TPTP Problem File: SLH0521^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Quasi_Borel_Spaces/0000_StandardBorel/prob_00306_010606__15062766_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1350 ( 526 unt; 78 typ; 0 def)
% Number of atoms : 3663 (1311 equ; 0 cnn)
% Maximal formula atoms : 36 ( 2 avg)
% Number of connectives : 10438 ( 389 ~; 100 |; 176 &;8052 @)
% ( 0 <=>;1721 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 833 ( 833 >; 0 *; 0 +; 0 <<)
% Number of symbols : 75 ( 72 usr; 15 con; 0-3 aty)
% Number of variables : 3739 ( 208 ^;3375 !; 156 ?;3739 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:01:12.396
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (72)
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nonnegative____Real__Oennreal,type,
extend2057119558705770725nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Extended__Nonnegative__Real_Oenn2real,type,
extend1669699412028896998n2real: extend8495563244428889912nnreal > real ).
thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
extend7643940197134561352nnreal: real > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
comp_E7860224481218928525nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
comp_E4178961840025359489l_real: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
comp_E6146708109475903745nnreal: ( extend8495563244428889912nnreal > real ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > real ).
thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_E3822617923592311797l_real: ( extend8495563244428889912nnreal > real ) > ( real > extend8495563244428889912nnreal ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
comp_r6281409797179841921nnreal: ( real > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > real ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
comp_r6279034453215524981l_real: ( real > extend8495563244428889912nnreal ) > ( real > real ) > real > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
comp_r8246780722666069237nnreal: ( real > real ) > ( extend8495563244428889912nnreal > real ) > extend8495563244428889912nnreal > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Oid_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
id_Ext6301196394018042846nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Oid_001t__Extended____Nonnegative____Real__Oennreal,type,
id_Ext8331313139072774535nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
id_real: real > real ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Harmonic__Numbers_Oharm_001t__Real__Oreal,type,
harmonic_harm_real: nat > real ).
thf(sy_c_If_001t__Extended____Nonnegative____Real__Oennreal,type,
if_Ext9135588136721118450nnreal: $o > extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nonnegative____Real__Oennreal,type,
semiri6283507881447550617nnreal: nat > extend8495563244428889912nnreal ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nonnegative____Real__Oennreal,type,
order_7545170809120406815nnreal: ( extend8495563244428889912nnreal > $o ) > extend8495563244428889912nnreal ).
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__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oordering__top_001t__Extended____Nonnegative____Real__Oennreal,type,
orderi7292949391079830343nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ) > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nonnegative____Real__Oennreal,type,
top_to1496364449551166952nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
top_to7994903218803871134nnreal: set_Ex3793607809372303086nnreal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_preal__to__real____,type,
preal_to_real: extend8495563244428889912nnreal > real ).
thf(sy_v_r____,type,
r: extend8495563244428889912nnreal ).
thf(sy_v_real__to__preal____,type,
real_to_preal: real > extend8495563244428889912nnreal ).
% Relevant facts (1264)
thf(fact_0__092_060open_062ennreal_A_Ienn2real_Ar_J_A_061_A_Iif_Ar_A_061_A_092_060top_062_Athen_A0_Aelse_Ar_J_092_060close_062,axiom,
( ( ( r = top_to1496364449551166952nnreal )
=> ( ( extend7643940197134561352nnreal @ ( extend1669699412028896998n2real @ r ) )
= zero_z7100319975126383169nnreal ) )
& ( ( r != top_to1496364449551166952nnreal )
=> ( ( extend7643940197134561352nnreal @ ( extend1669699412028896998n2real @ r ) )
= r ) ) ) ).
% \<open>ennreal (enn2real r) = (if r = \<top> then 0 else r)\<close>
thf(fact_1_fun_Omap__id,axiom,
! [T: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_E7860224481218928525nnreal @ id_Ext8331313139072774535nnreal @ T )
= T ) ).
% fun.map_id
thf(fact_2_comp__id,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_E7860224481218928525nnreal @ F @ id_Ext8331313139072774535nnreal )
= F ) ).
% comp_id
thf(fact_3_id__comp,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_E7860224481218928525nnreal @ id_Ext8331313139072774535nnreal @ G )
= G ) ).
% id_comp
thf(fact_4_id__apply,axiom,
( id_real
= ( ^ [X: real] : X ) ) ).
% id_apply
thf(fact_5_id__apply,axiom,
( id_Ext8331313139072774535nnreal
= ( ^ [X: extend8495563244428889912nnreal] : X ) ) ).
% id_apply
thf(fact_6_comp__apply,axiom,
( comp_E7860224481218928525nnreal
= ( ^ [F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_7_comp__apply,axiom,
( comp_E3822617923592311797l_real
= ( ^ [F2: extend8495563244428889912nnreal > real,G2: real > extend8495563244428889912nnreal,X: real] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_8_comp__apply,axiom,
( comp_r6281409797179841921nnreal
= ( ^ [F2: real > extend8495563244428889912nnreal,G2: extend8495563244428889912nnreal > real,X: extend8495563244428889912nnreal] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_9_fun_Omap__id0,axiom,
( ( comp_E7860224481218928525nnreal @ id_Ext8331313139072774535nnreal )
= id_Ext6301196394018042846nnreal ) ).
% fun.map_id0
thf(fact_10_comp__eq__id__dest,axiom,
! [A: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ A @ B )
= ( comp_E7860224481218928525nnreal @ id_Ext8331313139072774535nnreal @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_11_comp__eq__id__dest,axiom,
! [A: extend8495563244428889912nnreal > real,B: real > extend8495563244428889912nnreal,C: real > real,V: real] :
( ( ( comp_E3822617923592311797l_real @ A @ B )
= ( comp_real_real_real @ id_real @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_12_comp__eq__id__dest,axiom,
! [A: real > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ A @ B )
= ( comp_E7860224481218928525nnreal @ id_Ext8331313139072774535nnreal @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_13_pointfree__idE,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ F @ G )
= id_Ext8331313139072774535nnreal )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_14_pointfree__idE,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,X2: real] :
( ( ( comp_E3822617923592311797l_real @ F @ G )
= id_real )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_15_pointfree__idE,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,X2: extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= id_Ext8331313139072774535nnreal )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_16_isomorphism__expand,axiom,
! [F: real > real,G: real > real] :
( ( ( ( comp_real_real_real @ F @ G )
= id_real )
& ( ( comp_real_real_real @ G @ F )
= id_real ) )
= ( ! [X: real] :
( ( F @ ( G @ X ) )
= X )
& ! [X: real] :
( ( G @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_17_isomorphism__expand,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( ( comp_E7860224481218928525nnreal @ F @ G )
= id_Ext8331313139072774535nnreal )
& ( ( comp_E7860224481218928525nnreal @ G @ F )
= id_Ext8331313139072774535nnreal ) )
= ( ! [X: extend8495563244428889912nnreal] :
( ( F @ ( G @ X ) )
= X )
& ! [X: extend8495563244428889912nnreal] :
( ( G @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_18_isomorphism__expand,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal] :
( ( ( ( comp_E3822617923592311797l_real @ F @ G )
= id_real )
& ( ( comp_r6281409797179841921nnreal @ G @ F )
= id_Ext8331313139072774535nnreal ) )
= ( ! [X: real] :
( ( F @ ( G @ X ) )
= X )
& ! [X: extend8495563244428889912nnreal] :
( ( G @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_19_isomorphism__expand,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real] :
( ( ( ( comp_r6281409797179841921nnreal @ F @ G )
= id_Ext8331313139072774535nnreal )
& ( ( comp_E3822617923592311797l_real @ G @ F )
= id_real ) )
= ( ! [X: extend8495563244428889912nnreal] :
( ( F @ ( G @ X ) )
= X )
& ! [X: real] :
( ( G @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_20_left__right__inverse__eq,axiom,
! [F: real > real,G: real > real,H: real > real] :
( ( ( comp_real_real_real @ F @ G )
= id_real )
=> ( ( ( comp_real_real_real @ G @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_21_left__right__inverse__eq,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= id_Ext8331313139072774535nnreal )
=> ( ( ( comp_E3822617923592311797l_real @ G @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_22_left__right__inverse__eq,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ F @ G )
= id_Ext8331313139072774535nnreal )
=> ( ( ( comp_E7860224481218928525nnreal @ G @ H )
= id_Ext8331313139072774535nnreal )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_23_left__right__inverse__eq,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real] :
( ( ( comp_E3822617923592311797l_real @ F @ G )
= id_real )
=> ( ( ( comp_r6281409797179841921nnreal @ G @ H )
= id_Ext8331313139072774535nnreal )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_24_id__def,axiom,
( id_Ext8331313139072774535nnreal
= ( ^ [X: extend8495563244428889912nnreal] : X ) ) ).
% id_def
thf(fact_25_id__def,axiom,
( id_real
= ( ^ [X: real] : X ) ) ).
% id_def
thf(fact_26_eq__id__iff,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ! [X: extend8495563244428889912nnreal] :
( ( F @ X )
= X ) )
= ( F = id_Ext8331313139072774535nnreal ) ) ).
% eq_id_iff
thf(fact_27_eq__id__iff,axiom,
! [F: real > real] :
( ( ! [X: real] :
( ( F @ X )
= X ) )
= ( F = id_real ) ) ).
% eq_id_iff
thf(fact_28_ennreal__neg,axiom,
! [X2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_neg
thf(fact_29_rewriteR__comp__comp2,axiom,
! [G: real > extend8495563244428889912nnreal,H: real > real,R1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R2: real > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,L: extend8495563244428889912nnreal > real] :
( ( ( comp_r6279034453215524981l_real @ G @ H )
= ( comp_E4178961840025359489l_real @ R1 @ R2 ) )
=> ( ( ( comp_E6146708109475903745nnreal @ F @ R1 )
= L )
=> ( ( comp_real_real_real @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_E3822617923592311797l_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_30_rewriteR__comp__comp2,axiom,
! [G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R1: real > real,R2: extend8495563244428889912nnreal > real,F: real > extend8495563244428889912nnreal,L: real > extend8495563244428889912nnreal] :
( ( ( comp_E6146708109475903745nnreal @ G @ H )
= ( comp_r8246780722666069237nnreal @ R1 @ R2 ) )
=> ( ( ( comp_r6279034453215524981l_real @ F @ R1 )
= L )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_31_rewriteR__comp__comp2,axiom,
! [G: real > real,H: extend8495563244428889912nnreal > real,R1: extend8495563244428889912nnreal > real,R2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,L: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r8246780722666069237nnreal @ G @ H )
= ( comp_E6146708109475903745nnreal @ R1 @ R2 ) )
=> ( ( ( comp_r6281409797179841921nnreal @ F @ R1 )
= L )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_r6279034453215524981l_real @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_32_rewriteR__comp__comp2,axiom,
! [G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R1: extend8495563244428889912nnreal > real,R2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,L: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_E6146708109475903745nnreal @ G @ H )
= ( comp_E6146708109475903745nnreal @ R1 @ R2 ) )
=> ( ( ( comp_r6281409797179841921nnreal @ F @ R1 )
= L )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_33_rewriteR__comp__comp2,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal,R1: real > extend8495563244428889912nnreal,R2: real > real,F: extend8495563244428889912nnreal > real,L: real > real] :
( ( ( comp_E4178961840025359489l_real @ G @ H )
= ( comp_r6279034453215524981l_real @ R1 @ R2 ) )
=> ( ( ( comp_E3822617923592311797l_real @ F @ R1 )
= L )
=> ( ( comp_E3822617923592311797l_real @ ( comp_E6146708109475903745nnreal @ F @ G ) @ H )
= ( comp_real_real_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_34_rewriteR__comp__comp2,axiom,
! [G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R1: real > extend8495563244428889912nnreal,R2: extend8495563244428889912nnreal > real,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: real > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ G @ H )
= ( comp_r6281409797179841921nnreal @ R1 @ R2 ) )
=> ( ( ( comp_E4178961840025359489l_real @ F @ R1 )
= L )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_35_rewriteR__comp__comp2,axiom,
! [G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R1: real > extend8495563244428889912nnreal,R2: extend8495563244428889912nnreal > real,F: extend8495563244428889912nnreal > real,L: real > real] :
( ( ( comp_r6281409797179841921nnreal @ G @ H )
= ( comp_r6281409797179841921nnreal @ R1 @ R2 ) )
=> ( ( ( comp_E3822617923592311797l_real @ F @ R1 )
= L )
=> ( ( comp_r8246780722666069237nnreal @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_r8246780722666069237nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_36_rewriteR__comp__comp2,axiom,
! [G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,L: extend8495563244428889912nnreal > real] :
( ( ( comp_r6281409797179841921nnreal @ G @ H )
= ( comp_E7860224481218928525nnreal @ R1 @ R2 ) )
=> ( ( ( comp_E6146708109475903745nnreal @ F @ R1 )
= L )
=> ( ( comp_r8246780722666069237nnreal @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_E6146708109475903745nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_37_rewriteR__comp__comp2,axiom,
! [G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ G @ H )
= ( comp_E7860224481218928525nnreal @ R1 @ R2 ) )
=> ( ( ( comp_E7860224481218928525nnreal @ F @ R1 )
= L )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_38_rewriteR__comp__comp2,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R1: real > extend8495563244428889912nnreal,R2: extend8495563244428889912nnreal > real,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: real > extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ G @ H )
= ( comp_r6281409797179841921nnreal @ R1 @ R2 ) )
=> ( ( ( comp_E4178961840025359489l_real @ F @ R1 )
= L )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_39_rewriteL__comp__comp2,axiom,
! [F: real > real,G: extend8495563244428889912nnreal > real,L1: extend8495563244428889912nnreal > real,L2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal,R: real > extend8495563244428889912nnreal] :
( ( ( comp_r8246780722666069237nnreal @ F @ G )
= ( comp_E6146708109475903745nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E4178961840025359489l_real @ L2 @ H )
= R )
=> ( ( comp_real_real_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) )
= ( comp_E3822617923592311797l_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_40_rewriteL__comp__comp2,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,L1: real > extend8495563244428889912nnreal,L2: real > real,H: extend8495563244428889912nnreal > real,R: extend8495563244428889912nnreal > real] :
( ( ( comp_E4178961840025359489l_real @ F @ G )
= ( comp_r6279034453215524981l_real @ L1 @ L2 ) )
=> ( ( ( comp_r8246780722666069237nnreal @ L2 @ H )
= R )
=> ( ( comp_E7860224481218928525nnreal @ F @ ( comp_r6281409797179841921nnreal @ G @ H ) )
= ( comp_r6281409797179841921nnreal @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_41_rewriteL__comp__comp2,axiom,
! [F: real > extend8495563244428889912nnreal,G: real > real,L1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r6279034453215524981l_real @ F @ G )
= ( comp_E4178961840025359489l_real @ L1 @ L2 ) )
=> ( ( ( comp_r6281409797179841921nnreal @ L2 @ H )
= R )
=> ( ( comp_r6281409797179841921nnreal @ F @ ( comp_r8246780722666069237nnreal @ G @ H ) )
= ( comp_E7860224481218928525nnreal @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_42_rewriteL__comp__comp2,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,L1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_E4178961840025359489l_real @ F @ G )
= ( comp_E4178961840025359489l_real @ L1 @ L2 ) )
=> ( ( ( comp_r6281409797179841921nnreal @ L2 @ H )
= R )
=> ( ( comp_E7860224481218928525nnreal @ F @ ( comp_r6281409797179841921nnreal @ G @ H ) )
= ( comp_E7860224481218928525nnreal @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_43_rewriteL__comp__comp2,axiom,
! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L1: real > real,L2: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal,R: real > real] :
( ( ( comp_E6146708109475903745nnreal @ F @ G )
= ( comp_r8246780722666069237nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E3822617923592311797l_real @ L2 @ H )
= R )
=> ( ( comp_E3822617923592311797l_real @ F @ ( comp_E4178961840025359489l_real @ G @ H ) )
= ( comp_real_real_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_44_rewriteL__comp__comp2,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,L1: real > extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R: extend8495563244428889912nnreal > real] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= ( comp_r6281409797179841921nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E6146708109475903745nnreal @ L2 @ H )
= R )
=> ( ( comp_r6281409797179841921nnreal @ F @ ( comp_E6146708109475903745nnreal @ G @ H ) )
= ( comp_r6281409797179841921nnreal @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_45_rewriteL__comp__comp2,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,L1: real > extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal,R: real > real] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= ( comp_r6281409797179841921nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E3822617923592311797l_real @ L2 @ H )
= R )
=> ( ( comp_r6279034453215524981l_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) )
= ( comp_r6279034453215524981l_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_46_rewriteL__comp__comp2,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,L1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal,R: real > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= ( comp_E7860224481218928525nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E4178961840025359489l_real @ L2 @ H )
= R )
=> ( ( comp_r6279034453215524981l_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) )
= ( comp_E4178961840025359489l_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_47_rewriteL__comp__comp2,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,L1: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= ( comp_E7860224481218928525nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E7860224481218928525nnreal @ L2 @ H )
= R )
=> ( ( comp_r6281409797179841921nnreal @ F @ ( comp_E6146708109475903745nnreal @ G @ H ) )
= ( comp_E7860224481218928525nnreal @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_48_rewriteL__comp__comp2,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L1: real > extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R: extend8495563244428889912nnreal > real] :
( ( ( comp_E7860224481218928525nnreal @ F @ G )
= ( comp_r6281409797179841921nnreal @ L1 @ L2 ) )
=> ( ( ( comp_E6146708109475903745nnreal @ L2 @ H )
= R )
=> ( ( comp_E7860224481218928525nnreal @ F @ ( comp_E7860224481218928525nnreal @ G @ H ) )
= ( comp_r6281409797179841921nnreal @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_49_rewriteR__comp__comp,axiom,
! [G: real > extend8495563244428889912nnreal,H: real > real,R: real > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real] :
( ( ( comp_r6279034453215524981l_real @ G @ H )
= R )
=> ( ( comp_real_real_real @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_E3822617923592311797l_real @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_50_rewriteR__comp__comp,axiom,
! [G: real > real,H: extend8495563244428889912nnreal > real,R: extend8495563244428889912nnreal > real,F: real > extend8495563244428889912nnreal] :
( ( ( comp_r8246780722666069237nnreal @ G @ H )
= R )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_r6279034453215524981l_real @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_51_rewriteR__comp__comp,axiom,
! [G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R: extend8495563244428889912nnreal > real,F: real > extend8495563244428889912nnreal] :
( ( ( comp_E6146708109475903745nnreal @ G @ H )
= R )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_52_rewriteR__comp__comp,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal,R: real > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real] :
( ( ( comp_E4178961840025359489l_real @ G @ H )
= R )
=> ( ( comp_E3822617923592311797l_real @ ( comp_E6146708109475903745nnreal @ F @ G ) @ H )
= ( comp_E3822617923592311797l_real @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_53_rewriteR__comp__comp,axiom,
! [G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real] :
( ( ( comp_r6281409797179841921nnreal @ G @ H )
= R )
=> ( ( comp_r8246780722666069237nnreal @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_E6146708109475903745nnreal @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_54_rewriteR__comp__comp,axiom,
! [G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ G @ H )
= R )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_55_rewriteR__comp__comp,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ G @ H )
= R )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_56_rewriteR__comp__comp,axiom,
! [G: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal,R: real > real,F: real > extend8495563244428889912nnreal] :
( ( ( comp_E3822617923592311797l_real @ G @ H )
= R )
=> ( ( comp_E4178961840025359489l_real @ ( comp_r6281409797179841921nnreal @ F @ G ) @ H )
= ( comp_r6279034453215524981l_real @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_57_rewriteR__comp__comp,axiom,
! [G: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal,R: real > real,F: real > real] :
( ( ( comp_E3822617923592311797l_real @ G @ H )
= R )
=> ( ( comp_E3822617923592311797l_real @ ( comp_r8246780722666069237nnreal @ F @ G ) @ H )
= ( comp_real_real_real @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_58_rewriteL__comp__comp,axiom,
! [F: real > real,G: extend8495563244428889912nnreal > real,L: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal] :
( ( ( comp_r8246780722666069237nnreal @ F @ G )
= L )
=> ( ( comp_real_real_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) )
= ( comp_E3822617923592311797l_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_59_rewriteL__comp__comp,axiom,
! [F: real > extend8495563244428889912nnreal,G: real > real,L: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real] :
( ( ( comp_r6279034453215524981l_real @ F @ G )
= L )
=> ( ( comp_r6281409797179841921nnreal @ F @ ( comp_r8246780722666069237nnreal @ G @ H ) )
= ( comp_r6281409797179841921nnreal @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_60_rewriteL__comp__comp,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,L: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real] :
( ( ( comp_E4178961840025359489l_real @ F @ G )
= L )
=> ( ( comp_E7860224481218928525nnreal @ F @ ( comp_r6281409797179841921nnreal @ G @ H ) )
= ( comp_r6281409797179841921nnreal @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_61_rewriteL__comp__comp,axiom,
! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal] :
( ( ( comp_E6146708109475903745nnreal @ F @ G )
= L )
=> ( ( comp_E3822617923592311797l_real @ F @ ( comp_E4178961840025359489l_real @ G @ H ) )
= ( comp_E3822617923592311797l_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_62_rewriteL__comp__comp,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,L: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= L )
=> ( ( comp_r6279034453215524981l_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) )
= ( comp_E4178961840025359489l_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_63_rewriteL__comp__comp,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,L: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ F @ G )
= L )
=> ( ( comp_r6281409797179841921nnreal @ F @ ( comp_E6146708109475903745nnreal @ G @ H ) )
= ( comp_E7860224481218928525nnreal @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_64_rewriteL__comp__comp,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ F @ G )
= L )
=> ( ( comp_E7860224481218928525nnreal @ F @ ( comp_E7860224481218928525nnreal @ G @ H ) )
= ( comp_E7860224481218928525nnreal @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_65_rewriteL__comp__comp,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,L: real > real,H: extend8495563244428889912nnreal > real] :
( ( ( comp_E3822617923592311797l_real @ F @ G )
= L )
=> ( ( comp_E6146708109475903745nnreal @ F @ ( comp_r6281409797179841921nnreal @ G @ H ) )
= ( comp_r8246780722666069237nnreal @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_66_rewriteL__comp__comp,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,L: real > real,H: real > real] :
( ( ( comp_E3822617923592311797l_real @ F @ G )
= L )
=> ( ( comp_E3822617923592311797l_real @ F @ ( comp_r6279034453215524981l_real @ G @ H ) )
= ( comp_real_real_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_67_comp__eq__dest__lhs,axiom,
! [A: real > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_68_comp__eq__dest__lhs,axiom,
! [A: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_69_comp__eq__dest__lhs,axiom,
! [A: extend8495563244428889912nnreal > real,B: real > extend8495563244428889912nnreal,C: real > real,V: real] :
( ( ( comp_E3822617923592311797l_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_70_comp__eq__elim,axiom,
! [A: real > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real,C: real > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > real] :
( ( ( comp_r6281409797179841921nnreal @ A @ B )
= ( comp_r6281409797179841921nnreal @ C @ D ) )
=> ! [V2: extend8495563244428889912nnreal] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_71_comp__eq__elim,axiom,
! [A: real > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ A @ B )
= ( comp_E7860224481218928525nnreal @ C @ D ) )
=> ! [V2: extend8495563244428889912nnreal] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_72_comp__eq__elim,axiom,
! [A: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: real > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > real] :
( ( ( comp_E7860224481218928525nnreal @ A @ B )
= ( comp_r6281409797179841921nnreal @ C @ D ) )
=> ! [V2: extend8495563244428889912nnreal] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_73_comp__eq__elim,axiom,
! [A: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ A @ B )
= ( comp_E7860224481218928525nnreal @ C @ D ) )
=> ! [V2: extend8495563244428889912nnreal] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_74_comp__eq__elim,axiom,
! [A: extend8495563244428889912nnreal > real,B: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal > real,D: real > extend8495563244428889912nnreal] :
( ( ( comp_E3822617923592311797l_real @ A @ B )
= ( comp_E3822617923592311797l_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_75_comp__eq__dest,axiom,
! [A: real > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real,C: real > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > real,V: extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ A @ B )
= ( comp_r6281409797179841921nnreal @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_76_comp__eq__dest,axiom,
! [A: real > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
( ( ( comp_r6281409797179841921nnreal @ A @ B )
= ( comp_E7860224481218928525nnreal @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_77_comp__eq__dest,axiom,
! [A: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: real > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > real,V: extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ A @ B )
= ( comp_r6281409797179841921nnreal @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_78_comp__eq__dest,axiom,
! [A: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal > extend8495563244428889912nnreal,D: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
( ( ( comp_E7860224481218928525nnreal @ A @ B )
= ( comp_E7860224481218928525nnreal @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_79_comp__eq__dest,axiom,
! [A: extend8495563244428889912nnreal > real,B: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal > real,D: real > extend8495563244428889912nnreal,V: real] :
( ( ( comp_E3822617923592311797l_real @ A @ B )
= ( comp_E3822617923592311797l_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_80_comp__assoc,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal] :
( ( comp_E4178961840025359489l_real @ ( comp_r6281409797179841921nnreal @ F @ G ) @ H )
= ( comp_r6279034453215524981l_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) ) ) ).
% comp_assoc
thf(fact_81_comp__assoc,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real] :
( ( comp_r8246780722666069237nnreal @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_E6146708109475903745nnreal @ F @ ( comp_r6281409797179841921nnreal @ G @ H ) ) ) ).
% comp_assoc
thf(fact_82_comp__assoc,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,H: real > real] :
( ( comp_real_real_real @ ( comp_E3822617923592311797l_real @ F @ G ) @ H )
= ( comp_E3822617923592311797l_real @ F @ ( comp_r6279034453215524981l_real @ G @ H ) ) ) ).
% comp_assoc
thf(fact_83_comp__assoc,axiom,
! [F: real > extend8495563244428889912nnreal,G: real > real,H: extend8495563244428889912nnreal > real] :
( ( comp_r6281409797179841921nnreal @ ( comp_r6279034453215524981l_real @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ F @ ( comp_r8246780722666069237nnreal @ G @ H ) ) ) ).
% comp_assoc
thf(fact_84_comp__assoc,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real] :
( ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ F @ ( comp_r6281409797179841921nnreal @ G @ H ) ) ) ).
% comp_assoc
thf(fact_85_comp__assoc,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ G ) @ H )
= ( comp_r6281409797179841921nnreal @ F @ ( comp_E6146708109475903745nnreal @ G @ H ) ) ) ).
% comp_assoc
thf(fact_86_comp__assoc,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ G ) @ H )
= ( comp_E7860224481218928525nnreal @ F @ ( comp_E7860224481218928525nnreal @ G @ H ) ) ) ).
% comp_assoc
thf(fact_87_comp__assoc,axiom,
! [F: real > real,G: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal] :
( ( comp_E3822617923592311797l_real @ ( comp_r8246780722666069237nnreal @ F @ G ) @ H )
= ( comp_real_real_real @ F @ ( comp_E3822617923592311797l_real @ G @ H ) ) ) ).
% comp_assoc
thf(fact_88_comp__assoc,axiom,
! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal] :
( ( comp_E3822617923592311797l_real @ ( comp_E6146708109475903745nnreal @ F @ G ) @ H )
= ( comp_E3822617923592311797l_real @ F @ ( comp_E4178961840025359489l_real @ G @ H ) ) ) ).
% comp_assoc
thf(fact_89_comp__def,axiom,
( comp_r6281409797179841921nnreal
= ( ^ [F2: real > extend8495563244428889912nnreal,G2: extend8495563244428889912nnreal > real,X: extend8495563244428889912nnreal] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_90_comp__def,axiom,
( comp_E7860224481218928525nnreal
= ( ^ [F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_91_comp__def,axiom,
( comp_E3822617923592311797l_real
= ( ^ [F2: extend8495563244428889912nnreal > real,G2: real > extend8495563244428889912nnreal,X: real] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_92_fun_Omap__comp,axiom,
! [G: extend8495563244428889912nnreal > real,F: real > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal > real] :
( ( comp_E6146708109475903745nnreal @ G @ ( comp_r6281409797179841921nnreal @ F @ V ) )
= ( comp_r8246780722666069237nnreal @ ( comp_E3822617923592311797l_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_93_fun_Omap__comp,axiom,
! [G: real > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,V: real > extend8495563244428889912nnreal] :
( ( comp_r6279034453215524981l_real @ G @ ( comp_E3822617923592311797l_real @ F @ V ) )
= ( comp_E4178961840025359489l_real @ ( comp_r6281409797179841921nnreal @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_94_fun_Omap__comp,axiom,
! [G: real > real,F: extend8495563244428889912nnreal > real,V: real > extend8495563244428889912nnreal] :
( ( comp_real_real_real @ G @ ( comp_E3822617923592311797l_real @ F @ V ) )
= ( comp_E3822617923592311797l_real @ ( comp_r8246780722666069237nnreal @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_95_fun_Omap__comp,axiom,
! [G: real > extend8495563244428889912nnreal,F: real > real,V: extend8495563244428889912nnreal > real] :
( ( comp_r6281409797179841921nnreal @ G @ ( comp_r8246780722666069237nnreal @ F @ V ) )
= ( comp_r6281409797179841921nnreal @ ( comp_r6279034453215524981l_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_96_fun_Omap__comp,axiom,
! [G: real > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,V: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_r6281409797179841921nnreal @ G @ ( comp_E6146708109475903745nnreal @ F @ V ) )
= ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_97_fun_Omap__comp,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal > real] :
( ( comp_E7860224481218928525nnreal @ G @ ( comp_r6281409797179841921nnreal @ F @ V ) )
= ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_98_fun_Omap__comp,axiom,
! [G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comp_E7860224481218928525nnreal @ G @ ( comp_E7860224481218928525nnreal @ F @ V ) )
= ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_99_fun_Omap__comp,axiom,
! [G: extend8495563244428889912nnreal > real,F: real > extend8495563244428889912nnreal,V: real > real] :
( ( comp_E3822617923592311797l_real @ G @ ( comp_r6279034453215524981l_real @ F @ V ) )
= ( comp_real_real_real @ ( comp_E3822617923592311797l_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_100_fun_Omap__comp,axiom,
! [G: extend8495563244428889912nnreal > real,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: real > extend8495563244428889912nnreal] :
( ( comp_E3822617923592311797l_real @ G @ ( comp_E4178961840025359489l_real @ F @ V ) )
= ( comp_E3822617923592311797l_real @ ( comp_E6146708109475903745nnreal @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_101_ennreal__enn2real__if,axiom,
! [R: extend8495563244428889912nnreal] :
( ( ( R = top_to1496364449551166952nnreal )
=> ( ( extend7643940197134561352nnreal @ ( extend1669699412028896998n2real @ R ) )
= zero_z7100319975126383169nnreal ) )
& ( ( R != top_to1496364449551166952nnreal )
=> ( ( extend7643940197134561352nnreal @ ( extend1669699412028896998n2real @ R ) )
= R ) ) ) ).
% ennreal_enn2real_if
thf(fact_102_ennreal__zero__neq__top,axiom,
zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).
% ennreal_zero_neq_top
thf(fact_103_top__neq__ennreal,axiom,
! [R: real] :
( top_to1496364449551166952nnreal
!= ( extend7643940197134561352nnreal @ R ) ) ).
% top_neq_ennreal
thf(fact_104_enn2real__top,axiom,
( ( extend1669699412028896998n2real @ top_to1496364449551166952nnreal )
= zero_zero_real ) ).
% enn2real_top
thf(fact_105_enn2real__0,axiom,
( ( extend1669699412028896998n2real @ zero_z7100319975126383169nnreal )
= zero_zero_real ) ).
% enn2real_0
thf(fact_106_enn2real__eq__0__iff,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ( extend1669699412028896998n2real @ X2 )
= zero_zero_real )
= ( ( X2 = zero_z7100319975126383169nnreal )
| ( X2 = top_to1496364449551166952nnreal ) ) ) ).
% enn2real_eq_0_iff
thf(fact_107_ennreal__0,axiom,
( ( extend7643940197134561352nnreal @ zero_zero_real )
= zero_z7100319975126383169nnreal ) ).
% ennreal_0
thf(fact_108_enn2real__ennreal,axiom,
! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( extend1669699412028896998n2real @ ( extend7643940197134561352nnreal @ R ) )
= R ) ) ).
% enn2real_ennreal
thf(fact_109_ennreal__cong,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ( extend7643940197134561352nnreal @ X2 )
= ( extend7643940197134561352nnreal @ Y ) ) ) ).
% ennreal_cong
thf(fact_110_type__copy__map__cong0,axiom,
! [M: real > extend8495563244428889912nnreal,G: real > real,X2: real,N: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_real_real_real @ ( comp_E3822617923592311797l_real @ F @ M ) @ G @ X2 )
= ( comp_E3822617923592311797l_real @ ( comp_E6146708109475903745nnreal @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_111_type__copy__map__cong0,axiom,
! [M: real > real,G: extend8495563244428889912nnreal > real,X2: extend8495563244428889912nnreal,N: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_r6279034453215524981l_real @ F @ M ) @ G @ X2 )
= ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_112_type__copy__map__cong0,axiom,
! [M: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,X2: extend8495563244428889912nnreal,N: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ F @ M ) @ G @ X2 )
= ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_113_type__copy__map__cong0,axiom,
! [M: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,N: real > real,H: extend8495563244428889912nnreal > real,F: real > extend8495563244428889912nnreal] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ M ) @ G @ X2 )
= ( comp_r6281409797179841921nnreal @ ( comp_r6279034453215524981l_real @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_114_type__copy__map__cong0,axiom,
! [M: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,N: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ M ) @ G @ X2 )
= ( comp_E7860224481218928525nnreal @ ( comp_r6281409797179841921nnreal @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_115_type__copy__map__cong0,axiom,
! [M: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,N: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ M ) @ G @ X2 )
= ( comp_r6281409797179841921nnreal @ ( comp_E4178961840025359489l_real @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_116_type__copy__map__cong0,axiom,
! [M: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,N: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ M ) @ G @ X2 )
= ( comp_E7860224481218928525nnreal @ ( comp_E7860224481218928525nnreal @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_117_type__copy__map__cong0,axiom,
! [M: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,X2: real,N: real > extend8495563244428889912nnreal,H: real > real,F: extend8495563244428889912nnreal > real] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_E3822617923592311797l_real @ ( comp_E6146708109475903745nnreal @ F @ M ) @ G @ X2 )
= ( comp_real_real_real @ ( comp_E3822617923592311797l_real @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_118_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_119_dual__order_Orefl,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).
% dual_order.refl
thf(fact_120_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_121_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_122_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_123_order__refl,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ X2 ) ).
% order_refl
thf(fact_124_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_125_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_126_le__zero__eq,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N2 @ zero_z7100319975126383169nnreal )
= ( N2 = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_127_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_128_ennreal__inj,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( extend7643940197134561352nnreal @ A )
= ( extend7643940197134561352nnreal @ B ) )
= ( A = B ) ) ) ) ).
% ennreal_inj
thf(fact_129_ennreal__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal )
= ( X2 = zero_zero_real ) ) ) ).
% ennreal_eq_zero_iff
thf(fact_130_order__antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_131_order__antisym__conv,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_132_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_133_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_134_linorder__le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_135_linorder__le__cases,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_136_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_137_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_138_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_139_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_140_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_141_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_143_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_144_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_145_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > int,C: int] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_146_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_147_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_154_ord__eq__le__subst,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_155_ord__eq__le__subst,axiom,
! [A: int,F: extend8495563244428889912nnreal > int,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_156_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_157_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_158_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_160_linorder__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_161_linorder__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
| ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_162_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_163_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_164_order__eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_165_order__eq__refl,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X2 = Y )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_166_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_167_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_168_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_169_order__subst2,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_171_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_172_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_173_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_174_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_175_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > int,C: int] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_176_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_177_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_178_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst1,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_180_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_182_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_184_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_185_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: int > extend8495563244428889912nnreal,B: int,C: int] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_186_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_187_order__subst1,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_188_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_189_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y3 = Z ) )
= ( ^ [A3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A3 @ B2 )
& ( ord_le3935885782089961368nnreal @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_190_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_191_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_192_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_193_antisym,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_194_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_195_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_196_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_197_dual__order_Otrans,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_198_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_199_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_200_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_201_dual__order_Oantisym,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_202_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_203_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_204_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_205_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y3 = Z ) )
= ( ^ [A3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A3 )
& ( ord_le3935885782089961368nnreal @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_206_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_207_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_208_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_209_linorder__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_210_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_211_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_212_order__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_213_order__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_214_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_215_order__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_216_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_217_order_Otrans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% order.trans
thf(fact_218_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_219_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_220_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_221_order__antisym,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_222_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_223_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_224_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_225_ord__le__eq__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( B = C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_226_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_227_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_228_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_229_ord__eq__le__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A = B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_230_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_231_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_232_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
& ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_233_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y3 = Z ) )
= ( ^ [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y4 )
& ( ord_le3935885782089961368nnreal @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_234_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_235_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
& ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_236_le__cases3,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_237_le__cases3,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ~ ( ord_le3935885782089961368nnreal @ Y @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X2 @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ Y ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
=> ~ ( ord_le3935885782089961368nnreal @ Y @ X2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ X2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z2 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_238_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_239_le__cases3,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_240_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_241_nle__le,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A @ B ) )
= ( ( ord_le3935885782089961368nnreal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_242_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_243_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_244_DEADID_Oin__rel,axiom,
( ( ^ [Y3: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y3 = Z ) )
= ( ^ [A3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
? [Z3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Z3 @ top_to7994903218803871134nnreal )
& ( ( id_Ext8331313139072774535nnreal @ Z3 )
= A3 )
& ( ( id_Ext8331313139072774535nnreal @ Z3 )
= B2 ) ) ) ) ).
% DEADID.in_rel
thf(fact_245_DEADID_Oin__rel,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
? [Z3: real] :
( ( member_real @ Z3 @ top_top_set_real )
& ( ( id_real @ Z3 )
= A3 )
& ( ( id_real @ Z3 )
= B2 ) ) ) ) ).
% DEADID.in_rel
thf(fact_246_enn2real__nonneg,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_less_eq_real @ zero_zero_real @ ( extend1669699412028896998n2real @ X2 ) ) ).
% enn2real_nonneg
thf(fact_247_zero__le,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 ) ).
% zero_le
thf(fact_248_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_249_top_Oextremum__uniqueI,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A )
=> ( A = top_to1496364449551166952nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_250_top_Oextremum__unique,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A )
= ( A = top_to1496364449551166952nnreal ) ) ).
% top.extremum_unique
thf(fact_251_top__greatest,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ top_to1496364449551166952nnreal ) ).
% top_greatest
thf(fact_252_ennreal__eq__0__iff,axiom,
! [X2: real] :
( ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal )
= ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% ennreal_eq_0_iff
thf(fact_253_ennreal3__cases,axiom,
! [X2: extend8495563244428889912nnreal,Xa: extend8495563244428889912nnreal,Xaa: extend8495563244428889912nnreal] :
( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( X2
= ( extend7643940197134561352nnreal @ R3 ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ Ra ) )
=> ! [Raa: real] :
( ( ord_less_eq_real @ zero_zero_real @ Raa )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ Raa ) ) ) ) ) ) )
=> ( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( X2
= ( extend7643940197134561352nnreal @ R3 ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ Ra ) )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) ) )
=> ( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( X2
= ( extend7643940197134561352nnreal @ R3 ) )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
=> ( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( X2
= ( extend7643940197134561352nnreal @ R3 ) )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) )
=> ( ( ( X2 = top_to1496364449551166952nnreal )
=> ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ R3 ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
=> ( ( ( X2 = top_to1496364449551166952nnreal )
=> ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ R3 ) )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) )
=> ( ( ( X2 = top_to1496364449551166952nnreal )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ R3 ) ) ) ) )
=> ~ ( ( X2 = top_to1496364449551166952nnreal )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) ) ) ) ) ) ) ) ).
% ennreal3_cases
thf(fact_254_ennreal2__cases,axiom,
! [X2: extend8495563244428889912nnreal,Xa: extend8495563244428889912nnreal] :
( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( X2
= ( extend7643940197134561352nnreal @ R3 ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( Xa
!= ( extend7643940197134561352nnreal @ Ra ) ) ) ) )
=> ( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( ( X2
= ( extend7643940197134561352nnreal @ R3 ) )
=> ( Xa != top_to1496364449551166952nnreal ) ) )
=> ( ( ( X2 = top_to1496364449551166952nnreal )
=> ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( Xa
!= ( extend7643940197134561352nnreal @ R3 ) ) ) )
=> ~ ( ( X2 = top_to1496364449551166952nnreal )
=> ( Xa != top_to1496364449551166952nnreal ) ) ) ) ) ).
% ennreal2_cases
thf(fact_255_ennreal__cases,axiom,
! [X2: extend8495563244428889912nnreal] :
( ! [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
=> ( X2
!= ( extend7643940197134561352nnreal @ R3 ) ) )
=> ( X2 = top_to1496364449551166952nnreal ) ) ).
% ennreal_cases
thf(fact_256_zero__reorient,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X2 )
= ( X2 = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_257_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_258_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_259_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_260_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_261_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_262_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_263_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_264_enn2real__le,axiom,
! [E: extend8495563244428889912nnreal,R: real] :
( ( ord_less_eq_real @ ( extend1669699412028896998n2real @ E ) @ R )
=> ( ( E != top_to1496364449551166952nnreal )
=> ( ord_le3935885782089961368nnreal @ E @ ( extend7643940197134561352nnreal @ R ) ) ) ) ).
% enn2real_le
thf(fact_265_enn2real__leI,axiom,
! [B4: real,X2: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ ( extend7643940197134561352nnreal @ B4 ) )
=> ( ord_less_eq_real @ ( extend1669699412028896998n2real @ X2 ) @ B4 ) ) ) ).
% enn2real_leI
thf(fact_266_less__top__ennreal,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
= ( ? [R4: real] :
( ( ord_less_eq_real @ zero_zero_real @ R4 )
& ( X2
= ( extend7643940197134561352nnreal @ R4 ) ) ) ) ) ).
% less_top_ennreal
thf(fact_267_ennreal__enn2real,axiom,
! [R: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ R @ top_to1496364449551166952nnreal )
=> ( ( extend7643940197134561352nnreal @ ( extend1669699412028896998n2real @ R ) )
= R ) ) ).
% ennreal_enn2real
thf(fact_268_enn2real__eq__posreal__iff,axiom,
! [C: real,X2: extend8495563244428889912nnreal] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( extend1669699412028896998n2real @ X2 )
= C )
= ( X2
= ( extend7643940197134561352nnreal @ C ) ) ) ) ).
% enn2real_eq_posreal_iff
thf(fact_269_ennreal__minus__top,axiom,
! [A: real] :
( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) ).
% ennreal_minus_top
thf(fact_270_ennreal__le__iff,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ).
% ennreal_le_iff
thf(fact_271_Greatest__equality,axiom,
! [P: real > $o,X2: real] :
( ( P @ X2 )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( order_Greatest_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_272_Greatest__equality,axiom,
! [P: extend8495563244428889912nnreal > $o,X2: extend8495563244428889912nnreal] :
( ( P @ X2 )
=> ( ! [Y2: extend8495563244428889912nnreal] :
( ( P @ Y2 )
=> ( ord_le3935885782089961368nnreal @ Y2 @ X2 ) )
=> ( ( order_7545170809120406815nnreal @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_273_Greatest__equality,axiom,
! [P: int > $o,X2: int] :
( ( P @ X2 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( order_Greatest_int @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_274_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_275_GreatestI2__order,axiom,
! [P: real > $o,X2: real,Q: real > $o] :
( ( P @ X2 )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_276_GreatestI2__order,axiom,
! [P: extend8495563244428889912nnreal > $o,X2: extend8495563244428889912nnreal,Q: extend8495563244428889912nnreal > $o] :
( ( P @ X2 )
=> ( ! [Y2: extend8495563244428889912nnreal] :
( ( P @ Y2 )
=> ( ord_le3935885782089961368nnreal @ Y2 @ X2 ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( P @ X3 )
=> ( ! [Y5: extend8495563244428889912nnreal] :
( ( P @ Y5 )
=> ( ord_le3935885782089961368nnreal @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_7545170809120406815nnreal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_277_GreatestI2__order,axiom,
! [P: int > $o,X2: int,Q: int > $o] :
( ( P @ X2 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_278_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_279_comp__cong,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,X2: extend8495563244428889912nnreal,F3: real > extend8495563244428889912nnreal,G3: extend8495563244428889912nnreal > real,X4: extend8495563244428889912nnreal] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp_r6281409797179841921nnreal @ F @ G @ X2 )
= ( comp_r6281409797179841921nnreal @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_280_comp__cong,axiom,
! [F: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,X2: extend8495563244428889912nnreal,F3: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G3: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X4: extend8495563244428889912nnreal] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp_r6281409797179841921nnreal @ F @ G @ X2 )
= ( comp_E7860224481218928525nnreal @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_281_comp__cong,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,F3: real > extend8495563244428889912nnreal,G3: extend8495563244428889912nnreal > real,X4: extend8495563244428889912nnreal] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp_E7860224481218928525nnreal @ F @ G @ X2 )
= ( comp_r6281409797179841921nnreal @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_282_comp__cong,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,F3: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G3: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X4: extend8495563244428889912nnreal] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp_E7860224481218928525nnreal @ F @ G @ X2 )
= ( comp_E7860224481218928525nnreal @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_283_comp__cong,axiom,
! [F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,X2: real,F3: extend8495563244428889912nnreal > real,G3: real > extend8495563244428889912nnreal,X4: real] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp_E3822617923592311797l_real @ F @ G @ X2 )
= ( comp_E3822617923592311797l_real @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_284_not__gr__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) )
= ( N2 = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_285_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_286_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_287_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_288_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_289_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_290_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_291_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_292_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_293_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_294_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_295_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_296_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_297_ennreal__minus__zero,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% ennreal_minus_zero
thf(fact_298_zero__minus__ennreal,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% zero_minus_ennreal
thf(fact_299_ennreal__top__minus,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ top_to1496364449551166952nnreal @ X2 )
= top_to1496364449551166952nnreal ) ).
% ennreal_top_minus
thf(fact_300_ennreal__minus__eq__top,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= top_to1496364449551166952nnreal )
= ( A = top_to1496364449551166952nnreal ) ) ).
% ennreal_minus_eq_top
thf(fact_301_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_302_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_303_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_304_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_305_ennreal__diff__self,axiom,
! [A: extend8495563244428889912nnreal] :
( ( A != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ A @ A )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_diff_self
thf(fact_306_ennreal__less__zero__iff,axiom,
! [X2: real] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% ennreal_less_zero_iff
thf(fact_307_enn2real__less__iff,axiom,
! [X2: extend8495563244428889912nnreal,C: real] :
( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
=> ( ( ord_less_real @ ( extend1669699412028896998n2real @ X2 ) @ C )
= ( ord_le7381754540660121996nnreal @ X2 @ ( extend7643940197134561352nnreal @ C ) ) ) ) ).
% enn2real_less_iff
thf(fact_308_enn2real__le__iff,axiom,
! [X2: extend8495563244428889912nnreal,C: real] :
( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( extend1669699412028896998n2real @ X2 ) @ C )
= ( ord_le3935885782089961368nnreal @ X2 @ ( extend7643940197134561352nnreal @ C ) ) ) ) ) ).
% enn2real_le_iff
thf(fact_309_ennreal__mono__minus__cancel,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) )
=> ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ A )
=> ( ord_le3935885782089961368nnreal @ C @ B ) ) ) ) ) ).
% ennreal_mono_minus_cancel
thf(fact_310_ennreal__diff__le__mono__left,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% ennreal_diff_le_mono_left
thf(fact_311_diff__diff__commute__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% diff_diff_commute_ennreal
thf(fact_312_diff__le__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ A ) ).
% diff_le_self_ennreal
thf(fact_313_ennreal__mono__minus,axiom,
! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).
% ennreal_mono_minus
thf(fact_314_ennreal__minus__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ D @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ C @ D ) ) ) ) ).
% ennreal_minus_mono
thf(fact_315_order__less__imp__not__less,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ~ ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_316_order__less__imp__not__less,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_317_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_318_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_319_order__less__imp__not__eq2,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_320_order__less__imp__not__eq2,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_321_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_322_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_323_order__less__imp__not__eq,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_324_order__less__imp__not__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_325_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_326_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_327_linorder__less__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
| ( X2 = Y )
| ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_328_linorder__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_329_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_330_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_331_order__less__imp__triv,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,P: $o] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ( ord_le7381754540660121996nnreal @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_332_order__less__imp__triv,axiom,
! [X2: real,Y: real,P: $o] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_333_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_334_order__less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_335_order__less__not__sym,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ~ ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_336_order__less__not__sym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_337_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_338_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_339_order__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_340_order__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_341_order__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_342_order__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > int,C: int] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_343_order__less__subst2,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_344_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_345_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_346_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_347_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_348_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_349_order__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_350_order__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_351_order__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_352_order__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: int > extend8495563244428889912nnreal,B: int,C: int] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_353_order__less__subst1,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_354_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_355_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_356_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_357_order__less__subst1,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_358_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_359_order__less__irrefl,axiom,
! [X2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_360_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_361_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_362_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_363_ord__less__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_364_ord__less__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_365_ord__less__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_366_ord__less__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > int,C: int] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_367_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_368_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_369_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_370_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_371_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_372_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_373_ord__eq__less__subst,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_374_ord__eq__less__subst,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_375_ord__eq__less__subst,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_376_ord__eq__less__subst,axiom,
! [A: int,F: extend8495563244428889912nnreal > int,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_377_ord__eq__less__subst,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_378_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_379_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_380_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_381_ord__eq__less__subst,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_382_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_383_order__less__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ( ord_le7381754540660121996nnreal @ Y @ Z2 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_384_order__less__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_385_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_386_order__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_387_order__less__asym_H,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ~ ( ord_le7381754540660121996nnreal @ B @ A ) ) ).
% order_less_asym'
thf(fact_388_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_389_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_390_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_391_linorder__neq__iff,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X2 != Y )
= ( ( ord_le7381754540660121996nnreal @ X2 @ Y )
| ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_392_linorder__neq__iff,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
= ( ( ord_less_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_393_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_394_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_395_order__less__asym,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ~ ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_396_order__less__asym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_397_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_398_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_399_linorder__neqE,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X2 != Y )
=> ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_400_linorder__neqE,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_401_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_402_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_403_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_404_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_405_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_406_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_407_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_408_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_409_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_410_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_411_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_412_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_413_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_414_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_415_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_416_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_417_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_418_order_Ostrict__implies__not__eq,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_419_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_420_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_421_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_422_dual__order_Ostrict__trans,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ( ord_le7381754540660121996nnreal @ C @ B )
=> ( ord_le7381754540660121996nnreal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_423_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_424_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_425_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_426_not__less__iff__gr__or__eq,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) )
= ( ( ord_le7381754540660121996nnreal @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_427_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ( ord_less_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_428_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_429_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_430_order_Ostrict__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_431_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_432_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_433_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_434_linorder__less__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extend8495563244428889912nnreal] : ( P @ A4 @ A4 )
=> ( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_435_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_436_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_437_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_438_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_439_dual__order_Oirrefl,axiom,
! [A: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A @ A ) ).
% dual_order.irrefl
thf(fact_440_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_441_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_442_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_443_dual__order_Oasym,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ~ ( ord_le7381754540660121996nnreal @ A @ B ) ) ).
% dual_order.asym
thf(fact_444_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_445_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_446_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_447_linorder__cases,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_448_linorder__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_449_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_450_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_451_antisym__conv3,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ Y @ X2 )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_452_antisym__conv3,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_real @ Y @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_453_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_454_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_455_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_456_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_457_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_458_ord__less__eq__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( B = C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_459_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_460_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_461_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_462_ord__eq__less__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A = B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_463_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_464_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_465_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_466_order_Oasym,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ~ ( ord_le7381754540660121996nnreal @ B @ A ) ) ).
% order.asym
thf(fact_467_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_468_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_469_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_470_less__imp__neq,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_471_less__imp__neq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_472_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_473_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_474_dense,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ? [Z4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Z4 )
& ( ord_le7381754540660121996nnreal @ Z4 @ Y ) ) ) ).
% dense
thf(fact_475_dense,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X2 @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_476_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_477_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_478_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_479_lt__ex,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% lt_ex
thf(fact_480_lt__ex,axiom,
! [X2: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X2 ) ).
% lt_ex
thf(fact_481_diff__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_482_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_483_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_484_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_485_diff__gr0__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ).
% diff_gr0_ennreal
thf(fact_486_ennreal__minus__eq__0,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A @ B ) ) ).
% ennreal_minus_eq_0
thf(fact_487_diff__less__top__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal ) ) ).
% diff_less_top_ennreal
thf(fact_488_ennreal__minus__cancel__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= ( minus_8429688780609304081nnreal @ A @ C ) )
= ( ( B = C )
| ( ( ord_le3935885782089961368nnreal @ A @ B )
& ( ord_le3935885782089961368nnreal @ A @ C ) )
| ( A = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_489_ennreal__minus__cancel,axiom,
! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( C != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ( ( minus_8429688780609304081nnreal @ C @ A )
= ( minus_8429688780609304081nnreal @ C @ B ) )
=> ( A = B ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_490_less__numeral__extra_I3_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(3)
thf(fact_491_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_492_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_493_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_494_diff__eq__0__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( minus_8429688780609304081nnreal @ A @ B )
= zero_z7100319975126383169nnreal ) ) ) ).
% diff_eq_0_ennreal
thf(fact_495_diff__eq__0__iff__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
= ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
& ( ord_le3935885782089961368nnreal @ A @ B ) ) ) ).
% diff_eq_0_iff_ennreal
thf(fact_496_ennreal__lessI,axiom,
! [Q2: real,R: real] :
( ( ord_less_real @ zero_zero_real @ Q2 )
=> ( ( ord_less_real @ R @ Q2 )
=> ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q2 ) ) ) ) ).
% ennreal_lessI
thf(fact_497_ennreal__between,axiom,
! [E: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ E )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X2 )
=> ( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ X2 @ E ) @ X2 ) ) ) ) ).
% ennreal_between
thf(fact_498_ennreal__minus__pos__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) )
=> ( ord_le7381754540660121996nnreal @ B @ A ) ) ) ).
% ennreal_minus_pos_iff
thf(fact_499_diff__gt__0__iff__gt__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) )
= ( ( ( A = top_to1496364449551166952nnreal )
& ( B = top_to1496364449551166952nnreal ) )
| ( ord_le7381754540660121996nnreal @ B @ A ) ) ) ).
% diff_gt_0_iff_gt_ennreal
thf(fact_500_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_501_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_502_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_503_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_504_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_505_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_506_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_507_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_508_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_509_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_510_leD,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_real @ X2 @ Y ) ) ).
% leD
thf(fact_511_leD,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) ) ).
% leD
thf(fact_512_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_513_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_514_leI,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% leI
thf(fact_515_leI,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).
% leI
thf(fact_516_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_517_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_518_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_519_nless__le,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ A @ B ) )
= ( ~ ( ord_le3935885782089961368nnreal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_520_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_521_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_522_antisym__conv1,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_523_antisym__conv1,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_524_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_525_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_526_antisym__conv2,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_527_antisym__conv2,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_528_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_529_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_530_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_531_dense__ge,axiom,
! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z2 @ X3 )
=> ( ord_le3935885782089961368nnreal @ Y @ X3 ) )
=> ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_532_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_533_dense__le,axiom,
! [Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y )
=> ( ord_le3935885782089961368nnreal @ X3 @ Z2 ) )
=> ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ).
% dense_le
thf(fact_534_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_535_less__le__not__le,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y4 )
& ~ ( ord_le3935885782089961368nnreal @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_536_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_537_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_538_not__le__imp__less,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_real @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_539_not__le__imp__less,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ Y @ X2 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_540_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_541_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_542_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_543_order_Oorder__iff__strict,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_544_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_545_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_546_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_547_order_Ostrict__iff__order,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_548_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_549_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_550_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_551_order_Ostrict__trans1,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_552_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_553_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_554_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_555_order_Ostrict__trans2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le7381754540660121996nnreal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_556_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_557_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_558_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_559_order_Ostrict__iff__not,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A3: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A3 @ B2 )
& ~ ( ord_le3935885782089961368nnreal @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_560_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_561_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_562_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_563_dense__ge__bounded,axiom,
! [Z2: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z2 @ X2 )
=> ( ! [W: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z2 @ W )
=> ( ( ord_le7381754540660121996nnreal @ W @ X2 )
=> ( ord_le3935885782089961368nnreal @ Y @ W ) ) )
=> ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_564_dense__le__bounded,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_565_dense__le__bounded,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ! [W: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ W )
=> ( ( ord_le7381754540660121996nnreal @ W @ Y )
=> ( ord_le3935885782089961368nnreal @ W @ Z2 ) ) )
=> ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_566_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_567_dual__order_Oorder__iff__strict,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B2: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_568_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_569_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_570_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_571_dual__order_Ostrict__iff__order,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B2: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_572_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_573_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_574_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_575_dual__order_Ostrict__trans1,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le7381754540660121996nnreal @ C @ B )
=> ( ord_le7381754540660121996nnreal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_576_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_577_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_578_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_579_dual__order_Ostrict__trans2,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le7381754540660121996nnreal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_580_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_581_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_582_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_583_dual__order_Ostrict__iff__not,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B2: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A3 )
& ~ ( ord_le3935885782089961368nnreal @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_584_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_585_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_586_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_587_order_Ostrict__implies__order,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_588_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_589_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_590_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_591_dual__order_Ostrict__implies__order,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A )
=> ( ord_le3935885782089961368nnreal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_592_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_593_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_594_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_595_order__le__less,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_596_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_597_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_598_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_599_order__less__le,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_600_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_601_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_602_linorder__not__le,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
= ( ord_less_real @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_603_linorder__not__le,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y ) )
= ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_604_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_605_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_606_linorder__not__less,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_607_linorder__not__less,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) )
= ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_608_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_609_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_610_order__less__imp__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_611_order__less__imp__le,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_612_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_613_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_614_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_615_order__le__neq__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( A != B )
=> ( ord_le7381754540660121996nnreal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_616_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_617_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_618_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_619_order__neq__le__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A != B )
=> ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le7381754540660121996nnreal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_620_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_621_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_622_order__le__less__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_623_order__le__less__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ord_le7381754540660121996nnreal @ Y @ Z2 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_624_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_625_order__le__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_626_order__less__le__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_627_order__less__le__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_628_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_629_order__less__le__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_630_order__le__less__subst1,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_631_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_632_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_633_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_634_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_635_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_636_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_637_order__le__less__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: int > extend8495563244428889912nnreal,B: int,C: int] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_638_order__le__less__subst1,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_639_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_640_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_641_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_642_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_643_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_644_order__le__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_645_order__le__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_646_order__le__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_647_order__le__less__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > int,C: int] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_648_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_649_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_650_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_651_order__less__le__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_652_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_653_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_654_order__less__le__subst1,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_655_order__less__le__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_656_order__less__le__subst1,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_657_order__less__le__subst1,axiom,
! [A: int,F: extend8495563244428889912nnreal > int,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_658_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_659_order__less__le__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le7381754540660121996nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_660_order__less__le__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_661_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_662_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_663_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_664_order__less__le__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_665_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_666_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_667_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_668_order__less__le__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_669_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_670_linorder__le__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_671_linorder__le__less__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
| ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_672_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_673_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_674_order__le__imp__less__or__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_675_order__le__imp__less__or__eq,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( ord_le7381754540660121996nnreal @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_676_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_677_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_678_gr__zeroI,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( N2 != zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) ) ).
% gr_zeroI
thf(fact_679_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_680_not__less__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ N2 @ zero_z7100319975126383169nnreal ) ).
% not_less_zero
thf(fact_681_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_682_gr__implies__not__zero,axiom,
! [M3: extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ M3 @ N2 )
=> ( N2 != zero_z7100319975126383169nnreal ) ) ).
% gr_implies_not_zero
thf(fact_683_gr__implies__not__zero,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_684_zero__less__iff__neq__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 )
= ( N2 != zero_z7100319975126383169nnreal ) ) ).
% zero_less_iff_neq_zero
thf(fact_685_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_686_top_Oextremum__strict,axiom,
! [A: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ top_to1496364449551166952nnreal @ A ) ).
% top.extremum_strict
thf(fact_687_top_Onot__eq__extremum,axiom,
! [A: extend8495563244428889912nnreal] :
( ( A != top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal ) ) ).
% top.not_eq_extremum
thf(fact_688_neq__top__trans,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( Y != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( X2 != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_689_ennreal__less__iff,axiom,
! [R: real,Q2: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q2 ) )
= ( ord_less_real @ R @ Q2 ) ) ) ).
% ennreal_less_iff
thf(fact_690_enn2real__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
=> ( ord_less_eq_real @ ( extend1669699412028896998n2real @ A ) @ ( extend1669699412028896998n2real @ B ) ) ) ) ).
% enn2real_mono
thf(fact_691_enn2real__less,axiom,
! [E: extend8495563244428889912nnreal,R: real] :
( ( ord_less_real @ ( extend1669699412028896998n2real @ E ) @ R )
=> ( ( E != top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ E @ ( extend7643940197134561352nnreal @ R ) ) ) ) ).
% enn2real_less
thf(fact_692_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_693_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_694_enn2real__positive__iff,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ord_less_real @ zero_zero_real @ ( extend1669699412028896998n2real @ X2 ) )
= ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X2 )
& ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal ) ) ) ).
% enn2real_positive_iff
thf(fact_695_ennreal__leI,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y ) ) ) ).
% ennreal_leI
thf(fact_696_minus__top__ennreal,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ( X2 = top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X2 @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) )
& ( ( X2 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X2 @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) ) ) ).
% minus_top_ennreal
thf(fact_697_ennreal__less__top,axiom,
! [X2: real] : ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ top_to1496364449551166952nnreal ) ).
% ennreal_less_top
thf(fact_698_ennreal__zero__less__top,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).
% ennreal_zero_less_top
thf(fact_699_le__ennreal__iff,axiom,
! [R: real,X2: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ ( extend7643940197134561352nnreal @ R ) )
= ( ? [Q3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q3 )
& ( X2
= ( extend7643940197134561352nnreal @ Q3 ) )
& ( ord_less_eq_real @ Q3 @ R ) ) ) ) ) ).
% le_ennreal_iff
thf(fact_700_ennreal__le__iff2,axiom,
! [X2: real,Y: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ Y )
& ( ord_less_eq_real @ X2 @ Y ) )
| ( ( ord_less_eq_real @ X2 @ zero_zero_real )
& ( ord_less_eq_real @ Y @ zero_zero_real ) ) ) ) ).
% ennreal_le_iff2
thf(fact_701_ennreal__lt__0,axiom,
! [X2: real] :
( ( ord_less_real @ X2 @ zero_zero_real )
=> ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_lt_0
thf(fact_702_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B2 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_703_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B2 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_704_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_705_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_706_seq__mono__lemma,axiom,
! [M3: nat,D: nat > real,E: nat > real] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E @ N4 ) ) )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_eq_real @ ( E @ N4 ) @ ( E @ M3 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_real @ ( D @ N5 ) @ ( E @ M3 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_707_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_708_bgauge__existence__lemma,axiom,
! [S: set_real,Q2: real > real > $o] :
( ( ! [X: real] :
( ( member_real @ X @ S )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( Q2 @ D2 @ X ) ) ) )
= ( ! [X: real] :
? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( ( member_real @ X @ S )
=> ( Q2 @ D2 @ X ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_709_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I: nat] :
( ( ord_less_nat @ K @ I )
=> ( P @ I ) )
=> ( P @ K ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_710_UNIV__witness,axiom,
? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_711_UNIV__eq__I,axiom,
! [A2: set_real] :
( ! [X3: real] : ( member_real @ X3 @ A2 )
=> ( top_top_set_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_712_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S2 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ? [Y2: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y2 ) )
& ! [Z5: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ( ord_less_eq_real @ Y2 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_713_ennreal__minus,axiom,
! [Q2: real,R: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q2 )
=> ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q2 ) )
= ( extend7643940197134561352nnreal @ ( minus_minus_real @ R @ Q2 ) ) ) ) ).
% ennreal_minus
thf(fact_714_ennreal__minus__if,axiom,
! [A: real,B: real] :
( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
= ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( if_real @ ( ord_less_eq_real @ B @ A ) @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) @ A ) ) ) ).
% ennreal_minus_if
thf(fact_715_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_716_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B3: real,C2: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_717_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_718_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_719_diff__is__0__eq,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_720_diff__is__0__eq_H,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_721_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_722_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X6: real] :
( ( ( ord_less_eq_real @ A @ X6 )
& ( ord_less_real @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_723_complete__interval,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,P: extend8495563244428889912nnreal > $o] :
( ( ord_le7381754540660121996nnreal @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C2 )
& ( ord_le3935885782089961368nnreal @ C2 @ B )
& ! [X6: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A @ X6 )
& ( ord_le7381754540660121996nnreal @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A @ X3 )
& ( ord_le7381754540660121996nnreal @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_le3935885782089961368nnreal @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_724_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_725_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_726_subsetI,axiom,
! [A2: set_real,B4: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B4 ) )
=> ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% subsetI
thf(fact_727_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_728_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_729_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_730_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_731_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_732_zero__less__diff,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M3 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% zero_less_diff
thf(fact_733_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_734_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_735_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_736_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_737_diff__less,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ) ) ).
% diff_less
thf(fact_738_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_739_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_740_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_741_diffs0__imp__equal,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M3 )
= zero_zero_nat )
=> ( M3 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_742_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_743_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A5 )
=> ( member_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_744_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X: real] :
( ( member_real @ X @ A5 )
=> ( member_real @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_745_subsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% subsetD
thf(fact_746_in__mono,axiom,
! [A2: set_real,B4: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_747_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_748_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_749_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_750_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_751_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_752_diff__less__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_753_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_754_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_755_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_756_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_757_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_758_ex__gt__or__lt,axiom,
! [A: extend8495563244428889912nnreal] :
? [B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B3 )
| ( ord_le7381754540660121996nnreal @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_759_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_760_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_761_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_762_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_763_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_764_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_765_le__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_766_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_767_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_768_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_769_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_770_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_771_GreatestI__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_772_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_773_less__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_774_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_775_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_776_Greatest__le__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_777_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_778_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_779_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_780_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_781_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_782_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_783_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_784_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ~ ( P @ I ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_785_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_786_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_787_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_788_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_789_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X: real] : ( member_real @ X @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_790_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_791_minf_I8_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ~ ( ord_le3935885782089961368nnreal @ T @ X6 ) ) ).
% minf(8)
thf(fact_792_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_793_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_794_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_795_minf_I6_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ( ord_le3935885782089961368nnreal @ X6 @ T ) ) ).
% minf(6)
thf(fact_796_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_797_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_798_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_799_pinf_I8_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ( ord_le3935885782089961368nnreal @ T @ X6 ) ) ).
% pinf(8)
thf(fact_800_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_801_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_802_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_803_pinf_I6_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ~ ( ord_le3935885782089961368nnreal @ X6 @ T ) ) ).
% pinf(6)
thf(fact_804_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_805_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_806_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_807_verit__comp__simplify1_I3_J,axiom,
! [B6: extend8495563244428889912nnreal,A6: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ B6 @ A6 ) )
= ( ord_le7381754540660121996nnreal @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_808_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_809_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_810_DiffI,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B4 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_811_Diff__iff,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_812_DiffE,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% DiffE
thf(fact_813_DiffD1,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_814_DiffD2,axiom,
! [C: real,A2: set_real,B4: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B4 ) )
=> ~ ( member_real @ C @ B4 ) ) ).
% DiffD2
thf(fact_815_psubsetD,axiom,
! [A2: set_real,B4: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B4 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_816_psubset__imp__ex__mem,axiom,
! [A2: set_real,B4: set_real] :
( ( ord_less_set_real @ A2 @ B4 )
=> ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_817_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_818_verit__la__disequality,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A = B )
| ~ ( ord_le3935885782089961368nnreal @ A @ B )
| ~ ( ord_le3935885782089961368nnreal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_819_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_820_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_821_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_822_verit__comp__simplify1_I2_J,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_823_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_824_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_825_verit__comp__simplify1_I1_J,axiom,
! [A: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_826_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_827_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_828_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_829_pinf_I1_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P4: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q4: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_830_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_831_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_832_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_833_pinf_I2_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P4: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q4: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_834_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_835_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_836_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_837_pinf_I3_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_838_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_839_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_840_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_841_pinf_I4_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_842_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_843_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_844_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_845_pinf_I5_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ~ ( ord_le7381754540660121996nnreal @ X6 @ T ) ) ).
% pinf(5)
thf(fact_846_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_847_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_848_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_849_pinf_I7_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X6 )
=> ( ord_le7381754540660121996nnreal @ T @ X6 ) ) ).
% pinf(7)
thf(fact_850_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ Z4 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_851_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_852_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_853_minf_I1_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P4: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q4: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_854_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_855_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_856_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_857_minf_I2_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P4: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q4: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_858_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_859_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_860_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_861_minf_I3_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_862_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_863_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_864_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_865_minf_I4_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_866_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_867_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_868_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_869_minf_I5_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ( ord_le7381754540660121996nnreal @ X6 @ T ) ) ).
% minf(5)
thf(fact_870_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_871_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_872_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_873_minf_I7_J,axiom,
! [T: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X6: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X6 @ Z4 )
=> ~ ( ord_le7381754540660121996nnreal @ T @ X6 ) ) ).
% minf(7)
thf(fact_874_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z4 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_875_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_876_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_877_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_878_ennreal__le__epsilon,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ! [E2: real] :
( ( ord_le7381754540660121996nnreal @ Y @ top_to1496364449551166952nnreal )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_le3935885782089961368nnreal @ X2 @ ( plus_p1859984266308609217nnreal @ Y @ ( extend7643940197134561352nnreal @ E2 ) ) ) ) )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).
% ennreal_le_epsilon
thf(fact_879_top_Oordering__top__axioms,axiom,
orderi7292949391079830343nnreal @ ord_le3935885782089961368nnreal @ ord_le7381754540660121996nnreal @ top_to1496364449551166952nnreal ).
% top.ordering_top_axioms
thf(fact_880_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_881_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_882_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_883_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_884_minus__less__iff__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( ord_le7381754540660121996nnreal @ A @ ( plus_p1859984266308609217nnreal @ C @ B ) ) ) ) ) ).
% minus_less_iff_ennreal
thf(fact_885_harm__pos__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% harm_pos_iff
thf(fact_886_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_887_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_888_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_889_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_890_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_891_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_892_of__nat__eq__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( semiri6283507881447550617nnreal @ M3 )
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( M3 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_893_of__nat__eq__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M3 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M3 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_894_of__nat__eq__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M3 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_895_enn2real__of__nat,axiom,
! [N2: nat] :
( ( extend1669699412028896998n2real @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% enn2real_of_nat
thf(fact_896_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_897_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_898_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_899_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_900_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_901_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_902_add_Oright__neutral,axiom,
! [A: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% add.right_neutral
thf(fact_903_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_904_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_905_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_906_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_907_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_908_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_909_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_910_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_911_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_912_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_913_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_914_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_915_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_916_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_917_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_918_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_919_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_920_add__eq__0__iff__both__eq__0,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ X2 @ Y )
= zero_z7100319975126383169nnreal )
= ( ( X2 = zero_z7100319975126383169nnreal )
& ( Y = zero_z7100319975126383169nnreal ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_921_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_922_zero__eq__add__iff__both__eq__0,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal
= ( plus_p1859984266308609217nnreal @ X2 @ Y ) )
= ( ( X2 = zero_z7100319975126383169nnreal )
& ( Y = zero_z7100319975126383169nnreal ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_923_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_924_add__0,axiom,
! [A: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ zero_z7100319975126383169nnreal @ A )
= A ) ).
% add_0
thf(fact_925_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_926_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_927_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_928_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_929_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_930_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_931_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_932_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_933_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_934_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_935_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_936_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_937_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_938_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_939_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_940_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_941_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_942_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_943_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_944_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_945_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_946_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_947_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_948_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_949_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_950_of__nat__add,axiom,
! [M3: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_951_of__nat__add,axiom,
! [M3: nat,N2: nat] :
( ( semiri6283507881447550617nnreal @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_p1859984266308609217nnreal @ ( semiri6283507881447550617nnreal @ M3 ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% of_nat_add
thf(fact_952_of__nat__add,axiom,
! [M3: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_953_of__nat__add,axiom,
! [M3: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M3 @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_954_ennreal__add__eq__top,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ A @ B )
= top_to1496364449551166952nnreal )
= ( ( A = top_to1496364449551166952nnreal )
| ( B = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_add_eq_top
thf(fact_955_add__top__left__ennreal,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ top_to1496364449551166952nnreal @ X2 )
= top_to1496364449551166952nnreal ) ).
% add_top_left_ennreal
thf(fact_956_add__top__right__ennreal,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ X2 @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) ).
% add_top_right_ennreal
thf(fact_957_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_958_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_959_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_960_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_961_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_962_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_963_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_964_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_965_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_966_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_967_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_968_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_969_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_970_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_971_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_972_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_973_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_974_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_975_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_976_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_977_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_978_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_979_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_980_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_981_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_982_ennreal__le__of__nat__iff,axiom,
! [R: real,I2: nat] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( semiri6283507881447550617nnreal @ I2 ) )
= ( ord_less_eq_real @ R @ ( semiri5074537144036343181t_real @ I2 ) ) ) ).
% ennreal_le_of_nat_iff
thf(fact_983_ennreal__add__less__top,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ top_to1496364449551166952nnreal )
= ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
& ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) ) ) ).
% ennreal_add_less_top
thf(fact_984_add__diff__eq__iff__ennreal,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ X2 ) )
= Y )
= ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).
% add_diff_eq_iff_ennreal
thf(fact_985_ennreal__add__diff__cancel__right,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( Y != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Y ) @ Y )
= X2 ) ) ).
% ennreal_add_diff_cancel_right
thf(fact_986_ennreal__add__diff__cancel__left,axiom,
! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( Y != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ Y @ X2 ) @ Y )
= X2 ) ) ).
% ennreal_add_diff_cancel_left
thf(fact_987_of__nat__le__ennreal__iff,axiom,
! [R: real,I2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ I2 ) @ ( extend7643940197134561352nnreal @ R ) )
= ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ R ) ) ) ).
% of_nat_le_ennreal_iff
thf(fact_988_ennreal__of__nat__eq__real__of__nat,axiom,
( semiri6283507881447550617nnreal
= ( ^ [I4: nat] : ( extend7643940197134561352nnreal @ ( semiri5074537144036343181t_real @ I4 ) ) ) ) ).
% ennreal_of_nat_eq_real_of_nat
thf(fact_989_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_990_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_991_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_992_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_993_ennreal__top__neq__of__nat,axiom,
! [I2: nat] :
( top_to1496364449551166952nnreal
!= ( semiri6283507881447550617nnreal @ I2 ) ) ).
% ennreal_top_neq_of_nat
thf(fact_994_diff__add__eq__diff__diff__swap__ennreal,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ X2 @ ( plus_p1859984266308609217nnreal @ Y @ Z2 ) )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ X2 @ Y ) @ Z2 ) ) ).
% diff_add_eq_diff_diff_swap_ennreal
thf(fact_995_ennreal__Ex__less__of__nat,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
=> ? [N4: nat] : ( ord_le7381754540660121996nnreal @ X2 @ ( semiri6283507881447550617nnreal @ N4 ) ) ) ).
% ennreal_Ex_less_of_nat
thf(fact_996_of__nat__less__top,axiom,
! [I2: nat] : ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ I2 ) @ top_to1496364449551166952nnreal ) ).
% of_nat_less_top
thf(fact_997_add__diff__inverse__ennreal,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y )
=> ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ X2 ) )
= Y ) ) ).
% add_diff_inverse_ennreal
thf(fact_998_diff__add__cancel__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A ) @ A )
= B ) ) ).
% diff_add_cancel_ennreal
thf(fact_999_diff__add__assoc2__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ B ) ) ) ).
% diff_add_assoc2_ennreal
thf(fact_1000_ennreal__diff__add__assoc,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ C @ B ) @ A )
= ( plus_p1859984266308609217nnreal @ C @ ( minus_8429688780609304081nnreal @ B @ A ) ) ) ) ).
% ennreal_diff_add_assoc
thf(fact_1001_ennreal__ineq__diff__add,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( A
= ( plus_p1859984266308609217nnreal @ B @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ) ).
% ennreal_ineq_diff_add
thf(fact_1002_diff__add__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A ) @ A )
= B ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A ) @ A )
= A ) ) ) ).
% diff_add_self_ennreal
thf(fact_1003_add__diff__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ A ) )
= B ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ A ) )
= A ) ) ) ).
% add_diff_self_ennreal
thf(fact_1004_add__diff__le__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C ) @ ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ C ) ) ) ).
% add_diff_le_ennreal
thf(fact_1005_add__diff__eq__ennreal,axiom,
! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
=> ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Y ) @ Z2 ) ) ) ).
% add_diff_eq_ennreal
thf(fact_1006_diff__diff__ennreal_H,axiom,
! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
=> ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) @ X2 )
=> ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Z2 ) @ Y ) ) ) ) ).
% diff_diff_ennreal'
thf(fact_1007_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1008_diff__diff__ennreal_H_H,axiom,
! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
=> ( ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) @ X2 )
=> ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Z2 ) @ Y ) ) )
& ( ~ ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) @ X2 )
=> ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y @ Z2 ) )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% diff_diff_ennreal''
thf(fact_1009_ennreal__le__minus__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ C ) )
= ( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ B )
| ( ( A = zero_z7100319975126383169nnreal )
& ( ord_le3935885782089961368nnreal @ B @ C ) ) ) ) ).
% ennreal_le_minus_iff
thf(fact_1010_ennreal__minus__le__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( ( ord_le3935885782089961368nnreal @ A @ ( plus_p1859984266308609217nnreal @ B @ C ) )
& ( ( ( A = top_to1496364449551166952nnreal )
& ( B = top_to1496364449551166952nnreal ) )
=> ( C = top_to1496364449551166952nnreal ) ) ) ) ).
% ennreal_minus_le_iff
thf(fact_1011_less__diff__eq__ennreal,axiom,
! [B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ C @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ A @ ( minus_8429688780609304081nnreal @ B @ C ) )
= ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ B ) ) ) ).
% less_diff_eq_ennreal
thf(fact_1012_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1013_add__is__0,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1014_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1015_nat__add__left__cancel__less,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1016_diff__diff__left,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1017_add__gr__0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1018_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1019_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1020_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1021_ennreal__plus,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( extend7643940197134561352nnreal @ ( plus_plus_real @ A @ B ) )
= ( plus_p1859984266308609217nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ) ).
% ennreal_plus
thf(fact_1022_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M5: nat,N4: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1023_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1024_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1025_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N3: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1026_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1027_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1028_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1029_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_1030_conj__le__cong,axiom,
! [X2: int,X4: int,P: $o,P4: $o] :
( ( X2 = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1031_imp__le__cong,axiom,
! [X2: int,X4: int,P: $o,P4: $o] :
( ( X2 = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1032_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_1033_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1034_add__eq__self__zero,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= M3 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1035_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1036_add__leE,axiom,
! [M3: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1037_le__add1,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) ) ).
% le_add1
thf(fact_1038_le__add2,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M3 @ N2 ) ) ).
% le_add2
thf(fact_1039_add__leD1,axiom,
! [M3: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% add_leD1
thf(fact_1040_add__leD2,axiom,
! [M3: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1041_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K2 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1042_add__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1043_add__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1044_trans__le__add1,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1045_trans__le__add2,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1046_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1047_less__add__eq__less,axiom,
! [K2: nat,L: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1048_trans__less__add2,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1049_trans__less__add1,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1050_add__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1051_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1052_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1053_add__less__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1054_add__lessD1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
=> ( ord_less_nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_1055_diff__add__inverse2,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N2 ) @ N2 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1056_diff__add__inverse,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M3 ) @ N2 )
= M3 ) ).
% diff_add_inverse
thf(fact_1057_diff__cancel2,axiom,
! [M3: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% diff_cancel2
thf(fact_1058_Nat_Odiff__cancel,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1059_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I2 @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1060_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K2: nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1061_diff__add__0,axiom,
! [N2: nat,M3: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1062_le__diff__conv,axiom,
! [J: nat,K2: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1063_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1064_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1065_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1066_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1067_add__diff__inverse__nat,axiom,
! [M3: nat,N2: nat] :
( ~ ( ord_less_nat @ M3 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M3 @ N2 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1068_less__diff__conv,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1069_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1070_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1071_less__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1072_zle__int,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% zle_int
thf(fact_1073_add__mono__ennreal,axiom,
! [X2: extend8495563244428889912nnreal,Y: real,X4: extend8495563244428889912nnreal,Y6: real] :
( ( ord_le7381754540660121996nnreal @ X2 @ ( extend7643940197134561352nnreal @ Y ) )
=> ( ( ord_le7381754540660121996nnreal @ X4 @ ( extend7643940197134561352nnreal @ Y6 ) )
=> ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ X4 ) @ ( extend7643940197134561352nnreal @ ( plus_plus_real @ Y @ Y6 ) ) ) ) ) ).
% add_mono_ennreal
thf(fact_1074_ennreal__plus__if,axiom,
! [A: real,B: real] :
( ( plus_p1859984266308609217nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
= ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ A ) @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( plus_plus_real @ A @ B ) @ A ) @ B ) ) ) ).
% ennreal_plus_if
thf(fact_1075_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1076_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1077_enn2real__plus,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
=> ( ( extend1669699412028896998n2real @ ( plus_p1859984266308609217nnreal @ A @ B ) )
= ( plus_plus_real @ ( extend1669699412028896998n2real @ A ) @ ( extend1669699412028896998n2real @ B ) ) ) ) ) ).
% enn2real_plus
thf(fact_1078_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1079_eq__diff__eq_H,axiom,
! [X2: real,Y: real,Z2: real] :
( ( X2
= ( minus_minus_real @ Y @ Z2 ) )
= ( Y
= ( plus_plus_real @ X2 @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_1080_euler__mascheroni__sequence__decreasing,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( harmonic_harm_real @ N2 ) @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( minus_minus_real @ ( harmonic_harm_real @ M3 ) @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ) ) ).
% euler_mascheroni_sequence_decreasing
thf(fact_1081_euler__mascheroni__sequence__nonneg,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( harmonic_harm_real @ N2 ) @ ( ln_ln_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% euler_mascheroni_sequence_nonneg
thf(fact_1082_ln__le__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1083_ln__less__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X2 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1084_ln__inj__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X2 )
= ( ln_ln_real @ Y ) )
= ( X2 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1085_ln__less__self,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_less_self
thf(fact_1086_ln__bound,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_bound
thf(fact_1087_ln__le__harm,axiom,
! [N2: nat] : ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) @ ( harmonic_harm_real @ N2 ) ) ).
% ln_le_harm
thf(fact_1088_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= zero_zero_real )
= ( X2 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1089_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_1090_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1091_ln__ge__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_iff
thf(fact_1092_ln__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1093_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_1094_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_1095_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1096_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1097_ln__ge__zero__imp__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1098_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1099_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1100_ln__add__one__self__le__self,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self
thf(fact_1101_ln__eq__minus__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= ( minus_minus_real @ X2 @ one_one_real ) )
=> ( X2 = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1102_ln__le__minus__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1103_minus__one__less,axiom,
! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ one_one_real ) @ X2 ) ).
% minus_one_less
thf(fact_1104_preal__to__real__def,axiom,
( preal_to_real
= ( ^ [R4: extend8495563244428889912nnreal] : ( if_real @ ( R4 = extend2057119558705770725nnreal ) @ ( uminus_uminus_real @ one_one_real ) @ ( extend1669699412028896998n2real @ R4 ) ) ) ) ).
% preal_to_real_def
thf(fact_1105_real__to__preal__def,axiom,
( real_to_preal
= ( ^ [R4: real] :
( if_Ext9135588136721118450nnreal
@ ( R4
= ( uminus_uminus_real @ one_one_real ) )
@ extend2057119558705770725nnreal
@ ( extend7643940197134561352nnreal @ R4 ) ) ) ) ).
% real_to_preal_def
thf(fact_1106_ennreal__eq__1,axiom,
! [X2: real] :
( ( ( extend7643940197134561352nnreal @ X2 )
= one_on2969667320475766781nnreal )
= ( X2 = one_one_real ) ) ).
% ennreal_eq_1
thf(fact_1107_ennreal__1,axiom,
( ( extend7643940197134561352nnreal @ one_one_real )
= one_on2969667320475766781nnreal ) ).
% ennreal_1
thf(fact_1108_enn2real__1,axiom,
( ( extend1669699412028896998n2real @ one_on2969667320475766781nnreal )
= one_one_real ) ).
% enn2real_1
thf(fact_1109_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1110_real__add__minus__iff,axiom,
! [X2: real,A: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X2 = A ) ) ).
% real_add_minus_iff
thf(fact_1111_zle__add1__eq__le,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1112_zle__diff1__eq,axiom,
! [W3: int,Z2: int] :
( ( ord_less_eq_int @ W3 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W3 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1113_ennreal__le__1,axiom,
! [X2: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% ennreal_le_1
thf(fact_1114_ennreal__ge__1,axiom,
! [X2: real] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% ennreal_ge_1
thf(fact_1115_one__less__ennreal,axiom,
! [X2: real] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ).
% one_less_ennreal
thf(fact_1116_ennreal__less__one__iff,axiom,
! [X2: real] :
( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_real @ X2 @ one_one_real ) ) ).
% ennreal_less_one_iff
thf(fact_1117_ennreal__top__neq__one,axiom,
top_to1496364449551166952nnreal != one_on2969667320475766781nnreal ).
% ennreal_top_neq_one
thf(fact_1118_ennreal__add__left__cancel,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ A @ B )
= ( plus_p1859984266308609217nnreal @ A @ C ) )
= ( ( A = extend2057119558705770725nnreal )
| ( B = C ) ) ) ).
% ennreal_add_left_cancel
thf(fact_1119_infinity__ennreal__def,axiom,
extend2057119558705770725nnreal = top_to1496364449551166952nnreal ).
% infinity_ennreal_def
thf(fact_1120_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1121_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X: real,Y4: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y4 ) ) ) ) ).
% minus_real_def
thf(fact_1122_ennreal__add__left__cancel__le,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( plus_p1859984266308609217nnreal @ A @ C ) )
= ( ( A = extend2057119558705770725nnreal )
| ( ord_le3935885782089961368nnreal @ B @ C ) ) ) ).
% ennreal_add_left_cancel_le
thf(fact_1123_ennreal__add__left__cancel__less,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( plus_p1859984266308609217nnreal @ A @ C ) )
= ( ( A != extend2057119558705770725nnreal )
& ( ord_le7381754540660121996nnreal @ B @ C ) ) ) ).
% ennreal_add_left_cancel_less
thf(fact_1124_diff__diff__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( B != extend2057119558705770725nnreal )
=> ( ( minus_8429688780609304081nnreal @ B @ ( minus_8429688780609304081nnreal @ B @ A ) )
= A ) ) ) ).
% diff_diff_ennreal
thf(fact_1125_ennreal__add__diff__cancel,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( B != extend2057119558705770725nnreal )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ B )
= A ) ) ).
% ennreal_add_diff_cancel
thf(fact_1126_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1127_int__ge__induct,axiom,
! [K2: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1128_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_1129_zless__add1__eq,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W3 @ Z2 )
| ( W3 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1130_int__gr__induct,axiom,
! [K2: int,I2: int,P: int > $o] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1131_int__le__induct,axiom,
! [I2: int,K2: int,P: int > $o] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1132_int__less__induct,axiom,
! [I2: int,K2: int,P: int > $o] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1133_ennreal__one__less__top,axiom,
ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ top_to1496364449551166952nnreal ).
% ennreal_one_less_top
thf(fact_1134_real__0__le__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1135_real__add__le__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_1136_real__add__less__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_1137_real__0__less__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1138_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1139_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1140_zless__imp__add1__zle,axiom,
! [W3: int,Z2: int] :
( ( ord_less_int @ W3 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1141_add1__zle__eq,axiom,
! [W3: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W3 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1142_int__induct,axiom,
! [P: int > $o,K2: int,I2: int] :
( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
thf(fact_1143_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1144_ln__add__one__self__le__self2,axiom,
! [X2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self2
thf(fact_1145_ln__one__minus__pos__upper__bound,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1146_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I: nat] : ( ord_less_eq_nat @ ( L3 @ X6 @ I ) @ one_one_nat )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( X6 @ I )
= zero_zero_real ) )
=> ( ( L3 @ X6 @ I )
= zero_zero_nat ) )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( X6 @ I )
= one_one_real ) )
=> ( ( L3 @ X6 @ I )
= one_one_nat ) )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( L3 @ X6 @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I ) @ ( F @ X6 @ I ) ) )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( L3 @ X6 @ I )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I ) @ ( X6 @ I ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1147_kuhn__lemma,axiom,
! [P5: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P5 )
=> ( ! [X3: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I ) @ P5 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( Label @ X3 @ I3 )
= zero_zero_nat )
| ( ( Label @ X3 @ I3 )
= one_one_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I ) @ P5 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X3 @ I3 )
= zero_zero_nat )
=> ( ( Label @ X3 @ I3 )
= zero_zero_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I ) @ P5 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X3 @ I3 )
= P5 )
=> ( ( Label @ X3 @ I3 )
= one_one_nat ) ) ) )
=> ~ ! [Q5: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( ord_less_nat @ ( Q5 @ I ) @ P5 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ? [R3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q5 @ J3 ) @ ( R3 @ J3 ) )
& ( ord_less_eq_nat @ ( R3 @ J3 ) @ ( plus_plus_nat @ ( Q5 @ J3 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q5 @ J3 ) @ ( S3 @ J3 ) )
& ( ord_less_eq_nat @ ( S3 @ J3 ) @ ( plus_plus_nat @ ( Q5 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I )
!= ( Label @ S3 @ I ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1148_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1149_negative__eq__positive,axiom,
! [N2: nat,M3: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M3 ) )
= ( ( N2 = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1150_negative__zle,axiom,
! [N2: nat,M3: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zle
thf(fact_1151_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1152_not__int__zless__negative,axiom,
! [N2: nat,M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% not_int_zless_negative
thf(fact_1153_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1154_int__cases4,axiom,
! [M3: int] :
( ! [N4: nat] :
( M3
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1155_int__zle__neg,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) )
= ( ( N2 = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1156_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1157_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N4: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1158_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1159_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1160_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1161_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
& ( ( F @ I3 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1162_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1163_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1164_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
& ( ( F @ I3 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1165_incr__lemma,axiom,
! [D: int,Z2: int,X2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_1166_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1167_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1168_Suc__le__mono,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1169_Suc__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1170_Suc__mono,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1171_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1172_add__Suc__right,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M3 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1173_Suc__diff__diff,axiom,
! [M3: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1174_diff__Suc__Suc,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1175_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1176_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1177_artanh__minus__real,axiom,
! [X2: real] :
( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).
% artanh_minus_real
thf(fact_1178_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1179_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1180_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1181_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1182_negative__zless,axiom,
! [N2: nat,M3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zless
thf(fact_1183_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1184_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1185_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1186_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W3: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W3 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W3 ) @ ( times_times_int @ Z22 @ W3 ) ) ) ).
% int_distrib(3)
thf(fact_1187_int__distrib_I4_J,axiom,
! [W3: int,Z1: int,Z22: int] :
( ( times_times_int @ W3 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W3 @ Z1 ) @ ( times_times_int @ W3 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1188_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1189_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_1190_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_1191_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1192_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1193_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1194_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1195_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1196_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1197_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1198_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1199_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1200_Suc__leD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_leD
thf(fact_1201_le__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( M3
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1202_le__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1203_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1204_le__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M3 @ N2 )
| ( M3
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1205_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1206_not__less__eq__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1207_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1208_nat__induct__at__least,axiom,
! [M3: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ M3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1209_transitive__stepwise__le,axiom,
! [M3: nat,N2: nat,R5: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ! [X3: nat] : ( R5 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z4: nat] :
( ( R5 @ X3 @ Y2 )
=> ( ( R5 @ Y2 @ Z4 )
=> ( R5 @ X3 @ Z4 ) ) )
=> ( ! [N4: nat] : ( R5 @ N4 @ ( suc @ N4 ) )
=> ( R5 @ M3 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1210_not__less__less__Suc__eq,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1211_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1212_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I3 @ K ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1213_less__trans__Suc,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1214_Suc__less__SucD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1215_less__antisym,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
=> ( M3 = N2 ) ) ) ).
% less_antisym
thf(fact_1216_Suc__less__eq2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M3 )
= ( ? [M7: nat] :
( ( M3
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1217_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1218_not__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M3 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1219_less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1220_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1221_less__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1222_less__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M3 @ N2 )
=> ( M3 = N2 ) ) ) ).
% less_SucE
thf(fact_1223_Suc__lessI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ( suc @ M3 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1224_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1225_Suc__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_lessD
thf(fact_1226_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1227_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I2: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1228_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1229_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_1230_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1231_nat__arith_Osuc1,axiom,
! [A2: nat,K2: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1232_add__Suc,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M3 @ N2 ) ) ) ).
% add_Suc
thf(fact_1233_add__Suc__shift,axiom,
! [M3: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N2 )
= ( plus_plus_nat @ M3 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1234_zmult__zless__mono2,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I2 ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1235_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N2: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1236_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1237_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1238_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1239_diff__less__Suc,axiom,
! [M3: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N2 ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_1240_Suc__diff__Suc,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1241_Suc__diff__le,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ N2 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1242_less__natE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ~ ! [Q5: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M3 @ Q5 ) ) ) ) ).
% less_natE
thf(fact_1243_less__add__Suc1,axiom,
! [I2: nat,M3: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1244_less__add__Suc2,axiom,
! [I2: nat,M3: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M3 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1245_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1246_less__imp__Suc__add,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1247_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1248_one__is__add,axiom,
! [M3: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N2 ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1249_add__is__1,axiom,
! [M3: nat,N2: nat] :
( ( ( plus_plus_nat @ M3 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1250_Suc__leI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1251_Suc__le__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1252_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1253_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1254_Suc__le__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1255_le__less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1256_less__Suc__eq__le,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1257_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1258_le__imp__less__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1259_less__Suc__eq__0__disj,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( M3 = zero_zero_nat )
| ? [J4: nat] :
( ( M3
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1260_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1261_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1262_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1263_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( comp_r6281409797179841921nnreal @ real_to_preal @ preal_to_real @ r )
= ( id_Ext8331313139072774535nnreal @ r ) ) ).
%------------------------------------------------------------------------------