_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z4: dedekind_real] : ( Y5 = Z4 ) )
= ( ^ [A4: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A4 )
& ( ord_le2716243287969276982d_real @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_229_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [A4: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A4 )
& ( ord_less_eq_num @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_230_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_231_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_232_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A2: dedekind_preal,B: dedekind_preal] :
( ! [A: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: dedekind_preal,B3: dedekind_preal] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_233_linorder__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A2: dedekind_real,B: dedekind_real] :
( ! [A: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: dedekind_real,B3: dedekind_real] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_234_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B: num] :
( ! [A: num,B3: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: num,B3: num] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_235_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: int,B3: int] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_236_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: nat,B3: nat] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_237_order__trans,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ( ( ord_le5604041210740703414_preal @ Y4 @ Z5 )
=> ( ord_le5604041210740703414_preal @ X5 @ Z5 ) ) ) ).
% order_trans
thf(fact_238_order__trans,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ Z5 )
=> ( ord_le2716243287969276982d_real @ X5 @ Z5 ) ) ) ).
% order_trans
thf(fact_239_order__trans,axiom,
! [X5: num,Y4: num,Z5: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z5 )
=> ( ord_less_eq_num @ X5 @ Z5 ) ) ) ).
% order_trans
thf(fact_240_order__trans,axiom,
! [X5: int,Y4: int,Z5: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z5 )
=> ( ord_less_eq_int @ X5 @ Z5 ) ) ) ).
% order_trans
thf(fact_241_order__trans,axiom,
! [X5: nat,Y4: nat,Z5: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z5 )
=> ( ord_less_eq_nat @ X5 @ Z5 ) ) ) ).
% order_trans
thf(fact_242_order_Otrans,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A2 @ C ) ) ) ).
% order.trans
thf(fact_243_order_Otrans,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_244_order_Otrans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_245_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_246_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_247_order__antisym,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ( ( ord_le5604041210740703414_preal @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_248_order__antisym,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_249_order__antisym,axiom,
! [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_250_order__antisym,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_251_order__antisym,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_252_ord__le__eq__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( B = C )
=> ( ord_le5604041210740703414_preal @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_253_ord__le__eq__trans,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_le2716243287969276982d_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_254_ord__le__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_255_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_256_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_257_ord__eq__le__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A2 = B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_258_ord__eq__le__trans,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A2 = B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_259_ord__eq__le__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_260_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_261_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_262_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z4: dedekind_preal] : ( Y5 = Z4 ) )
= ( ^ [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
& ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_263_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_real,Z4: dedekind_real] : ( Y5 = Z4 ) )
= ( ^ [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
& ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_264_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z4: num] : ( Y5 = Z4 ) )
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_265_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_266_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_267_le__cases3,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ~ ( ord_le5604041210740703414_preal @ Y4 @ Z5 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y4 @ X5 )
=> ~ ( ord_le5604041210740703414_preal @ X5 @ Z5 ) )
=> ( ( ( ord_le5604041210740703414_preal @ X5 @ Z5 )
=> ~ ( ord_le5604041210740703414_preal @ Z5 @ Y4 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z5 @ Y4 )
=> ~ ( ord_le5604041210740703414_preal @ Y4 @ X5 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y4 @ Z5 )
=> ~ ( ord_le5604041210740703414_preal @ Z5 @ X5 ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z5 @ X5 )
=> ~ ( ord_le5604041210740703414_preal @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_268_le__cases3,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ~ ( ord_le2716243287969276982d_real @ Y4 @ Z5 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ~ ( ord_le2716243287969276982d_real @ X5 @ Z5 ) )
=> ( ( ( ord_le2716243287969276982d_real @ X5 @ Z5 )
=> ~ ( ord_le2716243287969276982d_real @ Z5 @ Y4 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z5 @ Y4 )
=> ~ ( ord_le2716243287969276982d_real @ Y4 @ X5 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y4 @ Z5 )
=> ~ ( ord_le2716243287969276982d_real @ Z5 @ X5 ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z5 @ X5 )
=> ~ ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_269_le__cases3,axiom,
! [X5: num,Y4: num,Z5: num] :
( ( ( ord_less_eq_num @ X5 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z5 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ Z5 ) )
=> ( ( ( ord_less_eq_num @ X5 @ Z5 )
=> ~ ( ord_less_eq_num @ Z5 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z5 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z5 )
=> ~ ( ord_less_eq_num @ Z5 @ X5 ) )
=> ~ ( ( ord_less_eq_num @ Z5 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_270_le__cases3,axiom,
! [X5: int,Y4: int,Z5: int] :
( ( ( ord_less_eq_int @ X5 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z5 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Z5 ) )
=> ( ( ( ord_less_eq_int @ X5 @ Z5 )
=> ~ ( ord_less_eq_int @ Z5 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z5 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z5 )
=> ~ ( ord_less_eq_int @ Z5 @ X5 ) )
=> ~ ( ( ord_less_eq_int @ Z5 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_271_le__cases3,axiom,
! [X5: nat,Y4: nat,Z5: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z5 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z5 ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z5 )
=> ~ ( ord_less_eq_nat @ Z5 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z5 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z5 )
=> ~ ( ord_less_eq_nat @ Z5 @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z5 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_272_nle__le,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A2 @ B ) )
= ( ( ord_le5604041210740703414_preal @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_273_nle__le,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ A2 @ B ) )
= ( ( ord_le2716243287969276982d_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_274_nle__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B ) )
= ( ( ord_less_eq_num @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_275_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_276_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_277_zero__reorient,axiom,
! [X5: dedekind_real] :
( ( zero_z580800474297136991d_real = X5 )
= ( X5 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_278_zero__reorient,axiom,
! [X5: int] :
( ( zero_zero_int = X5 )
= ( X5 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_279_lt__ex,axiom,
! [X5: dedekind_real] :
? [Y: dedekind_real] : ( ord_le2991122432403439658d_real @ Y @ X5 ) ).
% lt_ex
thf(fact_280_lt__ex,axiom,
! [X5: int] :
? [Y: int] : ( ord_less_int @ Y @ X5 ) ).
% lt_ex
thf(fact_281_gt__ex,axiom,
! [X5: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X5 @ X_1 ) ).
% gt_ex
thf(fact_282_gt__ex,axiom,
! [X5: int] :
? [X_1: int] : ( ord_less_int @ X5 @ X_1 ) ).
% gt_ex
thf(fact_283_gt__ex,axiom,
! [X5: nat] :
? [X_1: nat] : ( ord_less_nat @ X5 @ X_1 ) ).
% gt_ex
thf(fact_284_dense,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ? [Z2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Z2 )
& ( ord_le2991122432403439658d_real @ Z2 @ Y4 ) ) ) ).
% dense
thf(fact_285_less__imp__neq,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% less_imp_neq
thf(fact_286_less__imp__neq,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% less_imp_neq
thf(fact_287_less__imp__neq,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% less_imp_neq
thf(fact_288_less__imp__neq,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% less_imp_neq
thf(fact_289_less__imp__neq,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% less_imp_neq
thf(fact_290_order_Oasym,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A2 ) ) ).
% order.asym
thf(fact_291_order_Oasym,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A2 ) ) ).
% order.asym
thf(fact_292_order_Oasym,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order.asym
thf(fact_293_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_294_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_295_ord__eq__less__trans,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A2 = B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_296_ord__eq__less__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A2 = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_297_ord__eq__less__trans,axiom,
! [A2: num,B: num,C: num] :
( ( A2 = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_298_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_299_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_300_ord__less__eq__trans,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_le2991122432403439658d_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_301_ord__less__eq__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( B = C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_302_ord__less__eq__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_303_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_304_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_305_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_306_antisym__conv3,axiom,
! [Y4: dedekind_real,X5: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ Y4 @ X5 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_307_antisym__conv3,axiom,
! [Y4: dedekind_preal,X5: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y4 @ X5 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_308_antisym__conv3,axiom,
! [Y4: num,X5: num] :
( ~ ( ord_less_num @ Y4 @ X5 )
=> ( ( ~ ( ord_less_num @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_309_antisym__conv3,axiom,
! [Y4: int,X5: int] :
( ~ ( ord_less_int @ Y4 @ X5 )
=> ( ( ~ ( ord_less_int @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_310_antisym__conv3,axiom,
! [Y4: nat,X5: nat] :
( ~ ( ord_less_nat @ Y4 @ X5 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_311_linorder__cases,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ( X5 != Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ) ).
% linorder_cases
thf(fact_312_linorder__cases,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ( X5 != Y4 )
=> ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ) ).
% linorder_cases
thf(fact_313_linorder__cases,axiom,
! [X5: num,Y4: num] :
( ~ ( ord_less_num @ X5 @ Y4 )
=> ( ( X5 != Y4 )
=> ( ord_less_num @ Y4 @ X5 ) ) ) ).
% linorder_cases
thf(fact_314_linorder__cases,axiom,
! [X5: int,Y4: int] :
( ~ ( ord_less_int @ X5 @ Y4 )
=> ( ( X5 != Y4 )
=> ( ord_less_int @ Y4 @ X5 ) ) ) ).
% linorder_cases
thf(fact_315_linorder__cases,axiom,
! [X5: nat,Y4: nat] :
( ~ ( ord_less_nat @ X5 @ Y4 )
=> ( ( X5 != Y4 )
=> ( ord_less_nat @ Y4 @ X5 ) ) ) ).
% linorder_cases
thf(fact_316_dual__order_Oasym,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ~ ( ord_le2991122432403439658d_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_317_dual__order_Oasym,axiom,
! [B: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A2 )
=> ~ ( ord_le5708704896291381698_preal @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_318_dual__order_Oasym,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ~ ( ord_less_num @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_319_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_320_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_321_dual__order_Oirrefl,axiom,
! [A2: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_322_dual__order_Oirrefl,axiom,
! [A2: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_323_dual__order_Oirrefl,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_324_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_325_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_326_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_327_linorder__less__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A2: dedekind_real,B: dedekind_real] :
( ! [A: dedekind_real,B3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: dedekind_real] : ( P @ A @ A )
=> ( ! [A: dedekind_real,B3: dedekind_real] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_328_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A2: dedekind_preal,B: dedekind_preal] :
( ! [A: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: dedekind_preal] : ( P @ A @ A )
=> ( ! [A: dedekind_preal,B3: dedekind_preal] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_329_linorder__less__wlog,axiom,
! [P: num > num > $o,A2: num,B: num] :
( ! [A: num,B3: num] :
( ( ord_less_num @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: num] : ( P @ A @ A )
=> ( ! [A: num,B3: num] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_330_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A: int,B3: int] :
( ( ord_less_int @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: int] : ( P @ A @ A )
=> ( ! [A: int,B3: int] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_331_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( P @ A @ B3 ) )
=> ( ! [A: nat] : ( P @ A @ A )
=> ( ! [A: nat,B3: nat] :
( ( P @ B3 @ A )
=> ( P @ A @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_332_order_Ostrict__trans,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_333_order_Ostrict__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_334_order_Ostrict__trans,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_335_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_336_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_337_not__less__iff__gr__or__eq,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 ) )
= ( ( ord_le2991122432403439658d_real @ Y4 @ X5 )
| ( X5 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_338_not__less__iff__gr__or__eq,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 ) )
= ( ( ord_le5708704896291381698_preal @ Y4 @ X5 )
| ( X5 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_339_not__less__iff__gr__or__eq,axiom,
! [X5: num,Y4: num] :
( ( ~ ( ord_less_num @ X5 @ Y4 ) )
= ( ( ord_less_num @ Y4 @ X5 )
| ( X5 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_340_not__less__iff__gr__or__eq,axiom,
! [X5: int,Y4: int] :
( ( ~ ( ord_less_int @ X5 @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X5 )
| ( X5 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_341_not__less__iff__gr__or__eq,axiom,
! [X5: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X5 )
| ( X5 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_342_dual__order_Ostrict__trans,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_343_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A2 )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_344_dual__order_Ostrict__trans,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_345_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_346_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_347_order_Ostrict__implies__not__eq,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_348_order_Ostrict__implies__not__eq,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_349_order_Ostrict__implies__not__eq,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_350_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_351_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_352_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_353_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_354_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_355_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_356_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_357_linorder__neqE,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( X5 != Y4 )
=> ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_358_linorder__neqE,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( X5 != Y4 )
=> ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_359_linorder__neqE,axiom,
! [X5: num,Y4: num] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_num @ X5 @ Y4 )
=> ( ord_less_num @ Y4 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_360_linorder__neqE,axiom,
! [X5: int,Y4: int] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ Y4 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_361_linorder__neqE,axiom,
! [X5: nat,Y4: nat] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ Y4 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_362_order__less__asym,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ~ ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ).
% order_less_asym
thf(fact_363_order__less__asym,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ~ ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ).
% order_less_asym
thf(fact_364_order__less__asym,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X5 ) ) ).
% order_less_asym
thf(fact_365_order__less__asym,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X5 ) ) ).
% order_less_asym
thf(fact_366_order__less__asym,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X5 ) ) ).
% order_less_asym
thf(fact_367_linorder__neq__iff,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( X5 != Y4 )
= ( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
| ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_368_linorder__neq__iff,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( X5 != Y4 )
= ( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
| ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_369_linorder__neq__iff,axiom,
! [X5: num,Y4: num] :
( ( X5 != Y4 )
= ( ( ord_less_num @ X5 @ Y4 )
| ( ord_less_num @ Y4 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_370_linorder__neq__iff,axiom,
! [X5: int,Y4: int] :
( ( X5 != Y4 )
= ( ( ord_less_int @ X5 @ Y4 )
| ( ord_less_int @ Y4 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_371_linorder__neq__iff,axiom,
! [X5: nat,Y4: nat] :
( ( X5 != Y4 )
= ( ( ord_less_nat @ X5 @ Y4 )
| ( ord_less_nat @ Y4 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_372_order__less__asym_H,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_373_order__less__asym_H,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_374_order__less__asym_H,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ~ ( ord_less_num @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_375_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_376_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_377_order__less__trans,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ Z5 )
=> ( ord_le2991122432403439658d_real @ X5 @ Z5 ) ) ) ).
% order_less_trans
thf(fact_378_order__less__trans,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ Y4 @ Z5 )
=> ( ord_le5708704896291381698_preal @ X5 @ Z5 ) ) ) ).
% order_less_trans
thf(fact_379_order__less__trans,axiom,
! [X5: num,Y4: num,Z5: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z5 )
=> ( ord_less_num @ X5 @ Z5 ) ) ) ).
% order_less_trans
thf(fact_380_order__less__trans,axiom,
! [X5: int,Y4: int,Z5: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z5 )
=> ( ord_less_int @ X5 @ Z5 ) ) ) ).
% order_less_trans
thf(fact_381_order__less__trans,axiom,
! [X5: nat,Y4: nat,Z5: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z5 )
=> ( ord_less_nat @ X5 @ Z5 ) ) ) ).
% order_less_trans
thf(fact_382_ord__eq__less__subst,axiom,
! [A2: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_383_ord__eq__less__subst,axiom,
! [A2: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_384_ord__eq__less__subst,axiom,
! [A2: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_385_ord__eq__less__subst,axiom,
! [A2: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_386_ord__eq__less__subst,axiom,
! [A2: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_387_ord__eq__less__subst,axiom,
! [A2: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_388_ord__eq__less__subst,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_389_ord__eq__less__subst,axiom,
! [A2: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_390_ord__eq__less__subst,axiom,
! [A2: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_391_ord__eq__less__subst,axiom,
! [A2: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_392_ord__less__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_393_ord__less__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_394_ord__less__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_395_ord__less__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_396_ord__less__eq__subst,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_397_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_398_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_399_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_400_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_401_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_402_order__less__irrefl,axiom,
! [X5: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_403_order__less__irrefl,axiom,
! [X5: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_404_order__less__irrefl,axiom,
! [X5: num] :
~ ( ord_less_num @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_405_order__less__irrefl,axiom,
! [X5: int] :
~ ( ord_less_int @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_406_order__less__irrefl,axiom,
! [X5: nat] :
~ ( ord_less_nat @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_407_order__less__subst1,axiom,
! [A2: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_408_order__less__subst1,axiom,
! [A2: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_409_order__less__subst1,axiom,
! [A2: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_410_order__less__subst1,axiom,
! [A2: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_411_order__less__subst1,axiom,
! [A2: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_412_order__less__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_413_order__less__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_414_order__less__subst1,axiom,
! [A2: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_415_order__less__subst1,axiom,
! [A2: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_416_order__less__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_417_order__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_418_order__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_419_order__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_420_order__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_421_order__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_422_order__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_423_order__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_424_order__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_425_order__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_426_order__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_427_order__less__not__sym,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ~ ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ).
% order_less_not_sym
thf(fact_428_order__less__not__sym,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ~ ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ).
% order_less_not_sym
thf(fact_429_order__less__not__sym,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X5 ) ) ).
% order_less_not_sym
thf(fact_430_order__less__not__sym,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X5 ) ) ).
% order_less_not_sym
thf(fact_431_order__less__not__sym,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X5 ) ) ).
% order_less_not_sym
thf(fact_432_order__less__imp__triv,axiom,
! [X5: dedekind_real,Y4: dedekind_real,P: $o] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_433_order__less__imp__triv,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ Y4 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_434_order__less__imp__triv,axiom,
! [X5: num,Y4: num,P: $o] :
( ( ord_less_num @ X5 @ Y4 )
=> ( ( ord_less_num @ Y4 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_435_order__less__imp__triv,axiom,
! [X5: int,Y4: int,P: $o] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ( ord_less_int @ Y4 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_436_order__less__imp__triv,axiom,
! [X5: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_437_linorder__less__linear,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
| ( X5 = Y4 )
| ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ).
% linorder_less_linear
thf(fact_438_linorder__less__linear,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
| ( X5 = Y4 )
| ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ).
% linorder_less_linear
thf(fact_439_linorder__less__linear,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
| ( X5 = Y4 )
| ( ord_less_num @ Y4 @ X5 ) ) ).
% linorder_less_linear
thf(fact_440_linorder__less__linear,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
| ( X5 = Y4 )
| ( ord_less_int @ Y4 @ X5 ) ) ).
% linorder_less_linear
thf(fact_441_linorder__less__linear,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
| ( X5 = Y4 )
| ( ord_less_nat @ Y4 @ X5 ) ) ).
% linorder_less_linear
thf(fact_442_order__less__imp__not__eq,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_443_order__less__imp__not__eq,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_444_order__less__imp__not__eq,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_445_order__less__imp__not__eq,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_446_order__less__imp__not__eq,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( X5 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_447_order__less__imp__not__eq2,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( Y4 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_448_order__less__imp__not__eq2,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( Y4 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_449_order__less__imp__not__eq2,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ( Y4 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_450_order__less__imp__not__eq2,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( Y4 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_451_order__less__imp__not__eq2,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( Y4 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_452_order__less__imp__not__less,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ~ ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_453_order__less__imp__not__less,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ~ ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_454_order__less__imp__not__less,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_455_order__less__imp__not__less,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_456_order__less__imp__not__less,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_457_not__psubset__empty,axiom,
! [A3: set_Dedekind_preal] :
~ ( ord_le1802228187270208418_preal @ A3 @ bot_bo4848840305443107682_preal ) ).
% not_psubset_empty
thf(fact_458_ex__in__conv,axiom,
! [A3: set_Dedekind_real] :
( ( ? [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ A3 ) )
= ( A3 != bot_bo2548302898904406688d_real ) ) ).
% ex_in_conv
thf(fact_459_ex__in__conv,axiom,
! [A3: set_Dedekind_preal] :
( ( ? [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ A3 ) )
= ( A3 != bot_bo4848840305443107682_preal ) ) ).
% ex_in_conv
thf(fact_460_equals0I,axiom,
! [A3: set_Dedekind_real] :
( ! [Y: dedekind_real] :
~ ( member_Dedekind_real @ Y @ A3 )
=> ( A3 = bot_bo2548302898904406688d_real ) ) ).
% equals0I
thf(fact_461_equals0I,axiom,
! [A3: set_Dedekind_preal] :
( ! [Y: dedekind_preal] :
~ ( member6871284927547481791_preal @ Y @ A3 )
=> ( A3 = bot_bo4848840305443107682_preal ) ) ).
% equals0I
thf(fact_462_equals0D,axiom,
! [A3: set_Dedekind_real,A2: dedekind_real] :
( ( A3 = bot_bo2548302898904406688d_real )
=> ~ ( member_Dedekind_real @ A2 @ A3 ) ) ).
% equals0D
thf(fact_463_equals0D,axiom,
! [A3: set_Dedekind_preal,A2: dedekind_preal] :
( ( A3 = bot_bo4848840305443107682_preal )
=> ~ ( member6871284927547481791_preal @ A2 @ A3 ) ) ).
% equals0D
thf(fact_464_emptyE,axiom,
! [A2: dedekind_real] :
~ ( member_Dedekind_real @ A2 @ bot_bo2548302898904406688d_real ) ).
% emptyE
thf(fact_465_emptyE,axiom,
! [A2: dedekind_preal] :
~ ( member6871284927547481791_preal @ A2 @ bot_bo4848840305443107682_preal ) ).
% emptyE
thf(fact_466_empty__def,axiom,
( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] : $false ) ) ).
% empty_def
thf(fact_467_zero__le,axiom,
! [X5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X5 ) ).
% zero_le
thf(fact_468_leD,axiom,
! [Y4: dedekind_preal,X5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y4 @ X5 )
=> ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 ) ) ).
% leD
thf(fact_469_leD,axiom,
! [Y4: dedekind_real,X5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 ) ) ).
% leD
thf(fact_470_leD,axiom,
! [Y4: num,X5: num] :
( ( ord_less_eq_num @ Y4 @ X5 )
=> ~ ( ord_less_num @ X5 @ Y4 ) ) ).
% leD
thf(fact_471_leD,axiom,
! [Y4: int,X5: int] :
( ( ord_less_eq_int @ Y4 @ X5 )
=> ~ ( ord_less_int @ X5 @ Y4 ) ) ).
% leD
thf(fact_472_leD,axiom,
! [Y4: nat,X5: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ Y4 ) ) ).
% leD
thf(fact_473_leI,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ord_le5604041210740703414_preal @ Y4 @ X5 ) ) ).
% leI
thf(fact_474_leI,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ord_le2716243287969276982d_real @ Y4 @ X5 ) ) ).
% leI
thf(fact_475_leI,axiom,
! [X5: num,Y4: num] :
( ~ ( ord_less_num @ X5 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X5 ) ) ).
% leI
thf(fact_476_leI,axiom,
! [X5: int,Y4: int] :
( ~ ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X5 ) ) ).
% leI
thf(fact_477_leI,axiom,
! [X5: nat,Y4: nat] :
( ~ ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X5 ) ) ).
% leI
thf(fact_478_nless__le,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A2 @ B ) )
= ( ~ ( ord_le5604041210740703414_preal @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_479_nless__le,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ A2 @ B ) )
= ( ~ ( ord_le2716243287969276982d_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_480_nless__le,axiom,
! [A2: num,B: num] :
( ( ~ ( ord_less_num @ A2 @ B ) )
= ( ~ ( ord_less_eq_num @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_481_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_482_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_483_antisym__conv1,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_484_antisym__conv1,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_485_antisym__conv1,axiom,
! [X5: num,Y4: num] :
( ~ ( ord_less_num @ X5 @ Y4 )
=> ( ( ord_less_eq_num @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_486_antisym__conv1,axiom,
! [X5: int,Y4: int] :
( ~ ( ord_less_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_487_antisym__conv1,axiom,
! [X5: nat,Y4: nat] :
( ~ ( ord_less_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_488_antisym__conv2,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_489_antisym__conv2,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_490_antisym__conv2,axiom,
! [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
=> ( ( ~ ( ord_less_num @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_491_antisym__conv2,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ~ ( ord_less_int @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_492_antisym__conv2,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y4 ) )
= ( X5 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_493_dense__ge,axiom,
! [Z5: dedekind_real,Y4: dedekind_real] :
( ! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X )
=> ( ord_le2716243287969276982d_real @ Y4 @ X ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z5 ) ) ).
% dense_ge
thf(fact_494_dense__le,axiom,
! [Y4: dedekind_real,Z5: dedekind_real] :
( ! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y4 )
=> ( ord_le2716243287969276982d_real @ X @ Z5 ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z5 ) ) ).
% dense_le
thf(fact_495_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
& ~ ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_496_less__le__not__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
& ~ ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_497_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_498_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_499_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_500_not__le__imp__less,axiom,
! [Y4: dedekind_preal,X5: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y4 @ X5 )
=> ( ord_le5708704896291381698_preal @ X5 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_501_not__le__imp__less,axiom,
! [Y4: dedekind_real,X5: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ( ord_le2991122432403439658d_real @ X5 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_502_not__le__imp__less,axiom,
! [Y4: num,X5: num] :
( ~ ( ord_less_eq_num @ Y4 @ X5 )
=> ( ord_less_num @ X5 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_503_not__le__imp__less,axiom,
! [Y4: int,X5: int] :
( ~ ( ord_less_eq_int @ Y4 @ X5 )
=> ( ord_less_int @ X5 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_504_not__le__imp__less,axiom,
! [Y4: nat,X5: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ord_less_nat @ X5 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_505_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A4: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_506_order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A4: dedekind_real,B2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_507_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B2: num] :
( ( ord_less_num @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_508_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_509_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_510_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A4: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_511_order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A4: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_512_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B2: num] :
( ( ord_less_eq_num @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_513_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_514_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_515_order_Ostrict__trans1,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_516_order_Ostrict__trans1,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_517_order_Ostrict__trans1,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_518_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_519_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_520_order_Ostrict__trans2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_521_order_Ostrict__trans2,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_522_order_Ostrict__trans2,axiom,
! [A2: num,B: num,C: num] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_523_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_524_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_525_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A4: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B2 )
& ~ ( ord_le5604041210740703414_preal @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_526_order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A4: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B2 )
& ~ ( ord_le2716243287969276982d_real @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_527_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B2: num] :
( ( ord_less_eq_num @ A4 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_528_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_529_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_530_dense__ge__bounded,axiom,
! [Z5: dedekind_real,X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X5 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ X5 )
=> ( ord_le2716243287969276982d_real @ Y4 @ W2 ) ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z5 ) ) ) ).
% dense_ge_bounded
thf(fact_531_dense__le__bounded,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ! [W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ W2 )
=> ( ( ord_le2991122432403439658d_real @ W2 @ Y4 )
=> ( ord_le2716243287969276982d_real @ W2 @ Z5 ) ) )
=> ( ord_le2716243287969276982d_real @ Y4 @ Z5 ) ) ) ).
% dense_le_bounded
thf(fact_532_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B2: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_533_dual__order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [B2: dedekind_real,A4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_534_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A4: num] :
( ( ord_less_num @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_535_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_536_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_537_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_538_dual__order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_539_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A4: num] :
( ( ord_less_eq_num @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_540_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_541_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_542_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A2 )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_543_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A2 )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_544_dual__order_Ostrict__trans1,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_eq_num @ B @ A2 )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_545_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_546_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_547_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A2 )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_548_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_549_dual__order_Ostrict__trans2,axiom,
! [B: num,A2: num,C: num] :
( ( ord_less_num @ B @ A2 )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_550_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_551_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_552_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A4 )
& ~ ( ord_le5604041210740703414_preal @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_553_dual__order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A4 )
& ~ ( ord_le2716243287969276982d_real @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_554_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A4: num] :
( ( ord_less_eq_num @ B2 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_555_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_556_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_557_order_Ostrict__implies__order,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ord_le5604041210740703414_preal @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_558_order_Ostrict__implies__order,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ord_le2716243287969276982d_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_559_order_Ostrict__implies__order,axiom,
! [A2: num,B: num] :
( ( ord_less_num @ A2 @ B )
=> ( ord_less_eq_num @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_560_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_561_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_562_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A2 )
=> ( ord_le5604041210740703414_preal @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_563_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ( ord_le2716243287969276982d_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_564_dual__order_Ostrict__implies__order,axiom,
! [B: num,A2: num] :
( ( ord_less_num @ B @ A2 )
=> ( ord_less_eq_num @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_565_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_566_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_567_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_568_order__le__less,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_569_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_570_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_571_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_572_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_573_order__less__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_574_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_575_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_576_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_577_linorder__not__le,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X5 @ Y4 ) )
= ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ).
% linorder_not_le
thf(fact_578_linorder__not__le,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ X5 @ Y4 ) )
= ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ).
% linorder_not_le
thf(fact_579_linorder__not__le,axiom,
! [X5: num,Y4: num] :
( ( ~ ( ord_less_eq_num @ X5 @ Y4 ) )
= ( ord_less_num @ Y4 @ X5 ) ) ).
% linorder_not_le
thf(fact_580_linorder__not__le,axiom,
! [X5: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X5 @ Y4 ) )
= ( ord_less_int @ Y4 @ X5 ) ) ).
% linorder_not_le
thf(fact_581_linorder__not__le,axiom,
! [X5: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X5 @ Y4 ) )
= ( ord_less_nat @ Y4 @ X5 ) ) ).
% linorder_not_le
thf(fact_582_linorder__not__less,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y4 ) )
= ( ord_le5604041210740703414_preal @ Y4 @ X5 ) ) ).
% linorder_not_less
thf(fact_583_linorder__not__less,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 ) )
= ( ord_le2716243287969276982d_real @ Y4 @ X5 ) ) ).
% linorder_not_less
thf(fact_584_linorder__not__less,axiom,
! [X5: num,Y4: num] :
( ( ~ ( ord_less_num @ X5 @ Y4 ) )
= ( ord_less_eq_num @ Y4 @ X5 ) ) ).
% linorder_not_less
thf(fact_585_linorder__not__less,axiom,
! [X5: int,Y4: int] :
( ( ~ ( ord_less_int @ X5 @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X5 ) ) ).
% linorder_not_less
thf(fact_586_linorder__not__less,axiom,
! [X5: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X5 ) ) ).
% linorder_not_less
thf(fact_587_order__less__imp__le,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ord_le5604041210740703414_preal @ X5 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_588_order__less__imp__le,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_589_order__less__imp__le,axiom,
! [X5: num,Y4: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ( ord_less_eq_num @ X5 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_590_order__less__imp__le,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ X5 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_591_order__less__imp__le,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ X5 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_592_order__le__neq__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le5708704896291381698_preal @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_593_order__le__neq__trans,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le2991122432403439658d_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_594_order__le__neq__trans,axiom,
! [A2: num,B: num] :
( ( ord_less_eq_num @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_595_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_596_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_597_order__neq__le__trans,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( A2 != B )
=> ( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ord_le5708704896291381698_preal @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_598_order__neq__le__trans,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( A2 != B )
=> ( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ord_le2991122432403439658d_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_599_order__neq__le__trans,axiom,
! [A2: num,B: num] :
( ( A2 != B )
=> ( ( ord_less_eq_num @ A2 @ B )
=> ( ord_less_num @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_600_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_601_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_602_order__le__less__trans,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ Y4 @ Z5 )
=> ( ord_le5708704896291381698_preal @ X5 @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_603_order__le__less__trans,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ Z5 )
=> ( ord_le2991122432403439658d_real @ X5 @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_604_order__le__less__trans,axiom,
! [X5: num,Y4: num,Z5: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z5 )
=> ( ord_less_num @ X5 @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_605_order__le__less__trans,axiom,
! [X5: int,Y4: int,Z5: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z5 )
=> ( ord_less_int @ X5 @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_606_order__le__less__trans,axiom,
! [X5: nat,Y4: nat,Z5: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z5 )
=> ( ord_less_nat @ X5 @ Z5 ) ) ) ).
% order_le_less_trans
thf(fact_607_order__less__le__trans,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
=> ( ( ord_le5604041210740703414_preal @ Y4 @ Z5 )
=> ( ord_le5708704896291381698_preal @ X5 @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_608_order__less__le__trans,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ Z5 )
=> ( ord_le2991122432403439658d_real @ X5 @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_609_order__less__le__trans,axiom,
! [X5: num,Y4: num,Z5: num] :
( ( ord_less_num @ X5 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z5 )
=> ( ord_less_num @ X5 @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_610_order__less__le__trans,axiom,
! [X5: int,Y4: int,Z5: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z5 )
=> ( ord_less_int @ X5 @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_611_order__less__le__trans,axiom,
! [X5: nat,Y4: nat,Z5: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z5 )
=> ( ord_less_nat @ X5 @ Z5 ) ) ) ).
% order_less_le_trans
thf(fact_612_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_613_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_614_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_615_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_616_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_617_order__le__less__subst1,axiom,
! [A2: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_618_order__le__less__subst1,axiom,
! [A2: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_619_order__le__less__subst1,axiom,
! [A2: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_620_order__le__less__subst1,axiom,
! [A2: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_621_order__le__less__subst1,axiom,
! [A2: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_622_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_623_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_624_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_625_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_626_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_627_order__le__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_628_order__le__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_629_order__le__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_630_order__le__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_631_order__le__less__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_632_order__less__le__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_633_order__less__le__subst1,axiom,
! [A2: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_634_order__less__le__subst1,axiom,
! [A2: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_635_order__less__le__subst1,axiom,
! [A2: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_636_order__less__le__subst1,axiom,
! [A2: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_637_order__less__le__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_638_order__less__le__subst1,axiom,
! [A2: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_639_order__less__le__subst1,axiom,
! [A2: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_640_order__less__le__subst1,axiom,
! [A2: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_641_order__less__le__subst1,axiom,
! [A2: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_642_order__less__le__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_643_order__less__le__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_644_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > dedekind_preal,C: dedekind_preal] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_645_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_646_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_647_order__less__le__subst2,axiom,
! [A2: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_648_order__less__le__subst2,axiom,
! [A2: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_649_order__less__le__subst2,axiom,
! [A2: num,B: num,F: num > dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_650_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_651_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_652_linorder__le__less__linear,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
| ( ord_le5708704896291381698_preal @ Y4 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_653_linorder__le__less__linear,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
| ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_654_linorder__le__less__linear,axiom,
! [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
| ( ord_less_num @ Y4 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_655_linorder__le__less__linear,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
| ( ord_less_int @ Y4 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_656_linorder__le__less__linear,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
| ( ord_less_nat @ Y4 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_657_order__le__imp__less__or__eq,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y4 )
=> ( ( ord_le5708704896291381698_preal @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_658_order__le__imp__less__or__eq,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_659_order__le__imp__less__or__eq,axiom,
! [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
=> ( ( ord_less_num @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_660_order__le__imp__less__or__eq,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_int @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_661_order__le__imp__less__or__eq,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_nat @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_662_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_663_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_664_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_665_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_666_bot_Oextremum__uniqueI,axiom,
! [A2: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A2 @ bot_bo4848840305443107682_preal )
=> ( A2 = bot_bo4848840305443107682_preal ) ) ).
% bot.extremum_uniqueI
thf(fact_667_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_668_bot_Oextremum__unique,axiom,
! [A2: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A2 @ bot_bo4848840305443107682_preal )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% bot.extremum_unique
thf(fact_669_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_670_bot_Oextremum,axiom,
! [A2: set_Dedekind_preal] : ( ord_le7349499860212017814_preal @ bot_bo4848840305443107682_preal @ A2 ) ).
% bot.extremum
thf(fact_671_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_672_bot_Oextremum__strict,axiom,
! [A2: set_Dedekind_preal] :
~ ( ord_le1802228187270208418_preal @ A2 @ bot_bo4848840305443107682_preal ) ).
% bot.extremum_strict
thf(fact_673_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_674_less__numeral__extra_I3_J,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ) ).
% less_numeral_extra(3)
thf(fact_675_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_676_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_677_field__lbound__gt__zero,axiom,
! [D1: dedekind_real,D2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D1 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ D2 )
=> ? [E: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ E )
& ( ord_le2991122432403439658d_real @ E @ D1 )
& ( ord_le2991122432403439658d_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_678_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_preal,A5: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B4 @ A5 ) )
= ( ord_le5708704896291381698_preal @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_679_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_real,A5: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ B4 @ A5 ) )
= ( ord_le2991122432403439658d_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_680_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
= ( ord_less_num @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_681_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_682_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_683_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A2 @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A2 @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A2 @ X )
& ( ord_less_int @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_684_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_685_le__numeral__extra_I3_J,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ).
% le_numeral_extra(3)
thf(fact_686_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_687_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_688_bot_Oordering__top__axioms,axiom,
( orderi8630438233043249959_preal
@ ^ [X3: set_Dedekind_preal,Y3: set_Dedekind_preal] : ( ord_le7349499860212017814_preal @ Y3 @ X3 )
@ ^ [X3: set_Dedekind_preal,Y3: set_Dedekind_preal] : ( ord_le1802228187270208418_preal @ Y3 @ X3 )
@ bot_bo4848840305443107682_preal ) ).
% bot.ordering_top_axioms
thf(fact_689_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_690_Set_Ois__empty__def,axiom,
( is_emp2142545888348893928_preal
= ( ^ [A6: set_Dedekind_preal] : ( A6 = bot_bo4848840305443107682_preal ) ) ) ).
% Set.is_empty_def
thf(fact_691_Greatest__equality,axiom,
! [P: dedekind_preal > $o,X5: dedekind_preal] :
( ( P @ X5 )
=> ( ! [Y: dedekind_preal] :
( ( P @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X5 ) )
=> ( ( order_958373252487505263_preal @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_692_Greatest__equality,axiom,
! [P: dedekind_real > $o,X5: dedekind_real] :
( ( P @ X5 )
=> ( ! [Y: dedekind_real] :
( ( P @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X5 ) )
=> ( ( order_1105151008113787197d_real @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_693_Greatest__equality,axiom,
! [P: num > $o,X5: num] :
( ( P @ X5 )
=> ( ! [Y: num] :
( ( P @ Y )
=> ( ord_less_eq_num @ Y @ X5 ) )
=> ( ( order_Greatest_num @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_694_Greatest__equality,axiom,
! [P: int > $o,X5: int] :
( ( P @ X5 )
=> ( ! [Y: int] :
( ( P @ Y )
=> ( ord_less_eq_int @ Y @ X5 ) )
=> ( ( order_Greatest_int @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_695_Greatest__equality,axiom,
! [P: nat > $o,X5: nat] :
( ( P @ X5 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X5 ) )
=> ( ( order_Greatest_nat @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_696_GreatestI2__order,axiom,
! [P: dedekind_preal > $o,X5: dedekind_preal,Q: dedekind_preal > $o] :
( ( P @ X5 )
=> ( ! [Y: dedekind_preal] :
( ( P @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X5 ) )
=> ( ! [X: dedekind_preal] :
( ( P @ X )
=> ( ! [Y6: dedekind_preal] :
( ( P @ Y6 )
=> ( ord_le5604041210740703414_preal @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_958373252487505263_preal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_697_GreatestI2__order,axiom,
! [P: dedekind_real > $o,X5: dedekind_real,Q: dedekind_real > $o] :
( ( P @ X5 )
=> ( ! [Y: dedekind_real] :
( ( P @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X5 ) )
=> ( ! [X: dedekind_real] :
( ( P @ X )
=> ( ! [Y6: dedekind_real] :
( ( P @ Y6 )
=> ( ord_le2716243287969276982d_real @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1105151008113787197d_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_698_GreatestI2__order,axiom,
! [P: num > $o,X5: num,Q: num > $o] :
( ( P @ X5 )
=> ( ! [Y: num] :
( ( P @ Y )
=> ( ord_less_eq_num @ Y @ X5 ) )
=> ( ! [X: num] :
( ( P @ X )
=> ( ! [Y6: num] :
( ( P @ Y6 )
=> ( ord_less_eq_num @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_699_GreatestI2__order,axiom,
! [P: int > $o,X5: int,Q: int > $o] :
( ( P @ X5 )
=> ( ! [Y: int] :
( ( P @ Y )
=> ( ord_less_eq_int @ Y @ X5 ) )
=> ( ! [X: int] :
( ( P @ X )
=> ( ! [Y6: int] :
( ( P @ Y6 )
=> ( ord_less_eq_int @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_700_GreatestI2__order,axiom,
! [P: nat > $o,X5: nat,Q: nat > $o] :
( ( P @ X5 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X5 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_701_subsetI,axiom,
! [A3: set_Dedekind_real,B5: set_Dedekind_real] :
( ! [X: dedekind_real] :
( ( member_Dedekind_real @ X @ A3 )
=> ( member_Dedekind_real @ X @ B5 ) )
=> ( ord_le5071027967042688620d_real @ A3 @ B5 ) ) ).
% subsetI
thf(fact_702_subsetI,axiom,
! [A3: set_Dedekind_preal,B5: set_Dedekind_preal] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A3 )
=> ( member6871284927547481791_preal @ X @ B5 ) )
=> ( ord_le7349499860212017814_preal @ A3 @ B5 ) ) ).
% subsetI
thf(fact_703_in__mono,axiom,
! [A3: set_Dedekind_real,B5: set_Dedekind_real,X5: dedekind_real] :
( ( ord_le5071027967042688620d_real @ A3 @ B5 )
=> ( ( member_Dedekind_real @ X5 @ A3 )
=> ( member_Dedekind_real @ X5 @ B5 ) ) ) ).
% in_mono
thf(fact_704_in__mono,axiom,
! [A3: set_Dedekind_preal,B5: set_Dedekind_preal,X5: dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A3 @ B5 )
=> ( ( member6871284927547481791_preal @ X5 @ A3 )
=> ( member6871284927547481791_preal @ X5 @ B5 ) ) ) ).
% in_mono
thf(fact_705_subsetD,axiom,
! [A3: set_Dedekind_real,B5: set_Dedekind_real,C: dedekind_real] :
( ( ord_le5071027967042688620d_real @ A3 @ B5 )
=> ( ( member_Dedekind_real @ C @ A3 )
=> ( member_Dedekind_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_706_subsetD,axiom,
! [A3: set_Dedekind_preal,B5: set_Dedekind_preal,C: dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A3 @ B5 )
=> ( ( member6871284927547481791_preal @ C @ A3 )
=> ( member6871284927547481791_preal @ C @ B5 ) ) ) ).
% subsetD
thf(fact_707_psubsetD,axiom,
! [A3: set_Dedekind_real,B5: set_Dedekind_real,C: dedekind_real] :
( ( ord_le1729950913864802912d_real @ A3 @ B5 )
=> ( ( member_Dedekind_real @ C @ A3 )
=> ( member_Dedekind_real @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_708_psubsetD,axiom,
! [A3: set_Dedekind_preal,B5: set_Dedekind_preal,C: dedekind_preal] :
( ( ord_le1802228187270208418_preal @ A3 @ B5 )
=> ( ( member6871284927547481791_preal @ C @ A3 )
=> ( member6871284927547481791_preal @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_709_subset__eq,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A6: set_Dedekind_real,B6: set_Dedekind_real] :
! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ A6 )
=> ( member_Dedekind_real @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_710_subset__eq,axiom,
( ord_le7349499860212017814_preal
= ( ^ [A6: set_Dedekind_preal,B6: set_Dedekind_preal] :
! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A6 )
=> ( member6871284927547481791_preal @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_711_subset__iff,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A6: set_Dedekind_real,B6: set_Dedekind_real] :
! [T2: dedekind_real] :
( ( member_Dedekind_real @ T2 @ A6 )
=> ( member_Dedekind_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_712_subset__iff,axiom,
( ord_le7349499860212017814_preal
= ( ^ [A6: set_Dedekind_preal,B6: set_Dedekind_preal] :
! [T2: dedekind_preal] :
( ( member6871284927547481791_preal @ T2 @ A6 )
=> ( member6871284927547481791_preal @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_713_Collect__mono,axiom,
! [P: dedekind_preal > $o,Q: dedekind_preal > $o] :
( ! [X: dedekind_preal] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le7349499860212017814_preal @ ( collec1132657498972982273_preal @ P ) @ ( collec1132657498972982273_preal @ Q ) ) ) ).
% Collect_mono
thf(fact_714_less__set__def,axiom,
( ord_le1729950913864802912d_real
= ( ^ [A6: set_Dedekind_real,B6: set_Dedekind_real] :
( ord_le2006262546704330789real_o
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ A6 )
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_715_less__set__def,axiom,
( ord_le1802228187270208418_preal
= ( ^ [A6: set_Dedekind_preal,B6: set_Dedekind_preal] :
( ord_le1914443497901267899real_o
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ A6 )
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_716_Collect__subset,axiom,
! [A3: set_Dedekind_real,P: dedekind_real > $o] :
( ord_le5071027967042688620d_real
@ ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_717_Collect__subset,axiom,
! [A3: set_Dedekind_preal,P: dedekind_preal > $o] :
( ord_le7349499860212017814_preal
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_718_less__eq__set__def,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A6: set_Dedekind_real,B6: set_Dedekind_real] :
( ord_le60669685330047769real_o
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ A6 )
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_719_less__eq__set__def,axiom,
( ord_le7349499860212017814_preal
= ( ^ [A6: set_Dedekind_preal,B6: set_Dedekind_preal] :
( ord_le5990576202697070023real_o
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ A6 )
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_720_Collect__mono__iff,axiom,
! [P: dedekind_preal > $o,Q: dedekind_preal > $o] :
( ( ord_le7349499860212017814_preal @ ( collec1132657498972982273_preal @ P ) @ ( collec1132657498972982273_preal @ Q ) )
= ( ! [X3: dedekind_preal] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_721_real__le__refl,axiom,
! [W3: dedekind_real] : ( ord_le2716243287969276982d_real @ W3 @ W3 ) ).
% real_le_refl
thf(fact_722_real__le__trans,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real] :
( ( ord_le2716243287969276982d_real @ I @ J )
=> ( ( ord_le2716243287969276982d_real @ J @ K )
=> ( ord_le2716243287969276982d_real @ I @ K ) ) ) ).
% real_le_trans
thf(fact_723_real__le__antisym,axiom,
! [Z5: dedekind_real,W3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z5 @ W3 )
=> ( ( ord_le2716243287969276982d_real @ W3 @ Z5 )
=> ( Z5 = W3 ) ) ) ).
% real_le_antisym
thf(fact_724_verit__comp__simplify1_I2_J,axiom,
! [A2: dedekind_preal] : ( ord_le5604041210740703414_preal @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_725_verit__comp__simplify1_I2_J,axiom,
! [A2: dedekind_real] : ( ord_le2716243287969276982d_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_726_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_727_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_728_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_729_verit__la__disequality,axiom,
! [A2: dedekind_preal,B: dedekind_preal] :
( ( A2 = B )
| ~ ( ord_le5604041210740703414_preal @ A2 @ B )
| ~ ( ord_le5604041210740703414_preal @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_730_verit__la__disequality,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( A2 = B )
| ~ ( ord_le2716243287969276982d_real @ A2 @ B )
| ~ ( ord_le2716243287969276982d_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_731_verit__la__disequality,axiom,
! [A2: num,B: num] :
( ( A2 = B )
| ~ ( ord_less_eq_num @ A2 @ B )
| ~ ( ord_less_eq_num @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_732_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_733_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_734_verit__comp__simplify1_I1_J,axiom,
! [A2: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_735_verit__comp__simplify1_I1_J,axiom,
! [A2: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_736_verit__comp__simplify1_I1_J,axiom,
! [A2: num] :
~ ( ord_less_num @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_737_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_738_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_739_bot__empty__eq,axiom,
( bot_bo7891357765246109669real_o
= ( ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ bot_bo2548302898904406688d_real ) ) ) ).
% bot_empty_eq
thf(fact_740_bot__empty__eq,axiom,
( bot_bo4815710026557797371real_o
= ( ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ bot_bo4848840305443107682_preal ) ) ) ).
% bot_empty_eq
thf(fact_741_Collect__empty__eq__bot,axiom,
! [P: dedekind_preal > $o] :
( ( ( collec1132657498972982273_preal @ P )
= bot_bo4848840305443107682_preal )
= ( P = bot_bo4815710026557797371real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_742_subset__emptyI,axiom,
! [A3: set_Dedekind_real] :
( ! [X: dedekind_real] :
~ ( member_Dedekind_real @ X @ A3 )
=> ( ord_le5071027967042688620d_real @ A3 @ bot_bo2548302898904406688d_real ) ) ).
% subset_emptyI
thf(fact_743_subset__emptyI,axiom,
! [A3: set_Dedekind_preal] :
( ! [X: dedekind_preal] :
~ ( member6871284927547481791_preal @ X @ A3 )
=> ( ord_le7349499860212017814_preal @ A3 @ bot_bo4848840305443107682_preal ) ) ).
% subset_emptyI
thf(fact_744_pred__subset__eq,axiom,
! [R: set_Dedekind_real,S: set_Dedekind_real] :
( ( ord_le60669685330047769real_o
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ R )
@ ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ S ) )
= ( ord_le5071027967042688620d_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_745_pred__subset__eq,axiom,
! [R: set_Dedekind_preal,S: set_Dedekind_preal] :
( ( ord_le5990576202697070023real_o
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ R )
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ S ) )
= ( ord_le7349499860212017814_preal @ R @ S ) ) ).
% pred_subset_eq
thf(fact_746_conj__subset__def,axiom,
! [A3: set_Dedekind_preal,P: dedekind_preal > $o,Q: dedekind_preal > $o] :
( ( ord_le7349499860212017814_preal @ A3
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le7349499860212017814_preal @ A3 @ ( collec1132657498972982273_preal @ P ) )
& ( ord_le7349499860212017814_preal @ A3 @ ( collec1132657498972982273_preal @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_747_prop__restrict,axiom,
! [X5: dedekind_real,Z: set_Dedekind_real,X7: set_Dedekind_real,P: dedekind_real > $o] :
( ( member_Dedekind_real @ X5 @ Z )
=> ( ( ord_le5071027967042688620d_real @ Z
@ ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_748_prop__restrict,axiom,
! [X5: dedekind_preal,Z: set_Dedekind_preal,X7: set_Dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X5 @ Z )
=> ( ( ord_le7349499860212017814_preal @ Z
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_749_Collect__restrict,axiom,
! [X7: set_Dedekind_real,P: dedekind_real > $o] :
( ord_le5071027967042688620d_real
@ ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_750_Collect__restrict,axiom,
! [X7: set_Dedekind_preal,P: dedekind_preal > $o] :
( ord_le7349499860212017814_preal
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_751_subset__CollectI,axiom,
! [B5: set_Dedekind_real,A3: set_Dedekind_real,Q: dedekind_real > $o,P: dedekind_real > $o] :
( ( ord_le5071027967042688620d_real @ B5 @ A3 )
=> ( ! [X: dedekind_real] :
( ( member_Dedekind_real @ X @ B5 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le5071027967042688620d_real
@ ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_752_subset__CollectI,axiom,
! [B5: set_Dedekind_preal,A3: set_Dedekind_preal,Q: dedekind_preal > $o,P: dedekind_preal > $o] :
( ( ord_le7349499860212017814_preal @ B5 @ A3 )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B5 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le7349499860212017814_preal
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_753_subset__Collect__iff,axiom,
! [B5: set_Dedekind_real,A3: set_Dedekind_real,P: dedekind_real > $o] :
( ( ord_le5071027967042688620d_real @ B5 @ A3 )
=> ( ( ord_le5071027967042688620d_real @ B5
@ ( collec1873404778653292075d_real
@ ^ [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_754_subset__Collect__iff,axiom,
! [B5: set_Dedekind_preal,A3: set_Dedekind_preal,P: dedekind_preal > $o] :
( ( ord_le7349499860212017814_preal @ B5 @ A3 )
=> ( ( ord_le7349499860212017814_preal @ B5
@ ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_755_bdd__below_Opreordering__bdd__axioms,axiom,
( condit4019698779847608237_preal
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5604041210740703414_preal @ Y3 @ X3 )
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_756_bdd__below_Opreordering__bdd__axioms,axiom,
( condit6212436594666024831d_real
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2716243287969276982d_real @ Y3 @ X3 )
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_757_bdd__below_Opreordering__bdd__axioms,axiom,
( condit4492884260299903299dd_num
@ ^ [X3: num,Y3: num] : ( ord_less_eq_num @ Y3 @ X3 )
@ ^ [X3: num,Y3: num] : ( ord_less_num @ Y3 @ X3 ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_758_bdd__below_Opreordering__bdd__axioms,axiom,
( condit7933062003635074389dd_int
@ ^ [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ X3 )
@ ^ [X3: int,Y3: int] : ( ord_less_int @ Y3 @ X3 ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_759_bdd__below_Opreordering__bdd__axioms,axiom,
( condit7935552474144124665dd_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_760_dual__order_Oordering__axioms,axiom,
( orderi4673934450047582894_preal
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5604041210740703414_preal @ Y3 @ X3 )
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% dual_order.ordering_axioms
thf(fact_761_dual__order_Oordering__axioms,axiom,
( orderi8957706999020659774d_real
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2716243287969276982d_real @ Y3 @ X3 )
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% dual_order.ordering_axioms
thf(fact_762_dual__order_Oordering__axioms,axiom,
( ordering_num
@ ^ [X3: num,Y3: num] : ( ord_less_eq_num @ Y3 @ X3 )
@ ^ [X3: num,Y3: num] : ( ord_less_num @ Y3 @ X3 ) ) ).
% dual_order.ordering_axioms
thf(fact_763_dual__order_Oordering__axioms,axiom,
( ordering_int
@ ^ [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ X3 )
@ ^ [X3: int,Y3: int] : ( ord_less_int @ Y3 @ X3 ) ) ).
% dual_order.ordering_axioms
thf(fact_764_dual__order_Oordering__axioms,axiom,
( ordering_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 ) ) ).
% dual_order.ordering_axioms
thf(fact_765_dual__order_Opreordering__axioms,axiom,
( preord1889057019909701233_preal
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5604041210740703414_preal @ Y3 @ X3 )
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5708704896291381698_preal @ Y3 @ X3 ) ) ).
% dual_order.preordering_axioms
thf(fact_766_dual__order_Opreordering__axioms,axiom,
( preord5216658436359260091d_real
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2716243287969276982d_real @ Y3 @ X3 )
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2991122432403439658d_real @ Y3 @ X3 ) ) ).
% dual_order.preordering_axioms
thf(fact_767_dual__order_Opreordering__axioms,axiom,
( preordering_num
@ ^ [X3: num,Y3: num] : ( ord_less_eq_num @ Y3 @ X3 )
@ ^ [X3: num,Y3: num] : ( ord_less_num @ Y3 @ X3 ) ) ).
% dual_order.preordering_axioms
thf(fact_768_dual__order_Opreordering__axioms,axiom,
( preordering_int
@ ^ [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ X3 )
@ ^ [X3: int,Y3: int] : ( ord_less_int @ Y3 @ X3 ) ) ).
% dual_order.preordering_axioms
thf(fact_769_dual__order_Opreordering__axioms,axiom,
( preordering_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 ) ) ).
% dual_order.preordering_axioms
thf(fact_770_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec1132657498972982273_preal
@ ^ [S2: dedekind_preal] : P )
= top_to2164892472777521222_preal ) )
& ( ~ P
=> ( ( collec1132657498972982273_preal
@ ^ [S2: dedekind_preal] : P )
= bot_bo4848840305443107682_preal ) ) ) ).
% Collect_const
thf(fact_771_empty__not__UNIV,axiom,
bot_bo4848840305443107682_preal != top_to2164892472777521222_preal ).
% empty_not_UNIV
thf(fact_772_order_Opreordering__axioms,axiom,
preord1889057019909701233_preal @ ord_le5604041210740703414_preal @ ord_le5708704896291381698_preal ).
% order.preordering_axioms
thf(fact_773_order_Opreordering__axioms,axiom,
preord5216658436359260091d_real @ ord_le2716243287969276982d_real @ ord_le2991122432403439658d_real ).
% order.preordering_axioms
thf(fact_774_order_Opreordering__axioms,axiom,
preordering_num @ ord_less_eq_num @ ord_less_num ).
% order.preordering_axioms
thf(fact_775_order_Opreordering__axioms,axiom,
preordering_int @ ord_less_eq_int @ ord_less_int ).
% order.preordering_axioms
thf(fact_776_order_Opreordering__axioms,axiom,
preordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.preordering_axioms
thf(fact_777_order_Oordering__axioms,axiom,
orderi4673934450047582894_preal @ ord_le5604041210740703414_preal @ ord_le5708704896291381698_preal ).
% order.ordering_axioms
thf(fact_778_order_Oordering__axioms,axiom,
orderi8957706999020659774d_real @ ord_le2716243287969276982d_real @ ord_le2991122432403439658d_real ).
% order.ordering_axioms
thf(fact_779_order_Oordering__axioms,axiom,
ordering_num @ ord_less_eq_num @ ord_less_num ).
% order.ordering_axioms
thf(fact_780_order_Oordering__axioms,axiom,
ordering_int @ ord_less_eq_int @ ord_less_int ).
% order.ordering_axioms
thf(fact_781_order_Oordering__axioms,axiom,
ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.ordering_axioms
thf(fact_782_bdd__above_Opreordering__bdd__axioms,axiom,
condit4019698779847608237_preal @ ord_le5604041210740703414_preal @ ord_le5708704896291381698_preal ).
% bdd_above.preordering_bdd_axioms
thf(fact_783_bdd__above_Opreordering__bdd__axioms,axiom,
condit6212436594666024831d_real @ ord_le2716243287969276982d_real @ ord_le2991122432403439658d_real ).
% bdd_above.preordering_bdd_axioms
thf(fact_784_bdd__above_Opreordering__bdd__axioms,axiom,
condit4492884260299903299dd_num @ ord_less_eq_num @ ord_less_num ).
% bdd_above.preordering_bdd_axioms
thf(fact_785_bdd__above_Opreordering__bdd__axioms,axiom,
condit7933062003635074389dd_int @ ord_less_eq_int @ ord_less_int ).
% bdd_above.preordering_bdd_axioms
thf(fact_786_bdd__above_Opreordering__bdd__axioms,axiom,
condit7935552474144124665dd_nat @ ord_less_eq_nat @ ord_less_nat ).
% bdd_above.preordering_bdd_axioms
thf(fact_787_preordering__bdd_Oempty,axiom,
! [Less_eq: dedekind_preal > dedekind_preal > $o,Less: dedekind_preal > dedekind_preal > $o] :
( ( condit4019698779847608237_preal @ Less_eq @ Less )
=> ( condit1308845025010339211_preal @ Less_eq @ bot_bo4848840305443107682_preal ) ) ).
% preordering_bdd.empty
thf(fact_788_linorder__neqE__linordered__idom,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( X5 != Y4 )
=> ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_789_linorder__neqE__linordered__idom,axiom,
! [X5: int,Y4: int] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ Y4 @ X5 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_790_UNIV__I,axiom,
! [X5: dedekind_real] : ( member_Dedekind_real @ X5 @ top_to4551142129947008188d_real ) ).
% UNIV_I
thf(fact_791_UNIV__I,axiom,
! [X5: dedekind_preal] : ( member6871284927547481791_preal @ X5 @ top_to2164892472777521222_preal ) ).
% UNIV_I
thf(fact_792_top__empty__eq,axiom,
( top_to4979224932776997065real_o
= ( ^ [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ top_to4551142129947008188d_real ) ) ) ).
% top_empty_eq
thf(fact_793_top__empty__eq,axiom,
( top_to6295660221356622871real_o
= ( ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ top_to2164892472777521222_preal ) ) ) ).
% top_empty_eq
thf(fact_794_UNIV__witness,axiom,
? [X: dedekind_real] : ( member_Dedekind_real @ X @ top_to4551142129947008188d_real ) ).
% UNIV_witness
thf(fact_795_UNIV__witness,axiom,
? [X: dedekind_preal] : ( member6871284927547481791_preal @ X @ top_to2164892472777521222_preal ) ).
% UNIV_witness
thf(fact_796_top__set__def,axiom,
( top_to2164892472777521222_preal
= ( collec1132657498972982273_preal @ top_to6295660221356622871real_o ) ) ).
% top_set_def
thf(fact_797_UNIV__eq__I,axiom,
! [A3: set_Dedekind_real] :
( ! [X: dedekind_real] : ( member_Dedekind_real @ X @ A3 )
=> ( top_to4551142129947008188d_real = A3 ) ) ).
% UNIV_eq_I
thf(fact_798_UNIV__eq__I,axiom,
! [A3: set_Dedekind_preal] :
( ! [X: dedekind_preal] : ( member6871284927547481791_preal @ X @ A3 )
=> ( top_to2164892472777521222_preal = A3 ) ) ).
% UNIV_eq_I
thf(fact_799_UNIV__def,axiom,
( top_to2164892472777521222_preal
= ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] : $true ) ) ).
% UNIV_def
thf(fact_800_preordering__bdd_OE,axiom,
! [Less_eq: dedekind_real > dedekind_real > $o,Less: dedekind_real > dedekind_real > $o,A3: set_Dedekind_real] :
( ( condit6212436594666024831d_real @ Less_eq @ Less )
=> ( ( condit3870944721565864481d_real @ Less_eq @ A3 )
=> ~ ! [M4: dedekind_real] :
~ ! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ A3 )
=> ( Less_eq @ X2 @ M4 ) ) ) ) ).
% preordering_bdd.E
thf(fact_801_preordering__bdd_OE,axiom,
! [Less_eq: dedekind_preal > dedekind_preal > $o,Less: dedekind_preal > dedekind_preal > $o,A3: set_Dedekind_preal] :
( ( condit4019698779847608237_preal @ Less_eq @ Less )
=> ( ( condit1308845025010339211_preal @ Less_eq @ A3 )
=> ~ ! [M4: dedekind_preal] :
~ ! [X2: dedekind_preal] :
( ( member6871284927547481791_preal @ X2 @ A3 )
=> ( Less_eq @ X2 @ M4 ) ) ) ) ).
% preordering_bdd.E
thf(fact_802_preordering__bdd_OI,axiom,
! [Less_eq: dedekind_real > dedekind_real > $o,Less: dedekind_real > dedekind_real > $o,A3: set_Dedekind_real,M5: dedekind_real] :
( ( condit6212436594666024831d_real @ Less_eq @ Less )
=> ( ! [X: dedekind_real] :
( ( member_Dedekind_real @ X @ A3 )
=> ( Less_eq @ X @ M5 ) )
=> ( condit3870944721565864481d_real @ Less_eq @ A3 ) ) ) ).
% preordering_bdd.I
thf(fact_803_preordering__bdd_OI,axiom,
! [Less_eq: dedekind_preal > dedekind_preal > $o,Less: dedekind_preal > dedekind_preal > $o,A3: set_Dedekind_preal,M5: dedekind_preal] :
( ( condit4019698779847608237_preal @ Less_eq @ Less )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A3 )
=> ( Less_eq @ X @ M5 ) )
=> ( condit1308845025010339211_preal @ Less_eq @ A3 ) ) ) ).
% preordering_bdd.I
thf(fact_804_iso__tuple__UNIV__I,axiom,
! [X5: dedekind_real] : ( member_Dedekind_real @ X5 @ top_to4551142129947008188d_real ) ).
% iso_tuple_UNIV_I
thf(fact_805_iso__tuple__UNIV__I,axiom,
! [X5: dedekind_preal] : ( member6871284927547481791_preal @ X5 @ top_to2164892472777521222_preal ) ).
% iso_tuple_UNIV_I
thf(fact_806_order_Opartial__preordering__axioms,axiom,
partia2807573505362346828_preal @ ord_le5604041210740703414_preal ).
% order.partial_preordering_axioms
thf(fact_807_order_Opartial__preordering__axioms,axiom,
partia4124895370616003616d_real @ ord_le2716243287969276982d_real ).
% order.partial_preordering_axioms
thf(fact_808_order_Opartial__preordering__axioms,axiom,
partia3380149844792115556ng_num @ ord_less_eq_num ).
% order.partial_preordering_axioms
thf(fact_809_order_Opartial__preordering__axioms,axiom,
partia6820327588127286646ng_int @ ord_less_eq_int ).
% order.partial_preordering_axioms
thf(fact_810_order_Opartial__preordering__axioms,axiom,
partia6822818058636336922ng_nat @ ord_less_eq_nat ).
% order.partial_preordering_axioms
thf(fact_811_dual__order_Opartial__preordering__axioms,axiom,
( partia2807573505362346828_preal
@ ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_812_dual__order_Opartial__preordering__axioms,axiom,
( partia4124895370616003616d_real
@ ^ [X3: dedekind_real,Y3: dedekind_real] : ( ord_le2716243287969276982d_real @ Y3 @ X3 ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_813_dual__order_Opartial__preordering__axioms,axiom,
( partia3380149844792115556ng_num
@ ^ [X3: num,Y3: num] : ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_814_dual__order_Opartial__preordering__axioms,axiom,
( partia6820327588127286646ng_int
@ ^ [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_815_dual__order_Opartial__preordering__axioms,axiom,
( partia6822818058636336922ng_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_816_linordered__field__no__ub,axiom,
! [X2: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_817_linordered__field__no__lb,axiom,
! [X2: dedekind_real] :
? [Y: dedekind_real] : ( ord_le2991122432403439658d_real @ Y @ X2 ) ).
% linordered_field_no_lb
thf(fact_818_dbl__simps_I2_J,axiom,
( ( neg_nu8665065731627423525d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% dbl_simps(2)
thf(fact_819_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_820_le__rel__bool__arg__iff,axiom,
( ord_le6023059474077774659_preal
= ( ^ [X8: $o > dedekind_preal,Y7: $o > dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le5604041210740703414_preal @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_821_le__rel__bool__arg__iff,axiom,
( ord_le267469853237711487d_real
= ( ^ [X8: $o > dedekind_real,Y7: $o > dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le2716243287969276982d_real @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_822_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_num
= ( ^ [X8: $o > num,Y7: $o > num] :
( ( ord_less_eq_num @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_num @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_823_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_int
= ( ^ [X8: $o > int,Y7: $o > int] :
( ( ord_less_eq_int @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_int @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_824_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X8: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X8 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X8 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_825_mult__le__cancel__left,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_826_mult__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_827_mult__zero__left,axiom,
! [A2: dedekind_real] :
( ( times_2157731159493324635d_real @ zero_z580800474297136991d_real @ A2 )
= zero_z580800474297136991d_real ) ).
% mult_zero_left
thf(fact_828_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_829_mult__zero__right,axiom,
! [A2: dedekind_real] :
( ( times_2157731159493324635d_real @ A2 @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% mult_zero_right
thf(fact_830_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_831_mult__eq__0__iff,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A2 @ B )
= zero_z580800474297136991d_real )
= ( ( A2 = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% mult_eq_0_iff
thf(fact_832_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_833_mult__cancel__left,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A2 )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_834_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_835_mult__cancel__right,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A2 @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_836_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_837_real__of__preal__mult,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( times_3000655703912201937_preal @ X5 @ Y4 ) )
= ( times_2157731159493324635d_real @ ( dedeki829075424511776921_preal @ X5 ) @ ( dedeki829075424511776921_preal @ Y4 ) ) ) ).
% real_of_preal_mult
thf(fact_838_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A2 @ B ) @ C )
= ( times_3000655703912201937_preal @ A2 @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_839_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ C )
= ( times_2157731159493324635d_real @ A2 @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_840_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_841_mult_Oassoc,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A2 @ B ) @ C )
= ( times_3000655703912201937_preal @ A2 @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_842_mult_Oassoc,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ C )
= ( times_2157731159493324635d_real @ A2 @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_843_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_844_mult_Ocommute,axiom,
( times_3000655703912201937_preal
= ( ^ [A4: dedekind_preal,B2: dedekind_preal] : ( times_3000655703912201937_preal @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_845_mult_Ocommute,axiom,
( times_2157731159493324635d_real
= ( ^ [A4: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_846_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B2: int] : ( times_times_int @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_847_mult_Oleft__commute,axiom,
! [B: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ B @ ( times_3000655703912201937_preal @ A2 @ C ) )
= ( times_3000655703912201937_preal @ A2 @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_848_mult_Oleft__commute,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ B @ ( times_2157731159493324635d_real @ A2 @ C ) )
= ( times_2157731159493324635d_real @ A2 @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_849_mult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_850_mult__not__zero,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A2 @ B )
!= zero_z580800474297136991d_real )
=> ( ( A2 != zero_z580800474297136991d_real )
& ( B != zero_z580800474297136991d_real ) ) ) ).
% mult_not_zero
thf(fact_851_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_852_divisors__zero,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A2 @ B )
= zero_z580800474297136991d_real )
=> ( ( A2 = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divisors_zero
thf(fact_853_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_854_no__zero__divisors,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( A2 != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A2 @ B )
!= zero_z580800474297136991d_real ) ) ) ).
% no_zero_divisors
thf(fact_855_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_856_mult__left__cancel,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ C @ A2 )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_857_mult__left__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_858_mult__right__cancel,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A2 @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_859_mult__right__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_860_lambda__zero,axiom,
( ( ^ [H: dedekind_real] : zero_z580800474297136991d_real )
= ( times_2157731159493324635d_real @ zero_z580800474297136991d_real ) ) ).
% lambda_zero
thf(fact_861_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_862_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_863_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_864_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_865_zero__le__mult__iff,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_866_zero__le__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_867_mult__nonneg__nonpos2,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ B @ A2 ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_868_mult__nonneg__nonpos2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_869_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_870_mult__nonpos__nonneg,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_871_mult__nonpos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_872_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_873_mult__nonneg__nonpos,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_874_mult__nonneg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_875_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_876_mult__nonneg__nonneg,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_877_mult__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_878_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_879_split__mult__neg__le,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real ) ) ).
% split_mult_neg_le
thf(fact_880_split__mult__neg__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_881_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_882_mult__le__0__iff,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_883_mult__le__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_884_mult__right__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_885_mult__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_886_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_887_mult__right__mono__neg,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A2 )
=> ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_888_mult__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_889_mult__left__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_890_mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_891_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_892_mult__nonpos__nonpos,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_893_mult__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_894_mult__left__mono__neg,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A2 )
=> ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_895_mult__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_896_split__mult__pos__le,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
& ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real )
& ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real ) ) )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_897_split__mult__pos__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_898_zero__le__square,axiom,
! [A2: dedekind_real] : ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_899_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_900_mult__mono_H,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_901_mult__mono_H,axiom,
! [A2: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_902_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_903_mult__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_904_mult__mono,axiom,
! [A2: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_905_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_906_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_907_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_908_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_909_mult__less__cancel__right__disj,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
& ( ord_le2991122432403439658d_real @ A2 @ B ) )
| ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_910_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_911_mult__strict__right__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_912_mult__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_913_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_914_mult__strict__right__mono__neg,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_915_mult__strict__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_916_mult__less__cancel__left__disj,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
& ( ord_le2991122432403439658d_real @ A2 @ B ) )
| ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_917_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_918_mult__strict__left__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_919_mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_920_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_921_mult__strict__left__mono__neg,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A2 )
=> ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_922_mult__strict__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_923_mult__less__cancel__left__pos,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_924_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_925_mult__less__cancel__left__neg,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_926_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_927_zero__less__mult__pos2,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ B @ A2 ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_928_zero__less__mult__pos2,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_929_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_930_zero__less__mult__pos,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_931_zero__less__mult__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_932_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_933_zero__less__mult__iff,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
& ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) )
| ( ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_934_zero__less__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_935_mult__pos__neg2,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ B @ A2 ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_pos_neg2
thf(fact_936_mult__pos__neg2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_937_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_938_mult__pos__pos,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_939_mult__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_940_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_941_mult__pos__neg,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_pos_neg
thf(fact_942_mult__pos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_943_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_944_mult__neg__pos,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% mult_neg_pos
thf(fact_945_mult__neg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_946_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_947_mult__less__0__iff,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ zero_z580800474297136991d_real )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
& ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real ) )
| ( ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real )
& ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_948_mult__less__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_949_not__square__less__zero,axiom,
! [A2: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ A2 ) @ zero_z580800474297136991d_real ) ).
% not_square_less_zero
thf(fact_950_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_951_mult__neg__neg,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_952_mult__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_953_real__mult__order,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X5 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Y4 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ X5 @ Y4 ) ) ) ) ).
% real_mult_order
thf(fact_954_real__mult__less__mono2,axiom,
! [Z5: dedekind_real,X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z5 )
=> ( ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ Z5 @ X5 ) @ ( times_2157731159493324635d_real @ Z5 @ Y4 ) ) ) ) ).
% real_mult_less_mono2
thf(fact_955_mult__less__le__imp__less,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_956_mult__less__le__imp__less,axiom,
! [A2: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_957_mult__less__le__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_958_mult__le__less__imp__less,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D3 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_959_mult__le__less__imp__less,axiom,
! [A2: int,B: int,C: int,D3: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_960_mult__le__less__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_961_mult__right__le__imp__le,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_962_mult__right__le__imp__le,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_963_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_964_mult__left__le__imp__le,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_965_mult__left__le__imp__le,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_966_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_967_mult__le__cancel__left__pos,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ A2 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_968_mult__le__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_969_mult__le__cancel__left__neg,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ B @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_970_mult__le__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_971_mult__less__cancel__right,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_972_mult__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_973_mult__strict__mono_H,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D3 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_974_mult__strict__mono_H,axiom,
! [A2: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_975_mult__strict__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( 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 @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_976_mult__right__less__imp__less,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_977_mult__right__less__imp__less,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_978_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_979_mult__less__cancel__left,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_980_mult__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_981_mult__strict__mono,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real,D3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D3 )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_982_mult__strict__mono,axiom,
! [A2: int,B: int,C: int,D3: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D3 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_983_mult__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D3 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_984_mult__left__less__imp__less,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ ( times_2157731159493324635d_real @ C @ B ) )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_985_mult__left__less__imp__less,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_986_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_987_mult__le__cancel__right,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_988_mult__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_989_mult__le__cancel__iff2,axiom,
! [Z5: dedekind_real,X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z5 )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Z5 @ X5 ) @ ( times_2157731159493324635d_real @ Z5 @ Y4 ) )
= ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_990_mult__le__cancel__iff2,axiom,
! [Z5: int,X5: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z5 @ X5 ) @ ( times_times_int @ Z5 @ Y4 ) )
= ( ord_less_eq_int @ X5 @ Y4 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_991_mult__le__cancel__iff1,axiom,
! [Z5: dedekind_real,X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z5 )
=> ( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ X5 @ Z5 ) @ ( times_2157731159493324635d_real @ Y4 @ Z5 ) )
= ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_992_mult__le__cancel__iff1,axiom,
! [Z5: int,X5: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X5 @ Z5 ) @ ( times_times_int @ Y4 @ Z5 ) )
= ( ord_less_eq_int @ X5 @ Y4 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_993_mult__less__iff1,axiom,
! [Z5: dedekind_real,X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z5 )
=> ( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ X5 @ Z5 ) @ ( times_2157731159493324635d_real @ Y4 @ Z5 ) )
= ( ord_le2991122432403439658d_real @ X5 @ Y4 ) ) ) ).
% mult_less_iff1
thf(fact_994_mult__less__iff1,axiom,
! [Z5: int,X5: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_int @ ( times_times_int @ X5 @ Z5 ) @ ( times_times_int @ Y4 @ Z5 ) )
= ( ord_less_int @ X5 @ Y4 ) ) ) ).
% mult_less_iff1
thf(fact_995_mult__commute__abs,axiom,
! [C: dedekind_preal] :
( ( ^ [X3: dedekind_preal] : ( times_3000655703912201937_preal @ X3 @ C ) )
= ( times_3000655703912201937_preal @ C ) ) ).
% mult_commute_abs
thf(fact_996_mult__commute__abs,axiom,
! [C: dedekind_real] :
( ( ^ [X3: dedekind_real] : ( times_2157731159493324635d_real @ X3 @ C ) )
= ( times_2157731159493324635d_real @ C ) ) ).
% mult_commute_abs
thf(fact_997_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X3: int] : ( times_times_int @ X3 @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_998_field__le__mult__one__interval,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ! [Z2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Z2 )
=> ( ( ord_le2991122432403439658d_real @ Z2 @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Z2 @ X5 ) @ Y4 ) ) )
=> ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ).
% field_le_mult_one_interval
thf(fact_999_mult__less__cancel__right2,axiom,
! [A2: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ C )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ one_on6069100329679821595d_real ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1000_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_1001_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_1002_mult__1,axiom,
! [A2: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A2 )
= A2 ) ).
% mult_1
thf(fact_1003_mult__1,axiom,
! [A2: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_1004_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_1005_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_1006_mult_Oright__neutral,axiom,
! [A2: dedekind_preal] :
( ( times_3000655703912201937_preal @ A2 @ one_on9143529541772854033_preal )
= A2 ) ).
% mult.right_neutral
thf(fact_1007_mult_Oright__neutral,axiom,
! [A2: dedekind_real] :
( ( times_2157731159493324635d_real @ A2 @ one_on6069100329679821595d_real )
= A2 ) ).
% mult.right_neutral
thf(fact_1008_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_1009_mult__cancel__right2,axiom,
! [A2: dedekind_real,C: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A2 @ C )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A2 = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right2
thf(fact_1010_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_1011_mult__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right1
thf(fact_1012_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1013_mult__cancel__left2,axiom,
! [C: dedekind_real,A2: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A2 )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A2 = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left2
thf(fact_1014_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_1015_mult__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left1
thf(fact_1016_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1017_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( times_3000655703912201937_preal @ Y3 @ X3 ) ) ) ).
% preal_mult_commute
thf(fact_1018_real__mult__commute,axiom,
( times_2157731159493324635d_real
= ( ^ [Z6: dedekind_real,W: dedekind_real] : ( times_2157731159493324635d_real @ W @ Z6 ) ) ) ).
% real_mult_commute
thf(fact_1019_preal__mult__assoc,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X5 @ Y4 ) @ Z5 )
= ( times_3000655703912201937_preal @ X5 @ ( times_3000655703912201937_preal @ Y4 @ Z5 ) ) ) ).
% preal_mult_assoc
thf(fact_1020_real__mult__assoc,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,Z32: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ Z1 @ Z22 ) @ Z32 )
= ( times_2157731159493324635d_real @ Z1 @ ( times_2157731159493324635d_real @ Z22 @ Z32 ) ) ) ).
% real_mult_assoc
thf(fact_1021_preal__mult__1,axiom,
! [Z5: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z5 )
= Z5 ) ).
% preal_mult_1
thf(fact_1022_real__mult__1,axiom,
! [Z5: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z5 )
= Z5 ) ).
% real_mult_1
thf(fact_1023_one__reorient,axiom,
! [X5: dedekind_preal] :
( ( one_on9143529541772854033_preal = X5 )
= ( X5 = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_1024_one__reorient,axiom,
! [X5: dedekind_real] :
( ( one_on6069100329679821595d_real = X5 )
= ( X5 = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_1025_one__reorient,axiom,
! [X5: nat] :
( ( one_one_nat = X5 )
= ( X5 = one_one_nat ) ) ).
% one_reorient
thf(fact_1026_one__reorient,axiom,
! [X5: int] :
( ( one_one_int = X5 )
= ( X5 = one_one_int ) ) ).
% one_reorient
thf(fact_1027_le__numeral__extra_I4_J,axiom,
ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ).
% le_numeral_extra(4)
thf(fact_1028_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1029_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1030_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1031_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_1032_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1033_less__numeral__extra_I4_J,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ) ).
% less_numeral_extra(4)
thf(fact_1034_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1035_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1036_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_1037_mult_Ocomm__neutral,axiom,
! [A2: dedekind_preal] :
( ( times_3000655703912201937_preal @ A2 @ one_on9143529541772854033_preal )
= A2 ) ).
% mult.comm_neutral
thf(fact_1038_mult_Ocomm__neutral,axiom,
! [A2: dedekind_real] :
( ( times_2157731159493324635d_real @ A2 @ one_on6069100329679821595d_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_1039_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_1040_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_1041_comm__monoid__mult__class_Omult__1,axiom,
! [A2: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1042_comm__monoid__mult__class_Omult__1,axiom,
! [A2: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1043_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_1044_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_1045_real__mult__inverse__left__ex,axiom,
! [X5: dedekind_real] :
( ( X5 != zero_z580800474297136991d_real )
=> ~ ! [Y: dedekind_real] :
( ( times_2157731159493324635d_real @ Y @ X5 )
!= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left_ex
thf(fact_1046_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1047_lambda__one,axiom,
( ( ^ [X3: dedekind_preal] : X3 )
= ( times_3000655703912201937_preal @ one_on9143529541772854033_preal ) ) ).
% lambda_one
thf(fact_1048_lambda__one,axiom,
( ( ^ [X3: dedekind_real] : X3 )
= ( times_2157731159493324635d_real @ one_on6069100329679821595d_real ) ) ).
% lambda_one
thf(fact_1049_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_1050_zero__less__one__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one_class.zero_le_one
thf(fact_1051_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_1052_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_1053_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1054_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_1055_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_1056_not__one__le__zero,axiom,
~ ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_le_zero
thf(fact_1057_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1058_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1059_less__numeral__extra_I1_J,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% less_numeral_extra(1)
thf(fact_1060_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1061_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1062_zero__less__one,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one
thf(fact_1063_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1064_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1065_not__one__less__zero,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_less_zero
thf(fact_1066_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1067_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1068_less__1__mult,axiom,
! [M: dedekind_real,N: dedekind_real] :
( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ M )
=> ( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ N )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( times_2157731159493324635d_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1069_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1070_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1071_mult__left__le__one__le,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ X5 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ Y4 @ X5 ) @ X5 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1072_mult__left__le__one__le,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y4 @ X5 ) @ X5 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1073_mult__right__le__one__le,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ X5 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ X5 @ Y4 ) @ X5 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1074_mult__right__le__one__le,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X5 @ Y4 ) @ X5 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1075_mult__le__one,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ one_on6069100329679821595d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ one_on6069100329679821595d_real )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ B ) @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_le_one
thf(fact_1076_mult__le__one,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1077_mult__le__one,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1078_mult__left__le,axiom,
! [C: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ one_on6069100329679821595d_real )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_1079_mult__left__le,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_1080_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_1081_mult__le__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1082_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1083_mult__le__cancel__left2,axiom,
! [C: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ C )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ one_on6069100329679821595d_real ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1084_mult__le__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1085_mult__le__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1086_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1087_mult__le__cancel__right2,axiom,
! [A2: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( times_2157731159493324635d_real @ A2 @ C ) @ C )
= ( ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2716243287969276982d_real @ A2 @ one_on6069100329679821595d_real ) )
& ( ( ord_le2991122432403439658d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1088_mult__le__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1089_mult__less__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ ( times_2157731159493324635d_real @ C @ B ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1090_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1091_mult__less__cancel__left2,axiom,
! [C: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ C @ A2 ) @ C )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ A2 @ one_on6069100329679821595d_real ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1092_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_1093_mult__less__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ C @ ( times_2157731159493324635d_real @ B @ C ) )
= ( ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ B ) )
& ( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ B @ one_on6069100329679821595d_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1094_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1095_dbl__inc__simps_I2_J,axiom,
( ( neg_nu2849763295382964967d_real @ zero_z580800474297136991d_real )
= one_on6069100329679821595d_real ) ).
% dbl_inc_simps(2)
thf(fact_1096_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1097_convex__bound__lt,axiom,
! [X5: dedekind_real,A2: dedekind_real,Y4: dedekind_real,U: dedekind_real,V: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ A2 )
=> ( ( ord_le2991122432403439658d_real @ Y4 @ A2 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ U )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ V )
=> ( ( ( plus_p4060926892116697567d_real @ U @ V )
= one_on6069100329679821595d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ U @ X5 ) @ ( times_2157731159493324635d_real @ V @ Y4 ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1098_convex__bound__lt,axiom,
! [X5: int,A2: int,Y4: int,U: int,V: int] :
( ( ord_less_int @ X5 @ A2 )
=> ( ( ord_less_int @ Y4 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X5 ) @ ( times_times_int @ V @ Y4 ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1099_divide__le__eq__1__neg,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ ( divide9119111104558704680d_real @ B @ A2 ) @ one_on6069100329679821595d_real )
= ( ord_le2716243287969276982d_real @ A2 @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1100_divide__le__eq__1__pos,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 )
=> ( ( ord_le2716243287969276982d_real @ ( divide9119111104558704680d_real @ B @ A2 ) @ one_on6069100329679821595d_real )
= ( ord_le2716243287969276982d_real @ B @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1101_add__left__cancel,axiom,
! [A2: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A2 @ B )
= ( plus_p4060926892116697567d_real @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1102_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1103_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1104_add__right__cancel,axiom,
! [B: dedekind_real,A2: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A2 )
= ( plus_p4060926892116697567d_real @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1105_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1106_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1107_preal__add__less__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
= ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_less_cancel_right
thf(fact_1108_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
= ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_less_cancel_left
thf(fact_1109_preal__add__le__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
= ( ord_le5604041210740703414_preal @ R2 @ S3 ) ) ).
% preal_add_le_cancel_right
thf(fact_1110_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
= ( ord_le5604041210740703414_preal @ R2 @ S3 ) ) ).
% preal_add_le_cancel_left
thf(fact_1111_add__le__cancel__right,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A2 @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( ord_le2716243287969276982d_real @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1112_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1113_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1114_add__le__cancel__left,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A2 ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1115_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1116_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1117_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_1118_add__0,axiom,
! [A2: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A2 )
= A2 ) ).
% add_0
thf(fact_1119_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_1120_zero__eq__add__iff__both__eq__0,axiom,
! [X5: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X5 @ Y4 ) )
= ( ( X5 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1121_add__eq__0__iff__both__eq__0,axiom,
! [X5: nat,Y4: nat] :
( ( ( plus_plus_nat @ X5 @ Y4 )
= zero_zero_nat )
= ( ( X5 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1122_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1123_add__cancel__right__right,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ A2 @ B ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_right
thf(fact_1124_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1125_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1126_add__cancel__right__left,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ B @ A2 ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_left
thf(fact_1127_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1128_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1129_add__cancel__left__right,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A2 @ B )
= A2 )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_right
thf(fact_1130_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1131_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1132_add__cancel__left__left,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A2 )
= A2 )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_left
thf(fact_1133_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1134_double__zero__sym,axiom,
! [A2: dedekind_real] :
( ( zero_z580800474297136991d_real
= ( plus_p4060926892116697567d_real @ A2 @ A2 ) )
= ( A2 = zero_z580800474297136991d_real ) ) ).
% double_zero_sym
thf(fact_1135_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1136_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_1137_add_Oright__neutral,axiom,
! [A2: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A2 @ zero_z580800474297136991d_real )
= A2 ) ).
% add.right_neutral
thf(fact_1138_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_1139_add__less__cancel__left,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A2 ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_1140_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_1141_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_1142_add__less__cancel__right,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A2 @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( ord_le2991122432403439658d_real @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_1143_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_1144_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_1145_divide__eq__0__iff,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A2 @ B )
= zero_z580800474297136991d_real )
= ( ( A2 = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1146_divide__cancel__left,axiom,
! [C: dedekind_real,A2: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ C @ A2 )
= ( divide9119111104558704680d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A2 = B ) ) ) ).
% divide_cancel_left
thf(fact_1147_divide__cancel__right,axiom,
! [A2: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A2 @ C )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A2 = B ) ) ) ).
% divide_cancel_right
thf(fact_1148_division__ring__divide__zero,axiom,
! [A2: dedekind_real] :
( ( divide9119111104558704680d_real @ A2 @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% division_ring_divide_zero
thf(fact_1149_div__0,axiom,
! [A2: dedekind_real] :
( ( divide9119111104558704680d_real @ zero_z580800474297136991d_real @ A2 )
= zero_z580800474297136991d_real ) ).
% div_0
thf(fact_1150_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_1151_div__by__0,axiom,
! [A2: dedekind_real] :
( ( divide9119111104558704680d_real @ A2 @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% div_by_0
thf(fact_1152_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1153_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A2 @ A2 ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1154_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1155_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A2 @ A2 ) @ zero_z580800474297136991d_real )
= ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1156_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1157_le__add__same__cancel2,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ ( plus_p4060926892116697567d_real @ B @ A2 ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_1158_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1159_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1160_le__add__same__cancel1,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ ( plus_p4060926892116697567d_real @ A2 @ B ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_1161_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1162_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1163_add__le__same__cancel2,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A2 @ B ) @ B )
= ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real ) ) ).
% add_le_same_cancel2
thf(fact_1164_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1165_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1166_add__le__same__cancel1,axiom,
! [B: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ B @ A2 ) @ B )
= ( ord_le2716243287969276982d_real @ A2 @ zero_z580800474297136991d_real ) ) ).
% add_le_same_cancel1
thf(fact_1167_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1168_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1169_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A2 @ A2 ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1170_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1171_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A2 @ A2 ) @ zero_z580800474297136991d_real )
= ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1172_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1173_less__add__same__cancel2,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ ( plus_p4060926892116697567d_real @ B @ A2 ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_1174_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1175_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1176_less__add__same__cancel1,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ ( plus_p4060926892116697567d_real @ A2 @ B ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_1177_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1178_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1179_add__less__same__cancel2,axiom,
! [A2: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A2 @ B ) @ B )
= ( ord_le2991122432403439658d_real @ A2 @ zero_z580800474297136991d_real ) ) ).
% add_less_same_cancel2
thf(fact_1180_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1181_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1182_real__of__preal__add,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal] :
( ( dedeki829075424511776921_preal @ ( plus_p3173629198307831117_preal @ X5 @ Y4 ) )
= ( plus_p4060926892116697567d_real @ ( dedeki829075424511776921_preal @ X5 ) @ ( dedeki829075424511776921_preal @ Y4 ) ) ) ).
% real_of_preal_add
thf(fact_1183_preal__add__mult__distrib,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,W3: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X5 @ Y4 ) @ W3 )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X5 @ W3 ) @ ( times_3000655703912201937_preal @ Y4 @ W3 ) ) ) ).
% preal_add_mult_distrib
thf(fact_1184_preal__add__mult__distrib2,axiom,
! [W3: dedekind_preal,X5: dedekind_preal,Y4: dedekind_preal] :
( ( times_3000655703912201937_preal @ W3 @ ( plus_p3173629198307831117_preal @ X5 @ Y4 ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W3 @ X5 ) @ ( times_3000655703912201937_preal @ W3 @ Y4 ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1185_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X5: dedekind_preal,Y4: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X5 @ X1 ) @ ( times_3000655703912201937_preal @ Y4 @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X5 @ Y22 ) @ ( times_3000655703912201937_preal @ Y4 @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X5 @ X22 ) @ ( times_3000655703912201937_preal @ Y4 @ Y22 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X5 @ Y1 ) @ ( times_3000655703912201937_preal @ Y4 @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1186_preal__add__right__less__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_right_less_cancel
thf(fact_1187_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_left_less_cancel
thf(fact_1188_preal__self__less__add__left,axiom,
! [R2: dedekind_preal,S3: dedekind_preal] : ( ord_le5708704896291381698_preal @ R2 @ ( plus_p3173629198307831117_preal @ R2 @ S3 ) ) ).
% preal_self_less_add_left
thf(fact_1189_preal__add__less2__mono2,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) ) ) ).
% preal_add_less2_mono2
thf(fact_1190_preal__add__less2__mono1,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_1191_preal__eq__le__imp__le,axiom,
! [A2: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A2 @ B )
= ( plus_p3173629198307831117_preal @ C @ D3 ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A2 )
=> ( ord_le5604041210740703414_preal @ B @ D3 ) ) ) ).
% preal_eq_le_imp_le
thf(fact_1192_real__le__lemma,axiom,
! [U1: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,V1: dedekind_preal,X1: dedekind_preal,Y1: dedekind_preal,X22: dedekind_preal,Y22: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U1 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X1 @ V1 )
= ( plus_p3173629198307831117_preal @ U1 @ Y1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V2 )
= ( plus_p3173629198307831117_preal @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X1 @ Y22 ) @ ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ) ).
% real_le_lemma
thf(fact_1193_real__add__mult__distrib,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,W3: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ Z1 @ Z22 ) @ W3 )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ Z1 @ W3 ) @ ( times_2157731159493324635d_real @ Z22 @ W3 ) ) ) ).
% real_add_mult_distrib
thf(fact_1194_real__add__left__mono,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ Z5 @ X5 ) @ ( plus_p4060926892116697567d_real @ Z5 @ Y4 ) ) ) ).
% real_add_left_mono
thf(fact_1195_preal__add__right__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R2 @ T )
= ( plus_p3173629198307831117_preal @ S3 @ T ) )
=> ( R2 = S3 ) ) ).
% preal_add_right_cancel
thf(fact_1196_preal__add__left__cancel,axiom,
! [C: dedekind_preal,A2: dedekind_preal,B: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C @ A2 )
= ( plus_p3173629198307831117_preal @ C @ B ) )
=> ( A2 = B ) ) ).
% preal_add_left_cancel
thf(fact_1197_preal__trans__lemma,axiom,
! [X5: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y4: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X5 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y4 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X5 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y4 ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_1198_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X3: dedekind_preal,Y3: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y3 @ X3 ) ) ) ).
% preal_add_commute
thf(fact_1199_preal__add__assoc,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z5: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X5 @ Y4 ) @ Z5 )
= ( plus_p3173629198307831117_preal @ X5 @ ( plus_p3173629198307831117_preal @ Y4 @ Z5 ) ) ) ).
% preal_add_assoc
thf(fact_1200_real__of__preal__trichotomy,axiom,
! [X5: dedekind_real] :
? [M6: dedekind_preal] :
( ( X5
= ( dedeki829075424511776921_preal @ M6 ) )
| ( X5 = zero_z580800474297136991d_real )
| ( X5
= ( uminus7714077491378687647d_real @ ( dedeki829075424511776921_preal @ M6 ) ) ) ) ).
% real_of_preal_trichotomy
thf(fact_1201_real__less__sum__gt__zero,axiom,
! [W3: dedekind_real,S3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ W3 @ S3 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S3 @ ( uminus7714077491378687647d_real @ W3 ) ) ) ) ).
% real_less_sum_gt_zero
thf(fact_1202_real__sum__gt__zero__less,axiom,
! [S3: dedekind_real,W3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S3 @ ( uminus7714077491378687647d_real @ W3 ) ) )
=> ( ord_le2991122432403439658d_real @ W3 @ S3 ) ) ).
% real_sum_gt_zero_less
thf(fact_1203_add__inc,axiom,
! [X5: num,Y4: num] :
( ( plus_plus_num @ X5 @ ( inc @ Y4 ) )
= ( inc @ ( plus_plus_num @ X5 @ Y4 ) ) ) ).
% add_inc
thf(fact_1204_mult__inc,axiom,
! [X5: num,Y4: num] :
( ( times_times_num @ X5 @ ( inc @ Y4 ) )
= ( plus_plus_num @ ( times_times_num @ X5 @ Y4 ) @ X5 ) ) ).
% mult_inc
thf(fact_1205_num__induct,axiom,
! [P: num > $o,X5: num] :
( ( P @ one )
=> ( ! [X: num] :
( ( P @ X )
=> ( P @ ( inc @ X ) ) )
=> ( P @ X5 ) ) ) ).
% num_induct
thf(fact_1206_add__One,axiom,
! [X5: num] :
( ( plus_plus_num @ X5 @ one )
= ( inc @ X5 ) ) ).
% add_One
thf(fact_1207_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1208_le__num__One__iff,axiom,
! [X5: num] :
( ( ord_less_eq_num @ X5 @ one )
= ( X5 = one ) ) ).
% le_num_One_iff
thf(fact_1209_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1210_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1211_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1212_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1213_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1214_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1215_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1216_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1217_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1218_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1219_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1220_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1221_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1222_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1223_num_Oexhaust,axiom,
! [Y4: num] :
( ( Y4 != one )
=> ( ! [X23: num] :
( Y4
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y4
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1224_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1225_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1226_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1227_inc_Osimps_I2_J,axiom,
! [X5: num] :
( ( inc @ ( bit0 @ X5 ) )
= ( bit1 @ X5 ) ) ).
% inc.simps(2)
thf(fact_1228_inc_Osimps_I3_J,axiom,
! [X5: num] :
( ( inc @ ( bit1 @ X5 ) )
= ( bit0 @ ( inc @ X5 ) ) ) ).
% inc.simps(3)
thf(fact_1229_neg__zdiv__mult__2,axiom,
! [A2: int,B: 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 ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1230_pos__zdiv__mult__2,axiom,
! [A2: int,B: 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 ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ B @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1231_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_1232_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1233_zle__add1__eq__le,axiom,
! [W3: int,Z5: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z5 @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z5 ) ) ).
% zle_add1_eq_le
thf(fact_1234_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1235_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1236_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_1237_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_1238_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1239_add1__zle__eq,axiom,
! [W3: int,Z5: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z5 )
= ( ord_less_int @ W3 @ Z5 ) ) ).
% add1_zle_eq
thf(fact_1240_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1241_le__imp__0__less,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z5 ) ) ) ).
% le_imp_0_less
thf(fact_1242_zless__imp__add1__zle,axiom,
! [W3: int,Z5: int] :
( ( ord_less_int @ W3 @ Z5 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z5 ) ) ).
% zless_imp_add1_zle
thf(fact_1243_int__one__le__iff__zero__less,axiom,
! [Z5: int] :
( ( ord_less_eq_int @ one_one_int @ Z5 )
= ( ord_less_int @ zero_zero_int @ Z5 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1244_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1245_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D3 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1246_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I3: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I3: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1247_int__div__neg__eq,axiom,
! [A2: int,B: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A2 @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1248_int__div__pos__eq,axiom,
! [A2: int,B: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A2 @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1249_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1250_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1251_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1252_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1253_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1254_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1255_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1256_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1257_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1258_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1259_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1260_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1261_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1262_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1263_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1264_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1265_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1266_div__nonpos__pos__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1267_div__nonneg__neg__le0,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1268_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
% Conjectures (1)
thf(conj_0,conjecture,
( ord_le2991122432403439658d_real @ y
@ ( dedeki829075424511776921_preal
@ ( dedekind_psup
@ ( collec1132657498972982273_preal
@ ^ [W: dedekind_preal] : ( member_Dedekind_real @ ( dedeki829075424511776921_preal @ W ) @ p ) ) ) ) ) ).
%------------------------------------------------------------------------------