TPTP Problem File: ITP114^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP114^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_784__6255430_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Lower_Semicontinuous/prob_784__6255430_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.46 v8.1.0, 0.55 v7.5.0
% Syntax : Number of formulae : 424 ( 144 unt; 72 typ; 0 def)
% Number of atoms : 1154 ( 399 equ; 0 cnn)
% Maximal formula atoms : 81 ( 3 avg)
% Number of connectives : 3019 ( 124 ~; 19 |; 29 &;2177 @)
% ( 0 <=>; 670 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 271 ( 271 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 65 usr; 10 con; 0-2 aty)
% Number of variables : 999 ( 76 ^; 894 !; 29 ?; 999 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:46.608
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (65)
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
extend1289208545_ereal: extended_ereal ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Extended__Real_Oereal_OMInfty,type,
extended_MInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_OPInfty,type,
extended_PInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Oereal,type,
extended_ereal2: real > extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
extended_size_ereal: extended_ereal > nat ).
thf(sy_c_Extended__Real_Oereal__of__enat,type,
extend1771934483f_enat: extended_enat > extended_ereal ).
thf(sy_c_Extended__Real_Oreal__of__ereal,type,
extend1716541707_ereal: extended_ereal > real ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Extended____Real__Oereal,type,
abs_ab1260901297_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus1208298309_ereal: extended_ereal > extended_ereal ).
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____Nat__Oenat,type,
zero_z491942557d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z163181189_ereal: extended_ereal ).
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_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc_001tf__a_001t__Extended____Real__Oereal,type,
lower_1616484581_ereal: ( a > extended_ereal ) > $o ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__hull_001tf__a,type,
lower_881475195hull_a: ( a > extended_ereal ) > a > extended_ereal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera280919179d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Real__Oereal,type,
numera1793320307_ereal: num > extended_ereal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
ord_le291126163d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Real__Oereal_J,type,
ord_le637473275_ereal: ( $o > extended_ereal ) > ( $o > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J,type,
ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le1863327750d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nat__Oenat,type,
order_1628344639d_enat: ( extended_enat > $o ) > extended_enat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Real__Oereal,type,
order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).
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_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
order_2047034162d_enat: ( extended_enat > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Extended____Real__Oereal,type,
order_1147259034_ereal: ( extended_enat > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
order_442688004at_nat: ( extended_enat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
order_182315744t_real: ( extended_enat > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Extended____Nat__Oenat,type,
order_680161354d_enat: ( extended_ereal > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
order_1408494002_ereal: ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
order_523806444al_nat: ( extended_ereal > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
order_1800387528l_real: ( extended_ereal > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
order_1660553314d_enat: ( nat > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
order_736687562_ereal: ( nat > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat,type,
order_1631207636at_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
order_106095024t_real: ( nat > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
order_2106278841d_enat: ( extended_enat > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Extended____Real__Oereal,type,
order_1180956065_ereal: ( extended_enat > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
order_1622825661at_nat: ( extended_enat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
order_135436057t_real: ( extended_enat > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Extended____Nat__Oenat,type,
order_713858385d_enat: ( extended_ereal > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
order_555877177_ereal: ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
order_476926757al_nat: ( extended_ereal > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
order_1560271745l_real: ( extended_ereal > real ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
order_693207323d_enat: ( nat > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
order_689807875_ereal: ( nat > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
order_769474267at_nat: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Extended____Nat__Oenat,type,
order_1962034751d_enat: ( real > extended_enat ) > $o ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Extended____Real__Oereal,type,
order_946214183_ereal: ( real > extended_ereal ) > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_g,type,
g: a > extended_ereal ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (351)
thf(fact_0_assms,axiom,
! [X: a] : ( ord_le824540014_ereal @ ( g @ X ) @ ( f @ X ) ) ).
% assms
thf(fact_1_calculation,axiom,
ord_le824540014_ereal @ ( lower_881475195hull_a @ g @ x ) @ ( g @ x ) ).
% calculation
thf(fact_2_order__refl,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ X2 ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X2: extended_enat] : ( ord_le1863327750d_enat @ X2 @ X2 ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_6_ereal__complete__Inf,axiom,
! [S: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member1900190071_ereal @ Xa @ S )
=> ( ord_le824540014_ereal @ X3 @ Xa ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member1900190071_ereal @ Xa2 @ S )
=> ( ord_le824540014_ereal @ Z @ Xa2 ) )
=> ( ord_le824540014_ereal @ Z @ X3 ) ) ) ).
% ereal_complete_Inf
thf(fact_7_ereal__complete__Sup,axiom,
! [S: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member1900190071_ereal @ Xa @ S )
=> ( ord_le824540014_ereal @ Xa @ X3 ) )
& ! [Z: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member1900190071_ereal @ Xa2 @ S )
=> ( ord_le824540014_ereal @ Xa2 @ Z ) )
=> ( ord_le824540014_ereal @ X3 @ Z ) ) ) ).
% ereal_complete_Sup
thf(fact_8_order__subst1,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_9_order__subst1,axiom,
! [A: extended_ereal,F: extended_enat > extended_ereal,B: extended_enat,C: extended_enat] :
( ( ord_le824540014_ereal @ A @ ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_10_order__subst1,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( ord_le824540014_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_11_order__subst1,axiom,
! [A: extended_ereal,F: real > extended_ereal,B: real,C: real] :
( ( ord_le824540014_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_12_order__subst1,axiom,
! [A: extended_enat,F: extended_ereal > extended_enat,B: extended_ereal,C: extended_ereal] :
( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_13_order__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_14_order__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_15_order__subst1,axiom,
! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_16_order__subst1,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_17_order__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_18_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_19_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_enat,C: extended_enat] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_20_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_21_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_22_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_ereal,C: extended_ereal] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_23_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_24_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_25_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_26_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_27_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_28_verit__la__disequality,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( A = B )
| ~ ( ord_le824540014_ereal @ A @ B )
| ~ ( ord_le824540014_ereal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_29_verit__la__disequality,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = B )
| ~ ( ord_le1863327750d_enat @ A @ B )
| ~ ( ord_le1863327750d_enat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_30_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_31_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_32_ord__eq__le__subst,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_33_ord__eq__le__subst,axiom,
! [A: extended_enat,F: extended_ereal > extended_enat,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_34_ord__eq__le__subst,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_35_ord__eq__le__subst,axiom,
! [A: real,F: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_36_ord__eq__le__subst,axiom,
! [A: extended_ereal,F: extended_enat > extended_ereal,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_37_ord__eq__le__subst,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_38_ord__eq__le__subst,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_39_ord__eq__le__subst,axiom,
! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_40_ord__eq__le__subst,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_41_ord__eq__le__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_42_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_enat,C: extended_enat] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_46_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_ereal,C: extended_ereal] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_47_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_48_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_49_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_50_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_51_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_52_lsc__hull__le,axiom,
! [F: a > extended_ereal,X2: a] : ( ord_le824540014_ereal @ ( lower_881475195hull_a @ F @ X2 ) @ ( F @ X2 ) ) ).
% lsc_hull_le
thf(fact_53_dual__order_Oantisym,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le824540014_ereal @ B @ A )
=> ( ( ord_le824540014_ereal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_54_dual__order_Oantisym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le1863327750d_enat @ B @ A )
=> ( ( ord_le1863327750d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_55_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_56_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_57_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : Y2 = Z2 )
= ( ^ [A2: extended_ereal,B2: extended_ereal] :
( ( ord_le824540014_ereal @ B2 @ A2 )
& ( ord_le824540014_ereal @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_58_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: extended_enat,Z2: extended_enat] : Y2 = Z2 )
= ( ^ [A2: extended_enat,B2: extended_enat] :
( ( ord_le1863327750d_enat @ B2 @ A2 )
& ( ord_le1863327750d_enat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_59_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : Y2 = Z2 )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_60_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : Y2 = Z2 )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_61_dual__order_Otrans,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ B @ A )
=> ( ( ord_le824540014_ereal @ C @ B )
=> ( ord_le824540014_ereal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_62_dual__order_Otrans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le1863327750d_enat @ B @ A )
=> ( ( ord_le1863327750d_enat @ C @ B )
=> ( ord_le1863327750d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_63_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_64_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_65_linorder__wlog,axiom,
! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
( ! [A3: extended_ereal,B3: extended_ereal] :
( ( ord_le824540014_ereal @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: extended_ereal,B3: extended_ereal] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_66_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A3: extended_enat,B3: extended_enat] :
( ( ord_le1863327750d_enat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_67_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_68_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_69_dual__order_Orefl,axiom,
! [A: extended_ereal] : ( ord_le824540014_ereal @ A @ A ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le1863327750d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_73_order__trans,axiom,
! [X2: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ( ord_le824540014_ereal @ Y3 @ Z3 )
=> ( ord_le824540014_ereal @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_74_order__trans,axiom,
! [X2: extended_enat,Y3: extended_enat,Z3: extended_enat] :
( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ( ord_le1863327750d_enat @ Y3 @ Z3 )
=> ( ord_le1863327750d_enat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_75_order__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_76_order__trans,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_77_order__class_Oorder_Oantisym,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le824540014_ereal @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_78_order__class_Oorder_Oantisym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ord_le1863327750d_enat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_79_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_80_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_81_ord__le__eq__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( B = C )
=> ( ord_le824540014_ereal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_82_ord__le__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_83_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_84_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_85_ord__eq__le__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A = B )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ord_le824540014_ereal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_86_ord__eq__le__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_87_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_88_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_89_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : Y2 = Z2 )
= ( ^ [A2: extended_ereal,B2: extended_ereal] :
( ( ord_le824540014_ereal @ A2 @ B2 )
& ( ord_le824540014_ereal @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_90_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: extended_enat,Z2: extended_enat] : Y2 = Z2 )
= ( ^ [A2: extended_enat,B2: extended_enat] :
( ( ord_le1863327750d_enat @ A2 @ B2 )
& ( ord_le1863327750d_enat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_91_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : Y2 = Z2 )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : Y2 = Z2 )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_93_antisym__conv,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( ord_le824540014_ereal @ Y3 @ X2 )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_94_antisym__conv,axiom,
! [Y3: extended_enat,X2: extended_enat] :
( ( ord_le1863327750d_enat @ Y3 @ X2 )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_95_antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_96_antisym__conv,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq_real @ Y3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_97_le__cases3,axiom,
! [X2: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
( ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ~ ( ord_le824540014_ereal @ Y3 @ Z3 ) )
=> ( ( ( ord_le824540014_ereal @ Y3 @ X2 )
=> ~ ( ord_le824540014_ereal @ X2 @ Z3 ) )
=> ( ( ( ord_le824540014_ereal @ X2 @ Z3 )
=> ~ ( ord_le824540014_ereal @ Z3 @ Y3 ) )
=> ( ( ( ord_le824540014_ereal @ Z3 @ Y3 )
=> ~ ( ord_le824540014_ereal @ Y3 @ X2 ) )
=> ( ( ( ord_le824540014_ereal @ Y3 @ Z3 )
=> ~ ( ord_le824540014_ereal @ Z3 @ X2 ) )
=> ~ ( ( ord_le824540014_ereal @ Z3 @ X2 )
=> ~ ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_98_le__cases3,axiom,
! [X2: extended_enat,Y3: extended_enat,Z3: extended_enat] :
( ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ~ ( ord_le1863327750d_enat @ Y3 @ Z3 ) )
=> ( ( ( ord_le1863327750d_enat @ Y3 @ X2 )
=> ~ ( ord_le1863327750d_enat @ X2 @ Z3 ) )
=> ( ( ( ord_le1863327750d_enat @ X2 @ Z3 )
=> ~ ( ord_le1863327750d_enat @ Z3 @ Y3 ) )
=> ( ( ( ord_le1863327750d_enat @ Z3 @ Y3 )
=> ~ ( ord_le1863327750d_enat @ Y3 @ X2 ) )
=> ( ( ( ord_le1863327750d_enat @ Y3 @ Z3 )
=> ~ ( ord_le1863327750d_enat @ Z3 @ X2 ) )
=> ~ ( ( ord_le1863327750d_enat @ Z3 @ X2 )
=> ~ ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_99_le__cases3,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_100_le__cases3,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_101_order_Otrans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( ord_le824540014_ereal @ B @ C )
=> ( ord_le824540014_ereal @ A @ C ) ) ) ).
% order.trans
thf(fact_102_order_Otrans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le1863327750d_enat @ A @ B )
=> ( ( ord_le1863327750d_enat @ B @ C )
=> ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_103_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_104_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_105_le__cases,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ~ ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% le_cases
thf(fact_106_le__cases,axiom,
! [X2: extended_enat,Y3: extended_enat] :
( ~ ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ Y3 @ X2 ) ) ).
% le_cases
thf(fact_107_le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% le_cases
thf(fact_108_le__cases,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% le_cases
thf(fact_109_eq__refl,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( X2 = Y3 )
=> ( ord_le824540014_ereal @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_110_eq__refl,axiom,
! [X2: extended_enat,Y3: extended_enat] :
( ( X2 = Y3 )
=> ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_111_eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_112_eq__refl,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_113_linear,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y3 )
| ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% linear
thf(fact_114_linear,axiom,
! [X2: extended_enat,Y3: extended_enat] :
( ( ord_le1863327750d_enat @ X2 @ Y3 )
| ( ord_le1863327750d_enat @ Y3 @ X2 ) ) ).
% linear
thf(fact_115_linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linear
thf(fact_116_linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
| ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linear
thf(fact_117_antisym,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ( ord_le824540014_ereal @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% antisym
thf(fact_118_antisym,axiom,
! [X2: extended_enat,Y3: extended_enat] :
( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ( ord_le1863327750d_enat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% antisym
thf(fact_119_antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% antisym
thf(fact_120_antisym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% antisym
thf(fact_121_eq__iff,axiom,
( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : Y2 = Z2 )
= ( ^ [X4: extended_ereal,Y4: extended_ereal] :
( ( ord_le824540014_ereal @ X4 @ Y4 )
& ( ord_le824540014_ereal @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_122_eq__iff,axiom,
( ( ^ [Y2: extended_enat,Z2: extended_enat] : Y2 = Z2 )
= ( ^ [X4: extended_enat,Y4: extended_enat] :
( ( ord_le1863327750d_enat @ X4 @ Y4 )
& ( ord_le1863327750d_enat @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_123_eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : Y2 = Z2 )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_124_eq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : Y2 = Z2 )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_125_lsc__hull__iff__greatest,axiom,
! [G: a > extended_ereal,F: a > extended_ereal] :
( ( G
= ( lower_881475195hull_a @ F ) )
= ( ( lower_1616484581_ereal @ G )
& ! [X4: a] : ( ord_le824540014_ereal @ ( G @ X4 ) @ ( F @ X4 ) )
& ! [H: a > extended_ereal] :
( ( ( lower_1616484581_ereal @ H )
& ! [X4: a] : ( ord_le824540014_ereal @ ( H @ X4 ) @ ( F @ X4 ) ) )
=> ! [X4: a] : ( ord_le824540014_ereal @ ( H @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% lsc_hull_iff_greatest
thf(fact_126_lsc__hull__greatest,axiom,
! [G: a > extended_ereal,F: a > extended_ereal] :
( ( lower_1616484581_ereal @ G )
=> ( ! [X3: a] : ( ord_le824540014_ereal @ ( G @ X3 ) @ ( F @ X3 ) )
=> ! [X: a] : ( ord_le824540014_ereal @ ( G @ X ) @ ( lower_881475195hull_a @ F @ X ) ) ) ) ).
% lsc_hull_greatest
thf(fact_127_Greatest__equality,axiom,
! [P: extended_ereal > $o,X2: extended_ereal] :
( ( P @ X2 )
=> ( ! [Y: extended_ereal] :
( ( P @ Y )
=> ( ord_le824540014_ereal @ Y @ X2 ) )
=> ( ( order_1158471719_ereal @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_128_Greatest__equality,axiom,
! [P: extended_enat > $o,X2: extended_enat] :
( ( P @ X2 )
=> ( ! [Y: extended_enat] :
( ( P @ Y )
=> ( ord_le1863327750d_enat @ Y @ X2 ) )
=> ( ( order_1628344639d_enat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_129_Greatest__equality,axiom,
! [P: real > $o,X2: real] :
( ( P @ X2 )
=> ( ! [Y: real] :
( ( P @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( order_Greatest_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_130_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_131_GreatestI2__order,axiom,
! [P: extended_ereal > $o,X2: extended_ereal,Q: extended_ereal > $o] :
( ( P @ X2 )
=> ( ! [Y: extended_ereal] :
( ( P @ Y )
=> ( ord_le824540014_ereal @ Y @ X2 ) )
=> ( ! [X3: extended_ereal] :
( ( P @ X3 )
=> ( ! [Y5: extended_ereal] :
( ( P @ Y5 )
=> ( ord_le824540014_ereal @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1158471719_ereal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_132_GreatestI2__order,axiom,
! [P: extended_enat > $o,X2: extended_enat,Q: extended_enat > $o] :
( ( P @ X2 )
=> ( ! [Y: extended_enat] :
( ( P @ Y )
=> ( ord_le1863327750d_enat @ Y @ X2 ) )
=> ( ! [X3: extended_enat] :
( ( P @ X3 )
=> ( ! [Y5: extended_enat] :
( ( P @ Y5 )
=> ( ord_le1863327750d_enat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1628344639d_enat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_133_GreatestI2__order,axiom,
! [P: real > $o,X2: real,Q: real > $o] :
( ( P @ X2 )
=> ( ! [Y: real] :
( ( P @ Y )
=> ( ord_less_eq_real @ Y @ 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_134_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ 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_135_le__rel__bool__arg__iff,axiom,
( ord_le637473275_ereal
= ( ^ [X5: $o > extended_ereal,Y6: $o > extended_ereal] :
( ( ord_le824540014_ereal @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le824540014_ereal @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_136_le__rel__bool__arg__iff,axiom,
( ord_le291126163d_enat
= ( ^ [X5: $o > extended_enat,Y6: $o > extended_enat] :
( ( ord_le1863327750d_enat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le1863327750d_enat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_137_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_138_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_real
= ( ^ [X5: $o > real,Y6: $o > real] :
( ( ord_less_eq_real @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_real @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_139_lsc__lsc__hull,axiom,
! [F: a > extended_ereal] : ( lower_1616484581_ereal @ ( lower_881475195hull_a @ F ) ) ).
% lsc_lsc_hull
thf(fact_140_ereal__of__enat__le__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le824540014_ereal @ ( extend1771934483f_enat @ M ) @ ( extend1771934483f_enat @ N ) )
= ( ord_le1863327750d_enat @ M @ N ) ) ).
% ereal_of_enat_le_iff
thf(fact_141_antimono__def,axiom,
( order_1408494002_ereal
= ( ^ [F2: extended_ereal > extended_ereal] :
! [X4: extended_ereal,Y4: extended_ereal] :
( ( ord_le824540014_ereal @ X4 @ Y4 )
=> ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_142_antimono__def,axiom,
( order_680161354d_enat
= ( ^ [F2: extended_ereal > extended_enat] :
! [X4: extended_ereal,Y4: extended_ereal] :
( ( ord_le824540014_ereal @ X4 @ Y4 )
=> ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_143_antimono__def,axiom,
( order_523806444al_nat
= ( ^ [F2: extended_ereal > nat] :
! [X4: extended_ereal,Y4: extended_ereal] :
( ( ord_le824540014_ereal @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_144_antimono__def,axiom,
( order_1800387528l_real
= ( ^ [F2: extended_ereal > real] :
! [X4: extended_ereal,Y4: extended_ereal] :
( ( ord_le824540014_ereal @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_145_antimono__def,axiom,
( order_1147259034_ereal
= ( ^ [F2: extended_enat > extended_ereal] :
! [X4: extended_enat,Y4: extended_enat] :
( ( ord_le1863327750d_enat @ X4 @ Y4 )
=> ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_146_antimono__def,axiom,
( order_2047034162d_enat
= ( ^ [F2: extended_enat > extended_enat] :
! [X4: extended_enat,Y4: extended_enat] :
( ( ord_le1863327750d_enat @ X4 @ Y4 )
=> ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_147_antimono__def,axiom,
( order_442688004at_nat
= ( ^ [F2: extended_enat > nat] :
! [X4: extended_enat,Y4: extended_enat] :
( ( ord_le1863327750d_enat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_148_antimono__def,axiom,
( order_182315744t_real
= ( ^ [F2: extended_enat > real] :
! [X4: extended_enat,Y4: extended_enat] :
( ( ord_le1863327750d_enat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_149_antimono__def,axiom,
( order_736687562_ereal
= ( ^ [F2: nat > extended_ereal] :
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_150_antimono__def,axiom,
( order_1660553314d_enat
= ( ^ [F2: nat > extended_enat] :
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% antimono_def
thf(fact_151_antimonoI,axiom,
! [F: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_1408494002_ereal @ F ) ) ).
% antimonoI
thf(fact_152_antimonoI,axiom,
! [F: extended_ereal > extended_enat] :
( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_680161354d_enat @ F ) ) ).
% antimonoI
thf(fact_153_antimonoI,axiom,
! [F: extended_ereal > nat] :
( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_523806444al_nat @ F ) ) ).
% antimonoI
thf(fact_154_antimonoI,axiom,
! [F: extended_ereal > real] :
( ! [X3: extended_ereal,Y: extended_ereal] :
( ( ord_le824540014_ereal @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_1800387528l_real @ F ) ) ).
% antimonoI
thf(fact_155_antimonoI,axiom,
! [F: extended_enat > extended_ereal] :
( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_1147259034_ereal @ F ) ) ).
% antimonoI
thf(fact_156_antimonoI,axiom,
! [F: extended_enat > extended_enat] :
( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_2047034162d_enat @ F ) ) ).
% antimonoI
thf(fact_157_antimonoI,axiom,
! [F: extended_enat > nat] :
( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_442688004at_nat @ F ) ) ).
% antimonoI
thf(fact_158_antimonoI,axiom,
! [F: extended_enat > real] :
( ! [X3: extended_enat,Y: extended_enat] :
( ( ord_le1863327750d_enat @ X3 @ Y )
=> ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_182315744t_real @ F ) ) ).
% antimonoI
thf(fact_159_antimonoI,axiom,
! [F: nat > extended_ereal] :
( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_736687562_ereal @ F ) ) ).
% antimonoI
thf(fact_160_antimonoI,axiom,
! [F: nat > extended_enat] :
( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
=> ( order_1660553314d_enat @ F ) ) ).
% antimonoI
thf(fact_161_antimonoE,axiom,
! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
( ( order_1408494002_ereal @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_162_antimonoE,axiom,
! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
( ( order_680161354d_enat @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_163_antimonoE,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
( ( order_523806444al_nat @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_164_antimonoE,axiom,
! [F: extended_ereal > real,X2: extended_ereal,Y3: extended_ereal] :
( ( order_1800387528l_real @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_165_antimonoE,axiom,
! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
( ( order_1147259034_ereal @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_166_antimonoE,axiom,
! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
( ( order_2047034162d_enat @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_167_antimonoE,axiom,
! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
( ( order_442688004at_nat @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_168_antimonoE,axiom,
! [F: extended_enat > real,X2: extended_enat,Y3: extended_enat] :
( ( order_182315744t_real @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_169_antimonoE,axiom,
! [F: nat > extended_ereal,X2: nat,Y3: nat] :
( ( order_736687562_ereal @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_170_antimonoE,axiom,
! [F: nat > extended_enat,X2: nat,Y3: nat] :
( ( order_1660553314d_enat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_171_antimonoD,axiom,
! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
( ( order_1408494002_ereal @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_172_antimonoD,axiom,
! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
( ( order_680161354d_enat @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_173_antimonoD,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
( ( order_523806444al_nat @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_174_antimonoD,axiom,
! [F: extended_ereal > real,X2: extended_ereal,Y3: extended_ereal] :
( ( order_1800387528l_real @ F )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_175_antimonoD,axiom,
! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
( ( order_1147259034_ereal @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_176_antimonoD,axiom,
! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
( ( order_2047034162d_enat @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_177_antimonoD,axiom,
! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
( ( order_442688004at_nat @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_178_antimonoD,axiom,
! [F: extended_enat > real,X2: extended_enat,Y3: extended_enat] :
( ( order_182315744t_real @ F )
=> ( ( ord_le1863327750d_enat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_179_antimonoD,axiom,
! [F: nat > extended_ereal,X2: nat,Y3: nat] :
( ( order_736687562_ereal @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_180_antimonoD,axiom,
! [F: nat > extended_enat,X2: nat,Y3: nat] :
( ( order_1660553314d_enat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_181_decseqD,axiom,
! [F: nat > extended_ereal,I: nat,J: nat] :
( ( order_736687562_ereal @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le824540014_ereal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_182_decseqD,axiom,
! [F: nat > extended_enat,I: nat,J: nat] :
( ( order_1660553314d_enat @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le1863327750d_enat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_183_decseqD,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ( order_1631207636at_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_184_decseqD,axiom,
! [F: nat > real,I: nat,J: nat] :
( ( order_106095024t_real @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_185_decseq__def,axiom,
( order_736687562_ereal
= ( ^ [X5: nat > extended_ereal] :
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_le824540014_ereal @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% decseq_def
thf(fact_186_decseq__def,axiom,
( order_1660553314d_enat
= ( ^ [X5: nat > extended_enat] :
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_le1863327750d_enat @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% decseq_def
thf(fact_187_decseq__def,axiom,
( order_1631207636at_nat
= ( ^ [X5: nat > nat] :
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% decseq_def
thf(fact_188_decseq__def,axiom,
( order_106095024t_real
= ( ^ [X5: nat > real] :
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% decseq_def
thf(fact_189_ereal__of__enat__nonneg,axiom,
! [N: extended_enat] : ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extend1771934483f_enat @ N ) ) ).
% ereal_of_enat_nonneg
thf(fact_190_ereal__of__enat__ge__zero__cancel__iff,axiom,
! [N: extended_enat] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extend1771934483f_enat @ N ) )
= ( ord_le1863327750d_enat @ zero_z491942557d_enat @ N ) ) ).
% ereal_of_enat_ge_zero_cancel_iff
thf(fact_191_numeral__le__ereal__of__enat__iff,axiom,
! [M: num,N: extended_enat] :
( ( ord_le824540014_ereal @ ( numera1793320307_ereal @ M ) @ ( extend1771934483f_enat @ N ) )
= ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ N ) ) ).
% numeral_le_ereal_of_enat_iff
thf(fact_192_ereal__infty__less__eq_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ extend1289208545_ereal @ X2 )
= ( X2 = extend1289208545_ereal ) ) ).
% ereal_infty_less_eq(1)
thf(fact_193_strict__mono__less__eq,axiom,
! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
( ( order_555877177_ereal @ F )
=> ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_194_strict__mono__less__eq,axiom,
! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
( ( order_1180956065_ereal @ F )
=> ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_195_strict__mono__less__eq,axiom,
! [F: nat > extended_ereal,X2: nat,Y3: nat] :
( ( order_689807875_ereal @ F )
=> ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_196_strict__mono__less__eq,axiom,
! [F: real > extended_ereal,X2: real,Y3: real] :
( ( order_946214183_ereal @ F )
=> ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_197_strict__mono__less__eq,axiom,
! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
( ( order_713858385d_enat @ F )
=> ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_198_strict__mono__less__eq,axiom,
! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
( ( order_2106278841d_enat @ F )
=> ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_199_strict__mono__less__eq,axiom,
! [F: nat > extended_enat,X2: nat,Y3: nat] :
( ( order_693207323d_enat @ F )
=> ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_200_strict__mono__less__eq,axiom,
! [F: real > extended_enat,X2: real,Y3: real] :
( ( order_1962034751d_enat @ F )
=> ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_201_strict__mono__less__eq,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
( ( order_476926757al_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_202_strict__mono__less__eq,axiom,
! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
( ( order_1622825661at_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_203_ereal__of__enat__zero,axiom,
( ( extend1771934483f_enat @ zero_z491942557d_enat )
= zero_z163181189_ereal ) ).
% ereal_of_enat_zero
thf(fact_204_strict__mono__eq,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( order_769474267at_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% strict_mono_eq
thf(fact_205_Infty__neq__0_I1_J,axiom,
extend1289208545_ereal != zero_z163181189_ereal ).
% Infty_neq_0(1)
thf(fact_206_strict__mono__leD,axiom,
! [R: extended_ereal > extended_ereal,M: extended_ereal,N: extended_ereal] :
( ( order_555877177_ereal @ R )
=> ( ( ord_le824540014_ereal @ M @ N )
=> ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_207_strict__mono__leD,axiom,
! [R: extended_ereal > extended_enat,M: extended_ereal,N: extended_ereal] :
( ( order_713858385d_enat @ R )
=> ( ( ord_le824540014_ereal @ M @ N )
=> ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_208_strict__mono__leD,axiom,
! [R: extended_ereal > nat,M: extended_ereal,N: extended_ereal] :
( ( order_476926757al_nat @ R )
=> ( ( ord_le824540014_ereal @ M @ N )
=> ( ord_less_eq_nat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_209_strict__mono__leD,axiom,
! [R: extended_ereal > real,M: extended_ereal,N: extended_ereal] :
( ( order_1560271745l_real @ R )
=> ( ( ord_le824540014_ereal @ M @ N )
=> ( ord_less_eq_real @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_210_strict__mono__leD,axiom,
! [R: extended_enat > extended_ereal,M: extended_enat,N: extended_enat] :
( ( order_1180956065_ereal @ R )
=> ( ( ord_le1863327750d_enat @ M @ N )
=> ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_211_strict__mono__leD,axiom,
! [R: extended_enat > extended_enat,M: extended_enat,N: extended_enat] :
( ( order_2106278841d_enat @ R )
=> ( ( ord_le1863327750d_enat @ M @ N )
=> ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_212_strict__mono__leD,axiom,
! [R: extended_enat > nat,M: extended_enat,N: extended_enat] :
( ( order_1622825661at_nat @ R )
=> ( ( ord_le1863327750d_enat @ M @ N )
=> ( ord_less_eq_nat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_213_strict__mono__leD,axiom,
! [R: extended_enat > real,M: extended_enat,N: extended_enat] :
( ( order_135436057t_real @ R )
=> ( ( ord_le1863327750d_enat @ M @ N )
=> ( ord_less_eq_real @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_214_strict__mono__leD,axiom,
! [R: nat > extended_ereal,M: nat,N: nat] :
( ( order_689807875_ereal @ R )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_215_strict__mono__leD,axiom,
! [R: nat > extended_enat,M: nat,N: nat] :
( ( order_693207323d_enat @ R )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% strict_mono_leD
thf(fact_216_neq__PInf__trans,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( Y3 != extend1289208545_ereal )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( X2 != extend1289208545_ereal ) ) ) ).
% neq_PInf_trans
thf(fact_217_ereal__infty__less__eq2_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( A = extend1289208545_ereal )
=> ( B = extend1289208545_ereal ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_218_ereal__less__eq_I1_J,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ extend1289208545_ereal ) ).
% ereal_less_eq(1)
thf(fact_219_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( numera280919179d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_220_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_221_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_222_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le1863327750d_enat @ N @ zero_z491942557d_enat )
= ( N = zero_z491942557d_enat ) ) ).
% le_zero_eq
thf(fact_223_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_224_zero__le__numeral,axiom,
! [N: num] : ( ord_le1863327750d_enat @ zero_z491942557d_enat @ ( numera280919179d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_225_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_226_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_227_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le1863327750d_enat @ ( numera280919179d_enat @ N ) @ zero_z491942557d_enat ) ).
% not_numeral_le_zero
thf(fact_228_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_229_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_230_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_231_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le1863327750d_enat @ N @ zero_z491942557d_enat )
= ( N = zero_z491942557d_enat ) ) ).
% ile0_eq
thf(fact_232_i0__lb,axiom,
! [N: extended_enat] : ( ord_le1863327750d_enat @ zero_z491942557d_enat @ N ) ).
% i0_lb
thf(fact_233_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( numera280919179d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_234_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_235_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_236_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_237_real__of__ereal__le__0,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq_real @ ( extend1716541707_ereal @ X2 ) @ zero_zero_real )
= ( ( ord_le824540014_ereal @ X2 @ zero_z163181189_ereal )
| ( X2 = extend1289208545_ereal ) ) ) ).
% real_of_ereal_le_0
thf(fact_238_ereal__uminus__zero__iff,axiom,
! [A: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= zero_z163181189_ereal )
= ( A = zero_z163181189_ereal ) ) ).
% ereal_uminus_zero_iff
thf(fact_239_ereal__uminus__zero,axiom,
( ( uminus1208298309_ereal @ zero_z163181189_ereal )
= zero_z163181189_ereal ) ).
% ereal_uminus_zero
thf(fact_240_ereal__minus__le__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
= ( ord_le824540014_ereal @ B @ A ) ) ).
% ereal_minus_le_minus
thf(fact_241_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_242_real__of__ereal,axiom,
! [X2: extended_ereal] :
( ( extend1716541707_ereal @ ( uminus1208298309_ereal @ X2 ) )
= ( uminus_uminus_real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% real_of_ereal
thf(fact_243_ereal__infty__less__eq_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_244_ereal__uminus__le__0__iff,axiom,
! [A: extended_ereal] :
( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ zero_z163181189_ereal )
= ( ord_le824540014_ereal @ zero_z163181189_ereal @ A ) ) ).
% ereal_uminus_le_0_iff
thf(fact_245_ereal__0__le__uminus__iff,axiom,
! [A: extended_ereal] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A ) )
= ( ord_le824540014_ereal @ A @ zero_z163181189_ereal ) ) ).
% ereal_0_le_uminus_iff
thf(fact_246_real__of__ereal__0,axiom,
( ( extend1716541707_ereal @ zero_z163181189_ereal )
= zero_zero_real ) ).
% real_of_ereal_0
thf(fact_247_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_le1863327750d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_248_real__of__ereal_Osimps_I3_J,axiom,
( ( extend1716541707_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= zero_zero_real ) ).
% real_of_ereal.simps(3)
thf(fact_249_real__of__ereal__eq__0,axiom,
! [X2: extended_ereal] :
( ( ( extend1716541707_ereal @ X2 )
= zero_zero_real )
= ( ( X2 = extend1289208545_ereal )
| ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
| ( X2 = zero_z163181189_ereal ) ) ) ).
% real_of_ereal_eq_0
thf(fact_250_real__of__ereal_Osimps_I2_J,axiom,
( ( extend1716541707_ereal @ extend1289208545_ereal )
= zero_zero_real ) ).
% real_of_ereal.simps(2)
thf(fact_251_zero__enat__def,axiom,
( zero_z491942557d_enat
= ( extended_enat2 @ zero_zero_nat ) ) ).
% zero_enat_def
thf(fact_252_enat__0__iff_I1_J,axiom,
! [X2: nat] :
( ( ( extended_enat2 @ X2 )
= zero_z491942557d_enat )
= ( X2 = zero_zero_nat ) ) ).
% enat_0_iff(1)
thf(fact_253_enat__0__iff_I2_J,axiom,
! [X2: nat] :
( ( zero_z491942557d_enat
= ( extended_enat2 @ X2 ) )
= ( X2 = zero_zero_nat ) ) ).
% enat_0_iff(2)
thf(fact_254_MInfty__neq__PInfty_I1_J,axiom,
( extend1289208545_ereal
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_255_ereal__uminus__le__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ B )
= ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% ereal_uminus_le_reorder
thf(fact_256_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le1863327750d_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_257_numeral__eq__enat,axiom,
( numera280919179d_enat
= ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K2 ) ) ) ) ).
% numeral_eq_enat
thf(fact_258_real__of__ereal__pos,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% real_of_ereal_pos
thf(fact_259_Infty__neq__0_I3_J,axiom,
( ( uminus1208298309_ereal @ extend1289208545_ereal )
!= zero_z163181189_ereal ) ).
% Infty_neq_0(3)
thf(fact_260_ereal__infty__less__eq2_I2_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ A @ B )
=> ( ( B
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( A
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_261_ereal__less__eq_I2_J,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 ) ).
% ereal_less_eq(2)
thf(fact_262_real__of__ereal__positive__mono,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
=> ( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ( Y3 != extend1289208545_ereal )
=> ( ord_less_eq_real @ ( extend1716541707_ereal @ X2 ) @ ( extend1716541707_ereal @ Y3 ) ) ) ) ) ).
% real_of_ereal_positive_mono
thf(fact_263_not__MInfty__nonneg,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
=> ( X2
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% not_MInfty_nonneg
thf(fact_264_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_265_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_266_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_267_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_268_GreatestI__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_269_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_270_Greatest__le__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( ord_less_eq_nat @ K3 @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_271_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_272_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_273_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_769474267at_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_274_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_275_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_276_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_277_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_278_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_279_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_280_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= ( uminus1208298309_ereal @ B ) )
= ( A = B ) ) ).
% ereal_uminus_eq_iff
thf(fact_281_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus1208298309_ereal @ A )
= B )
= ( A
= ( uminus1208298309_ereal @ B ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_282_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_283_ereal__less__eq_I5_J,axiom,
! [R: real] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_eq_real @ zero_zero_real @ R ) ) ).
% ereal_less_eq(5)
thf(fact_284_ereal__less__eq_I4_J,axiom,
! [R: real] :
( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ zero_z163181189_ereal )
= ( ord_less_eq_real @ R @ zero_zero_real ) ) ).
% ereal_less_eq(4)
thf(fact_285_MInfty__eq__minfinity,axiom,
( extended_MInfty
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_eq_minfinity
thf(fact_286_ereal__cong,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ( extended_ereal2 @ X2 )
= ( extended_ereal2 @ Y3 ) ) ) ).
% ereal_cong
thf(fact_287_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_288_numeral__eq__ereal,axiom,
( numera1793320307_ereal
= ( ^ [W: num] : ( extended_ereal2 @ ( numeral_numeral_real @ W ) ) ) ) ).
% numeral_eq_ereal
thf(fact_289_ereal__eq__0_I1_J,axiom,
! [R: real] :
( ( ( extended_ereal2 @ R )
= zero_z163181189_ereal )
= ( R = zero_zero_real ) ) ).
% ereal_eq_0(1)
thf(fact_290_ereal__eq__0_I2_J,axiom,
! [R: real] :
( ( zero_z163181189_ereal
= ( extended_ereal2 @ R ) )
= ( R = zero_zero_real ) ) ).
% ereal_eq_0(2)
thf(fact_291_ereal__less__eq_I3_J,axiom,
! [R: real,P2: real] :
( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( ord_less_eq_real @ R @ P2 ) ) ).
% ereal_less_eq(3)
thf(fact_292_ereal_Odistinct_I3_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_MInfty ) ).
% ereal.distinct(3)
thf(fact_293_real__of__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( extend1716541707_ereal @ ( extended_ereal2 @ R ) )
= R ) ).
% real_of_ereal.simps(1)
thf(fact_294_ereal__le__real,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ! [Z4: real] :
( ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
=> ( ord_le824540014_ereal @ Y3 @ ( extended_ereal2 @ Z4 ) ) )
=> ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% ereal_le_real
thf(fact_295_PInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= extend1289208545_ereal ) ).
% PInfty_neq_ereal(1)
thf(fact_296_zero__ereal__def,axiom,
( zero_z163181189_ereal
= ( extended_ereal2 @ zero_zero_real ) ) ).
% zero_ereal_def
thf(fact_297_le__ereal__le,axiom,
! [A: extended_ereal,X2: real,Y3: real] :
( ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% le_ereal_le
thf(fact_298_ereal__le__le,axiom,
! [Y3: real,A: extended_ereal,X2: real] :
( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y3 ) @ A )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_le_le
thf(fact_299_real__of__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ extend1289208545_ereal )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( P @ A0 ) ) ) ) ).
% real_of_ereal.induct
thf(fact_300_real__of__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extend1289208545_ereal )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% real_of_ereal.cases
thf(fact_301_times__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal )
=> ( ! [R2: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
=> ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% times_ereal.induct
thf(fact_302_plus__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
=> ( ! [X_12: extended_ereal] : ( P @ extend1289208545_ereal @ X_12 )
=> ( ! [A3: extended_ereal] : ( P @ A3 @ extend1289208545_ereal )
=> ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ! [P3: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P3 ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% plus_ereal.induct
thf(fact_303_less__ereal_Oinduct,axiom,
! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
( ! [X3: real,Y: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extended_ereal2 @ Y ) )
=> ( ! [X_12: extended_ereal] : ( P @ extend1289208545_ereal @ X_12 )
=> ( ! [A3: extended_ereal] : ( P @ A3 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) @ extend1289208545_ereal )
=> ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% less_ereal.induct
thf(fact_304_abs__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( P @ extend1289208545_ereal )
=> ( P @ A0 ) ) ) ) ).
% abs_ereal.induct
thf(fact_305_ereal__all__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
! [X6: extended_ereal] : ( P4 @ X6 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1289208545_ereal )
& ! [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
& ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% ereal_all_split
thf(fact_306_abs__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( X2 = extend1289208545_ereal ) ) ) ).
% abs_ereal.cases
thf(fact_307_ereal__ex__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
? [X6: extended_ereal] : ( P4 @ X6 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1289208545_ereal )
| ? [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
| ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% ereal_ex_split
thf(fact_308_ereal3__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ! [Rb: real] :
( Xb
!= ( extended_ereal2 @ Rb ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3 = extend1289208545_ereal )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ? [R2: real] :
( Xa3
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb != extend1289208545_ereal ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3 = extend1289208545_ereal )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb != extend1289208545_ereal ) ) )
=> ~ ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( Xa3
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xb
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_309_ereal2__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xa3
!= ( extended_ereal2 @ Ra ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3 != extend1289208545_ereal ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa3
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( Xa3 != extend1289208545_ereal ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( Xa3
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ! [R2: real] :
( Xa3
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xa3 != extend1289208545_ereal ) )
=> ~ ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Xa3
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_310_ereal__cases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extend1289208545_ereal )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% ereal_cases
thf(fact_311_MInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% MInfty_neq_ereal(1)
thf(fact_312_ereal__top,axiom,
! [X2: extended_ereal] :
( ! [B4: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B4 ) @ X2 )
=> ( X2 = extend1289208545_ereal ) ) ).
% ereal_top
thf(fact_313_uminus__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( uminus1208298309_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_314_ereal__bot,axiom,
! [X2: extended_ereal] :
( ! [B4: real] : ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ B4 ) )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% ereal_bot
thf(fact_315_real__of__ereal_Oelims,axiom,
! [X2: extended_ereal,Y3: real] :
( ( ( extend1716541707_ereal @ X2 )
= Y3 )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y3 != R2 ) )
=> ( ( ( X2 = extend1289208545_ereal )
=> ( Y3 != zero_zero_real ) )
=> ~ ( ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( Y3 != zero_zero_real ) ) ) ) ) ).
% real_of_ereal.elims
thf(fact_316_ereal_Osize__gen_I3_J,axiom,
( ( extended_size_ereal @ extended_MInfty )
= zero_zero_nat ) ).
% ereal.size_gen(3)
thf(fact_317_ereal_Osize__gen_I1_J,axiom,
! [X1: real] :
( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
= zero_zero_nat ) ).
% ereal.size_gen(1)
thf(fact_318_uminus__ereal_Oelims,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ( uminus1208298309_ereal @ X2 )
= Y3 )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y3
!= ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) )
=> ( ( ( X2 = extended_PInfty )
=> ( Y3 != extended_MInfty ) )
=> ~ ( ( X2 = extended_MInfty )
=> ( Y3 != extended_PInfty ) ) ) ) ) ).
% uminus_ereal.elims
thf(fact_319_ereal_Odistinct_I1_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_PInfty ) ).
% ereal.distinct(1)
thf(fact_320_ereal_Osize__gen_I2_J,axiom,
( ( extended_size_ereal @ extended_PInfty )
= zero_zero_nat ) ).
% ereal.size_gen(2)
thf(fact_321_infinity__ereal__def,axiom,
extend1289208545_ereal = extended_PInfty ).
% infinity_ereal_def
thf(fact_322_ereal_Odistinct_I5_J,axiom,
extended_PInfty != extended_MInfty ).
% ereal.distinct(5)
thf(fact_323_uminus__ereal_Oinduct,axiom,
! [P: extended_ereal > $o,A0: extended_ereal] :
( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
=> ( ( P @ extended_PInfty )
=> ( ( P @ extended_MInfty )
=> ( P @ A0 ) ) ) ) ).
% uminus_ereal.induct
thf(fact_324_uminus__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extended_PInfty )
=> ( X2 = extended_MInfty ) ) ) ).
% uminus_ereal.cases
thf(fact_325_ereal_Oexhaust,axiom,
! [Y3: extended_ereal] :
( ! [X12: real] :
( Y3
!= ( extended_ereal2 @ X12 ) )
=> ( ( Y3 != extended_PInfty )
=> ( Y3 = extended_MInfty ) ) ) ).
% ereal.exhaust
thf(fact_326_ereal_Oinduct,axiom,
! [P: extended_ereal > $o,Ereal: extended_ereal] :
( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) )
=> ( ( P @ extended_PInfty )
=> ( ( P @ extended_MInfty )
=> ( P @ Ereal ) ) ) ) ).
% ereal.induct
thf(fact_327_uminus__ereal_Osimps_I2_J,axiom,
( ( uminus1208298309_ereal @ extended_PInfty )
= extended_MInfty ) ).
% uminus_ereal.simps(2)
thf(fact_328_uminus__ereal_Osimps_I3_J,axiom,
( ( uminus1208298309_ereal @ extended_MInfty )
= extended_PInfty ) ).
% uminus_ereal.simps(3)
thf(fact_329_ereal__divide__ereal,axiom,
! [R: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
= extend1289208545_ereal ) )
& ( ~ ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% ereal_divide_ereal
thf(fact_330_ereal__le__real__iff,axiom,
! [X2: real,Y3: extended_ereal] :
( ( ord_less_eq_real @ X2 @ ( extend1716541707_ereal @ Y3 ) )
= ( ( ( ( abs_ab1260901297_ereal @ Y3 )
!= extend1289208545_ereal )
=> ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ Y3 ) )
& ( ( ( abs_ab1260901297_ereal @ Y3 )
= extend1289208545_ereal )
=> ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ) ).
% ereal_le_real_iff
thf(fact_331_abs__ereal__uminus,axiom,
! [X2: extended_ereal] :
( ( abs_ab1260901297_ereal @ ( uminus1208298309_ereal @ X2 ) )
= ( abs_ab1260901297_ereal @ X2 ) ) ).
% abs_ereal_uminus
thf(fact_332_abs__ereal__zero,axiom,
( ( abs_ab1260901297_ereal @ zero_z163181189_ereal )
= zero_z163181189_ereal ) ).
% abs_ereal_zero
thf(fact_333_ereal__uminus__divide,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y3 )
= ( uminus1208298309_ereal @ ( divide595620860_ereal @ X2 @ Y3 ) ) ) ).
% ereal_uminus_divide
thf(fact_334_ereal__divide__zero__left,axiom,
! [A: extended_ereal] :
( ( divide595620860_ereal @ zero_z163181189_ereal @ A )
= zero_z163181189_ereal ) ).
% ereal_divide_zero_left
thf(fact_335_abs__ereal__ge0,axiom,
! [X2: extended_ereal] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
=> ( ( abs_ab1260901297_ereal @ X2 )
= X2 ) ) ).
% abs_ereal_ge0
thf(fact_336_ereal__divide__Infty_I1_J,axiom,
! [X2: extended_ereal] :
( ( divide595620860_ereal @ X2 @ extend1289208545_ereal )
= zero_z163181189_ereal ) ).
% ereal_divide_Infty(1)
thf(fact_337_ereal__divide__Infty_I2_J,axiom,
! [X2: extended_ereal] :
( ( divide595620860_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= zero_z163181189_ereal ) ).
% ereal_divide_Infty(2)
thf(fact_338_not__infty__ereal,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab1260901297_ereal @ X2 )
!= extend1289208545_ereal )
= ( ? [X7: real] :
( X2
= ( extended_ereal2 @ X7 ) ) ) ) ).
% not_infty_ereal
thf(fact_339_abs__neq__infinity__cases,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab1260901297_ereal @ X2 )
!= extend1289208545_ereal )
=> ~ ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) ) ) ).
% abs_neq_infinity_cases
thf(fact_340_abs__ereal_Osimps_I2_J,axiom,
( ( abs_ab1260901297_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
= extend1289208545_ereal ) ).
% abs_ereal.simps(2)
thf(fact_341_ereal__infinity__cases,axiom,
! [A: extended_ereal] :
( ( A != extend1289208545_ereal )
=> ( ( A
!= ( uminus1208298309_ereal @ extend1289208545_ereal ) )
=> ( ( abs_ab1260901297_ereal @ A )
!= extend1289208545_ereal ) ) ) ).
% ereal_infinity_cases
thf(fact_342_abs__eq__infinity__cases,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab1260901297_ereal @ X2 )
= extend1289208545_ereal )
=> ( ( X2 != extend1289208545_ereal )
=> ( X2
= ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% abs_eq_infinity_cases
thf(fact_343_ereal__abs__leI,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_le824540014_ereal @ X2 @ Y3 )
=> ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y3 )
=> ( ord_le824540014_ereal @ ( abs_ab1260901297_ereal @ X2 ) @ Y3 ) ) ) ).
% ereal_abs_leI
thf(fact_344_abs__ereal__pos,axiom,
! [X2: extended_ereal] : ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( abs_ab1260901297_ereal @ X2 ) ) ).
% abs_ereal_pos
thf(fact_345_zero__le__divide__ereal,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le824540014_ereal @ zero_z163181189_ereal @ A )
=> ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ B )
=> ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( divide595620860_ereal @ A @ B ) ) ) ) ).
% zero_le_divide_ereal
thf(fact_346_ereal__divide__eq__0__iff,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ( divide595620860_ereal @ X2 @ Y3 )
= zero_z163181189_ereal )
= ( ( X2 = zero_z163181189_ereal )
| ( ( abs_ab1260901297_ereal @ Y3 )
= extend1289208545_ereal ) ) ) ).
% ereal_divide_eq_0_iff
thf(fact_347_abs__ereal_Osimps_I3_J,axiom,
( ( abs_ab1260901297_ereal @ extend1289208545_ereal )
= extend1289208545_ereal ) ).
% abs_ereal.simps(3)
thf(fact_348_ereal__real_H,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab1260901297_ereal @ X2 )
!= extend1289208545_ereal )
=> ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
= X2 ) ) ).
% ereal_real'
thf(fact_349_ereal__real,axiom,
! [X2: extended_ereal] :
( ( ( ( abs_ab1260901297_ereal @ X2 )
= extend1289208545_ereal )
=> ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
= zero_z163181189_ereal ) )
& ( ( ( abs_ab1260901297_ereal @ X2 )
!= extend1289208545_ereal )
=> ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
= X2 ) ) ) ).
% ereal_real
thf(fact_350_real__le__ereal__iff,axiom,
! [Y3: extended_ereal,X2: real] :
( ( ord_less_eq_real @ ( extend1716541707_ereal @ Y3 ) @ X2 )
= ( ( ( ( abs_ab1260901297_ereal @ Y3 )
!= extend1289208545_ereal )
=> ( ord_le824540014_ereal @ Y3 @ ( extended_ereal2 @ X2 ) ) )
& ( ( ( abs_ab1260901297_ereal @ Y3 )
= extend1289208545_ereal )
=> ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ) ) ).
% real_le_ereal_iff
% Conjectures (1)
thf(conj_0,conjecture,
ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) ).
%------------------------------------------------------------------------------