TPTP Problem File: SLH0818^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00128_004667__11770484_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1167 ( 438 unt; 61 typ; 0 def)
% Number of atoms : 3296 ( 988 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8789 ( 361 ~; 114 |; 167 &;6702 @)
% ( 0 <=>;1445 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 757 ( 757 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 11 con; 0-2 aty)
% Number of variables : 2997 ( 238 ^;2673 !; 86 ?;2997 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:57:09.358
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
option_list_o: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (56)
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
extend1530274965995635425_ereal: extended_ereal ).
thf(sy_c_Groups_Oabel__semigroup_001t__Extended____Real__Oereal,type,
abel_s1811588628328619620_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Groups_Oabel__semigroup_001t__Nat__Onat,type,
abel_semigroup_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Oabel__semigroup__axioms_001t__Extended____Real__Oereal,type,
abel_s6691961667147211777_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Groups_Oabel__semigroup__axioms_001t__Nat__Onat,type,
abel_s2057502115565749341ms_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Extended____Real__Oereal,type,
abs_ab7465543570706387889_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ocomm__monoid_001t__Extended____Real__Oereal,type,
comm_m179930729793757699_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > extended_ereal > $o ).
thf(sy_c_Groups_Ocomm__monoid_001t__Nat__Onat,type,
comm_monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Ocomm__monoid__axioms_001t__Extended____Real__Oereal,type,
comm_m6066551062245714336_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > extended_ereal > $o ).
thf(sy_c_Groups_Ocomm__monoid__axioms_001t__Nat__Onat,type,
comm_m7489229083478665854ms_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Omonoid_001t__Extended____Real__Oereal,type,
monoid425827787017695951_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > extended_ereal > $o ).
thf(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Omonoid__axioms_001t__Extended____Real__Oereal,type,
monoid5257067646426653292_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > extended_ereal > $o ).
thf(sy_c_Groups_Omonoid__axioms_001t__Nat__Onat,type,
monoid_axioms_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Real__Oereal,type,
one_on4623092294121504201_ereal: extended_ereal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p7876563987511257093_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Osemigroup_001t__Extended____Real__Oereal,type,
semigr3155017398426397084_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Groups_Osemigroup_001t__Nat__Onat,type,
semigroup_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
times_7703590493115627913_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus27091377158695749_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Real__Oereal,type,
numera1204434989813589363_ereal: num > extended_ereal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Option_Ooption_ONone_001t__List__Olist_I_Eo_J,type,
none_list_o: option_list_o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mt__Extended____Real__Oereal_J,type,
ord_le5465781672467912687_ereal: ( $o > extended_ereal ) > ( $o > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mt__Num__Onum_J,type,
ord_less_o_num: ( $o > num ) > ( $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Real__Oereal,type,
ord_le1188267648640031866_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Extended____Real__Oereal_J_J,type,
ord_le8520952781634593988_ereal: ( $o > $o > extended_ereal ) > ( $o > $o > extended_ereal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_less_eq_o_o_nat: ( $o > $o > nat ) > ( $o > $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Num__Onum_J_J,type,
ord_less_eq_o_o_num: ( $o > $o > num ) > ( $o > $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Real__Oereal_J,type,
ord_le318542340408939003_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__Num__Onum_J,type,
ord_less_eq_o_num: ( $o > num ) > ( $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_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_Prefix__Free__Code__Combinators_Obit__count,type,
prefix3213528784805800034_count: option_list_o > extended_ereal ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__append,type,
prefix5314359684614007693append: option_list_o > option_list_o > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__comp_001_Eo,type,
prefix454693708527911765comp_o: option_list_o > option_list_o > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__prefix_001_Eo,type,
prefix8824957607401505554efix_o: option_list_o > option_list_o > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
divide8893690120176169980_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_v_x,type,
x: option_list_o ).
thf(sy_v_y,type,
y: option_list_o ).
% Relevant facts (1105)
thf(fact_0_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_2_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ C )
= ( plus_p7876563987511257093_ereal @ A @ ( plus_p7876563987511257093_ereal @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_3_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_4_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p7876563987511257093_ereal @ I @ K )
= ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_5_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_6_group__cancel_Oadd1,axiom,
! [A2: extended_ereal,K: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( A2
= ( plus_p7876563987511257093_ereal @ K @ A ) )
=> ( ( plus_p7876563987511257093_ereal @ A2 @ B )
= ( plus_p7876563987511257093_ereal @ K @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_7_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_8_group__cancel_Oadd2,axiom,
! [B2: extended_ereal,K: extended_ereal,B: extended_ereal,A: extended_ereal] :
( ( B2
= ( plus_p7876563987511257093_ereal @ K @ B ) )
=> ( ( plus_p7876563987511257093_ereal @ A @ B2 )
= ( plus_p7876563987511257093_ereal @ K @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_9_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_10_add_Oassoc,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ C )
= ( plus_p7876563987511257093_ereal @ A @ ( plus_p7876563987511257093_ereal @ B @ C ) ) ) ).
% add.assoc
thf(fact_11_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_12_add_Ocommute,axiom,
( plus_p7876563987511257093_ereal
= ( ^ [A3: extended_ereal,B3: extended_ereal] : ( plus_p7876563987511257093_ereal @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_13_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_14_add_Oleft__commute,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ B @ ( plus_p7876563987511257093_ereal @ A @ C ) )
= ( plus_p7876563987511257093_ereal @ A @ ( plus_p7876563987511257093_ereal @ B @ C ) ) ) ).
% add.left_commute
thf(fact_15_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_16_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_17_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_18_opt__comp__append__3,axiom,
! [X: option_list_o,Y: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X @ Y ) @ ( prefix5314359684614007693append @ X @ Z ) )
=> ( prefix454693708527911765comp_o @ Y @ Z ) ) ).
% opt_comp_append_3
thf(fact_19_opt__comp__append__2,axiom,
! [X: option_list_o,Y: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ X @ ( prefix5314359684614007693append @ Y @ Z ) )
=> ( prefix454693708527911765comp_o @ X @ Y ) ) ).
% opt_comp_append_2
thf(fact_20_opt__comp__append,axiom,
! [X: option_list_o,Y: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X @ Y ) @ Z )
=> ( prefix454693708527911765comp_o @ X @ Z ) ) ).
% opt_comp_append
thf(fact_21_opt__append_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
( ( prefix5314359684614007693append @ none_list_o @ Uv )
= none_list_o ) ).
% opt_append.simps(2)
thf(fact_22_opt__append_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
( ( prefix5314359684614007693append @ Uu @ none_list_o )
= none_list_o ) ).
% opt_append.simps(3)
thf(fact_23_add__0,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal @ A )
= A ) ).
% add_0
thf(fact_24_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_25_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_26_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_27_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_28_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_29_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_30_add_Oright__neutral,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ zero_z2744965634713055877_ereal )
= A ) ).
% add.right_neutral
thf(fact_31_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_32_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_33_zero__reorient,axiom,
! [X: extended_ereal] :
( ( zero_z2744965634713055877_ereal = X )
= ( X = zero_z2744965634713055877_ereal ) ) ).
% zero_reorient
thf(fact_34_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_35_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X2: option_list_o,Y2: option_list_o] : ( prefix454693708527911765comp_o @ Y2 @ X2 ) ) ) ).
% opt_comp_sym
thf(fact_36_comm__monoid__add__class_Oadd__0,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_37_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_38_add_Ocomm__neutral,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ zero_z2744965634713055877_ereal )
= A ) ).
% add.comm_neutral
thf(fact_39_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_40_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_41_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_42_bit__count_Osimps_I1_J,axiom,
( ( prefix3213528784805800034_count @ none_list_o )
= extend1530274965995635425_ereal ) ).
% bit_count.simps(1)
thf(fact_43_add_Omonoid__axioms,axiom,
monoid425827787017695951_ereal @ plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal ).
% add.monoid_axioms
thf(fact_44_add_Omonoid__axioms,axiom,
monoid_nat @ plus_plus_nat @ zero_zero_nat ).
% add.monoid_axioms
thf(fact_45_opt__comp__def,axiom,
( prefix454693708527911765comp_o
= ( ^ [X2: option_list_o,Y2: option_list_o] :
( ( prefix8824957607401505554efix_o @ X2 @ Y2 )
| ( prefix8824957607401505554efix_o @ Y2 @ X2 ) ) ) ) ).
% opt_comp_def
thf(fact_46_add_Ocomm__monoid__axioms,axiom,
comm_m179930729793757699_ereal @ plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal ).
% add.comm_monoid_axioms
thf(fact_47_add_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ plus_plus_nat @ zero_zero_nat ).
% add.comm_monoid_axioms
thf(fact_48_comm__monoid_Ocomm__neutral,axiom,
! [F: nat > nat > nat,Z: nat,A: nat] :
( ( comm_monoid_nat @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% comm_monoid.comm_neutral
thf(fact_49_comm__monoid_Ocomm__neutral,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal,A: extended_ereal] :
( ( comm_m179930729793757699_ereal @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% comm_monoid.comm_neutral
thf(fact_50_monoid_Oright__neutral,axiom,
! [F: nat > nat > nat,Z: nat,A: nat] :
( ( monoid_nat @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_51_monoid_Oright__neutral,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal,A: extended_ereal] :
( ( monoid425827787017695951_ereal @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_52_monoid_Oleft__neutral,axiom,
! [F: nat > nat > nat,Z: nat,A: nat] :
( ( monoid_nat @ F @ Z )
=> ( ( F @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_53_monoid_Oleft__neutral,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal,A: extended_ereal] :
( ( monoid425827787017695951_ereal @ F @ Z )
=> ( ( F @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_54_opt__prefix_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
~ ( prefix8824957607401505554efix_o @ Uu @ none_list_o ) ).
% opt_prefix.simps(3)
thf(fact_55_opt__prefix_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
~ ( prefix8824957607401505554efix_o @ none_list_o @ Uv ) ).
% opt_prefix.simps(2)
thf(fact_56_ereal__plus__eq__PInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ A @ B )
= extend1530274965995635425_ereal )
= ( ( A = extend1530274965995635425_ereal )
| ( B = extend1530274965995635425_ereal ) ) ) ).
% ereal_plus_eq_PInfty
thf(fact_57_ereal__PInfty__eq__plus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( extend1530274965995635425_ereal
= ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B = extend1530274965995635425_ereal ) ) ) ).
% ereal_PInfty_eq_plus
thf(fact_58_plus__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ extend1530274965995635425_ereal @ A )
= extend1530274965995635425_ereal ) ).
% plus_ereal.simps(2)
thf(fact_59_plus__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) ).
% plus_ereal.simps(3)
thf(fact_60_monoid_Oaxioms_I2_J,axiom,
! [F: nat > nat > nat,Z: nat] :
( ( monoid_nat @ F @ Z )
=> ( monoid_axioms_nat @ F @ Z ) ) ).
% monoid.axioms(2)
thf(fact_61_monoid_Oaxioms_I2_J,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal] :
( ( monoid425827787017695951_ereal @ F @ Z )
=> ( monoid5257067646426653292_ereal @ F @ Z ) ) ).
% monoid.axioms(2)
thf(fact_62_comm__monoid_Oaxioms_I2_J,axiom,
! [F: nat > nat > nat,Z: nat] :
( ( comm_monoid_nat @ F @ Z )
=> ( comm_m7489229083478665854ms_nat @ F @ Z ) ) ).
% comm_monoid.axioms(2)
thf(fact_63_comm__monoid_Oaxioms_I2_J,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal] :
( ( comm_m179930729793757699_ereal @ F @ Z )
=> ( comm_m6066551062245714336_ereal @ F @ Z ) ) ).
% comm_monoid.axioms(2)
thf(fact_64_Infty__neq__0_I1_J,axiom,
extend1530274965995635425_ereal != zero_z2744965634713055877_ereal ).
% Infty_neq_0(1)
thf(fact_65_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_66_ereal__infty__less__eq_I1_J,axiom,
! [X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ extend1530274965995635425_ereal @ X )
= ( X = extend1530274965995635425_ereal ) ) ).
% ereal_infty_less_eq(1)
thf(fact_67_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_68_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_69_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_70_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_71_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_72_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_73_verit__comp__simplify1_I2_J,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_74_verit__comp__simplify1_I2_J,axiom,
! [A: $o > num] : ( ord_less_eq_o_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_75_verit__comp__simplify1_I2_J,axiom,
! [A: $o > extended_ereal] : ( ord_le318542340408939003_ereal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_76_verit__comp__simplify1_I2_J,axiom,
! [A: extended_ereal] : ( ord_le1083603963089353582_ereal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_77_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_78_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_79_verit__la__disequality,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( A = B )
| ~ ( ord_le1083603963089353582_ereal @ A @ B )
| ~ ( ord_le1083603963089353582_ereal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_80_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_81_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_82_neq__PInf__trans,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( Y != extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( X != extend1530274965995635425_ereal ) ) ) ).
% neq_PInf_trans
thf(fact_83_ereal__infty__less__eq2_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( A = extend1530274965995635425_ereal )
=> ( B = extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_84_ereal__less__eq_I1_J,axiom,
! [X: extended_ereal] : ( ord_le1083603963089353582_ereal @ X @ extend1530274965995635425_ereal ) ).
% ereal_less_eq(1)
thf(fact_85_ereal__le__add__self,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ X @ ( plus_p7876563987511257093_ereal @ X @ Y ) ) ) ).
% ereal_le_add_self
thf(fact_86_ereal__le__add__mono1,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Z )
=> ( ord_le1083603963089353582_ereal @ X @ ( plus_p7876563987511257093_ereal @ Y @ Z ) ) ) ) ).
% ereal_le_add_mono1
thf(fact_87_ereal__le__add__mono2,axiom,
! [X: extended_ereal,Z: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Z )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ X @ ( plus_p7876563987511257093_ereal @ Y @ Z ) ) ) ) ).
% ereal_le_add_mono2
thf(fact_88_ereal__le__add__self2,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ X @ ( plus_p7876563987511257093_ereal @ Y @ X ) ) ) ).
% ereal_le_add_self2
thf(fact_89_ereal__add__nonneg__eq__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ( plus_p7876563987511257093_ereal @ A @ B )
= zero_z2744965634713055877_ereal )
= ( ( A = zero_z2744965634713055877_ereal )
& ( B = zero_z2744965634713055877_ereal ) ) ) ) ) ).
% ereal_add_nonneg_eq_0_iff
thf(fact_90_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_91_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_92_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ I @ J )
& ( K = L ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ I @ K ) @ ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_93_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_94_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( I = J )
& ( ord_le1083603963089353582_ereal @ K @ L ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ I @ K ) @ ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_95_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_96_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ I @ J )
& ( ord_le1083603963089353582_ereal @ K @ L ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ I @ K ) @ ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_97_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_98_add__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ C @ D )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ).
% add_mono
thf(fact_99_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_100_add__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ C @ A ) @ ( plus_p7876563987511257093_ereal @ C @ B ) ) ) ).
% add_left_mono
thf(fact_101_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_102_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_103_add__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ ( plus_p7876563987511257093_ereal @ B @ C ) ) ) ).
% add_right_mono
thf(fact_104_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_105_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_106_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_107_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_108_add__nonpos__eq__0__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1083603963089353582_ereal @ Y @ zero_z2744965634713055877_ereal )
=> ( ( ( plus_p7876563987511257093_ereal @ X @ Y )
= zero_z2744965634713055877_ereal )
= ( ( X = zero_z2744965634713055877_ereal )
& ( Y = zero_z2744965634713055877_ereal ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_109_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_110_add__nonneg__eq__0__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ( ( plus_p7876563987511257093_ereal @ X @ Y )
= zero_z2744965634713055877_ereal )
= ( ( X = zero_z2744965634713055877_ereal )
& ( Y = zero_z2744965634713055877_ereal ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_111_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_112_add__nonpos__nonpos,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ zero_z2744965634713055877_ereal ) ) ) ).
% add_nonpos_nonpos
thf(fact_113_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_114_add__nonneg__nonneg,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_115_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_116_add__increasing2,axiom,
! [C: extended_ereal,B: extended_ereal,A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1083603963089353582_ereal @ B @ A )
=> ( ord_le1083603963089353582_ereal @ B @ ( plus_p7876563987511257093_ereal @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_117_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_118_add__decreasing2,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ C @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_119_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_120_add__increasing,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ord_le1083603963089353582_ereal @ B @ ( plus_p7876563987511257093_ereal @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_121_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_122_add__decreasing,axiom,
! [A: extended_ereal,C: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1083603963089353582_ereal @ C @ B )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_123_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_124_dual__order_Orefl,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% dual_order.refl
thf(fact_125_dual__order_Orefl,axiom,
! [A: $o > num] : ( ord_less_eq_o_num @ A @ A ) ).
% dual_order.refl
thf(fact_126_dual__order_Orefl,axiom,
! [A: $o > extended_ereal] : ( ord_le318542340408939003_ereal @ A @ A ) ).
% dual_order.refl
thf(fact_127_dual__order_Orefl,axiom,
! [A: extended_ereal] : ( ord_le1083603963089353582_ereal @ A @ A ) ).
% dual_order.refl
thf(fact_128_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_129_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_130_order__refl,axiom,
! [X: $o > nat] : ( ord_less_eq_o_nat @ X @ X ) ).
% order_refl
thf(fact_131_order__refl,axiom,
! [X: $o > num] : ( ord_less_eq_o_num @ X @ X ) ).
% order_refl
thf(fact_132_order__refl,axiom,
! [X: $o > extended_ereal] : ( ord_le318542340408939003_ereal @ X @ X ) ).
% order_refl
thf(fact_133_order__refl,axiom,
! [X: extended_ereal] : ( ord_le1083603963089353582_ereal @ X @ X ) ).
% order_refl
thf(fact_134_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_135_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_136_monoid_Ointro,axiom,
! [F: nat > nat > nat,Z: nat] :
( ( semigroup_nat @ F )
=> ( ( monoid_axioms_nat @ F @ Z )
=> ( monoid_nat @ F @ Z ) ) ) ).
% monoid.intro
thf(fact_137_monoid_Ointro,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal] :
( ( semigr3155017398426397084_ereal @ F )
=> ( ( monoid5257067646426653292_ereal @ F @ Z )
=> ( monoid425827787017695951_ereal @ F @ Z ) ) ) ).
% monoid.intro
thf(fact_138_monoid__def,axiom,
( monoid_nat
= ( ^ [F2: nat > nat > nat,Z2: nat] :
( ( semigroup_nat @ F2 )
& ( monoid_axioms_nat @ F2 @ Z2 ) ) ) ) ).
% monoid_def
thf(fact_139_monoid__def,axiom,
( monoid425827787017695951_ereal
= ( ^ [F2: extended_ereal > extended_ereal > extended_ereal,Z2: extended_ereal] :
( ( semigr3155017398426397084_ereal @ F2 )
& ( monoid5257067646426653292_ereal @ F2 @ Z2 ) ) ) ) ).
% monoid_def
thf(fact_140_comm__monoid_Ointro,axiom,
! [F: nat > nat > nat,Z: nat] :
( ( abel_semigroup_nat @ F )
=> ( ( comm_m7489229083478665854ms_nat @ F @ Z )
=> ( comm_monoid_nat @ F @ Z ) ) ) ).
% comm_monoid.intro
thf(fact_141_comm__monoid_Ointro,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal] :
( ( abel_s1811588628328619620_ereal @ F )
=> ( ( comm_m6066551062245714336_ereal @ F @ Z )
=> ( comm_m179930729793757699_ereal @ F @ Z ) ) ) ).
% comm_monoid.intro
thf(fact_142_comm__monoid__def,axiom,
( comm_monoid_nat
= ( ^ [F2: nat > nat > nat,Z2: nat] :
( ( abel_semigroup_nat @ F2 )
& ( comm_m7489229083478665854ms_nat @ F2 @ Z2 ) ) ) ) ).
% comm_monoid_def
thf(fact_143_comm__monoid__def,axiom,
( comm_m179930729793757699_ereal
= ( ^ [F2: extended_ereal > extended_ereal > extended_ereal,Z2: extended_ereal] :
( ( abel_s1811588628328619620_ereal @ F2 )
& ( comm_m6066551062245714336_ereal @ F2 @ Z2 ) ) ) ) ).
% comm_monoid_def
thf(fact_144_ereal__add__strict__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( A != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ C @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_add_strict_mono
thf(fact_145_ereal__pos__distrib,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( C != extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ C @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C @ A ) @ ( times_7703590493115627913_ereal @ C @ B ) ) ) ) ) ).
% ereal_pos_distrib
thf(fact_146_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X3: $o > $o > nat,Y3: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X3 @ $false ) @ ( Y3 @ $false ) )
& ( ord_less_eq_o_nat @ ( X3 @ $true ) @ ( Y3 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_147_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_num
= ( ^ [X3: $o > $o > num,Y3: $o > $o > num] :
( ( ord_less_eq_o_num @ ( X3 @ $false ) @ ( Y3 @ $false ) )
& ( ord_less_eq_o_num @ ( X3 @ $true ) @ ( Y3 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_148_le__rel__bool__arg__iff,axiom,
( ord_le8520952781634593988_ereal
= ( ^ [X3: $o > $o > extended_ereal,Y3: $o > $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ ( X3 @ $false ) @ ( Y3 @ $false ) )
& ( ord_le318542340408939003_ereal @ ( X3 @ $true ) @ ( Y3 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_149_le__rel__bool__arg__iff,axiom,
( ord_le318542340408939003_ereal
= ( ^ [X3: $o > extended_ereal,Y3: $o > extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( X3 @ $false ) @ ( Y3 @ $false ) )
& ( ord_le1083603963089353582_ereal @ ( X3 @ $true ) @ ( Y3 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_150_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_num
= ( ^ [X3: $o > num,Y3: $o > num] :
( ( ord_less_eq_num @ ( X3 @ $false ) @ ( Y3 @ $false ) )
& ( ord_less_eq_num @ ( X3 @ $true ) @ ( Y3 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_151_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X3: $o > nat,Y3: $o > nat] :
( ( ord_less_eq_nat @ ( X3 @ $false ) @ ( Y3 @ $false ) )
& ( ord_less_eq_nat @ ( X3 @ $true ) @ ( Y3 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_152_order__antisym__conv,axiom,
! [Y: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X )
=> ( ( ord_less_eq_o_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_153_order__antisym__conv,axiom,
! [Y: $o > num,X: $o > num] :
( ( ord_less_eq_o_num @ Y @ X )
=> ( ( ord_less_eq_o_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_154_order__antisym__conv,axiom,
! [Y: $o > extended_ereal,X: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ Y @ X )
=> ( ( ord_le318542340408939003_ereal @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_155_order__antisym__conv,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( ( ord_le1083603963089353582_ereal @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_156_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_157_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_158_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_159_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_160_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_161_ereal__infty__less_I1_J,axiom,
! [X: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ extend1530274965995635425_ereal )
= ( X != extend1530274965995635425_ereal ) ) ).
% ereal_infty_less(1)
thf(fact_162_ereal__less__PInfty,axiom,
! [A: extended_ereal] :
( ( A != extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ A @ extend1530274965995635425_ereal ) ) ).
% ereal_less_PInfty
thf(fact_163_ereal__zero__times,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= zero_z2744965634713055877_ereal )
= ( ( A = zero_z2744965634713055877_ereal )
| ( B = zero_z2744965634713055877_ereal ) ) ) ).
% ereal_zero_times
thf(fact_164_ereal__zero__mult,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ zero_z2744965634713055877_ereal @ A )
= zero_z2744965634713055877_ereal ) ).
% ereal_zero_mult
thf(fact_165_ereal__mult__zero,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_mult_zero
thf(fact_166_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_167_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_168_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_169_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_170_ereal__complete__Sup,axiom,
! [S: set_Extended_ereal] :
? [X4: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
=> ( ord_le1083603963089353582_ereal @ Xa @ X4 ) )
& ! [Z3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S )
=> ( ord_le1083603963089353582_ereal @ Xa2 @ Z3 ) )
=> ( ord_le1083603963089353582_ereal @ X4 @ Z3 ) ) ) ).
% ereal_complete_Sup
thf(fact_171_ereal__complete__Inf,axiom,
! [S: set_Extended_ereal] :
? [X4: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
=> ( ord_le1083603963089353582_ereal @ X4 @ Xa ) )
& ! [Z3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S )
=> ( ord_le1083603963089353582_ereal @ Z3 @ Xa2 ) )
=> ( ord_le1083603963089353582_ereal @ Z3 @ X4 ) ) ) ).
% ereal_complete_Inf
thf(fact_172_less__eq__ereal__def,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X2: extended_ereal,Y2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_ereal_def
thf(fact_173_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_174_dense,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ? [Z4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Z4 )
& ( ord_le1188267648640031866_ereal @ Z4 @ Y ) ) ) ).
% dense
thf(fact_175_less__imp__neq,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_176_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_177_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_178_order_Oasym,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ~ ( ord_le1188267648640031866_ereal @ B @ A ) ) ).
% order.asym
thf(fact_179_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_180_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_181_ord__eq__less__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A = B )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ord_le1188267648640031866_ereal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_182_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_183_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_184_ord__less__eq__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( B = C )
=> ( ord_le1188267648640031866_ereal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_185_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_186_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_187_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_188_antisym__conv3,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ~ ( ord_le1188267648640031866_ereal @ Y @ X )
=> ( ( ~ ( ord_le1188267648640031866_ereal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_189_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_190_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_191_linorder__cases,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ( X != Y )
=> ( ord_le1188267648640031866_ereal @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_192_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_193_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_194_dual__order_Oasym,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ B @ A )
=> ~ ( ord_le1188267648640031866_ereal @ A @ B ) ) ).
% dual_order.asym
thf(fact_195_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_196_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_197_dual__order_Oirrefl,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ A @ A ) ).
% dual_order.irrefl
thf(fact_198_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_199_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_200_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_201_linorder__less__wlog,axiom,
! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
( ! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: extended_ereal] : ( P @ A4 @ A4 )
=> ( ! [A4: extended_ereal,B4: extended_ereal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_202_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_203_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_204_order_Ostrict__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ord_le1188267648640031866_ereal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_205_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_206_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_207_not__less__iff__gr__or__eq,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ~ ( ord_le1188267648640031866_ereal @ X @ Y ) )
= ( ( ord_le1188267648640031866_ereal @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_208_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_209_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_210_dual__order_Ostrict__trans,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ B @ A )
=> ( ( ord_le1188267648640031866_ereal @ C @ B )
=> ( ord_le1188267648640031866_ereal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_211_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_212_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_213_order_Ostrict__implies__not__eq,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_214_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_215_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_216_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_217_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_218_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_219_linorder__neqE,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( X != Y )
=> ( ~ ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ord_le1188267648640031866_ereal @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_220_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_221_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_222_order__less__asym,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ~ ( ord_le1188267648640031866_ereal @ Y @ X ) ) ).
% order_less_asym
thf(fact_223_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_224_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_225_linorder__neq__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( X != Y )
= ( ( ord_le1188267648640031866_ereal @ X @ Y )
| ( ord_le1188267648640031866_ereal @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_226_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_227_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_228_order__less__asym_H,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ~ ( ord_le1188267648640031866_ereal @ B @ A ) ) ).
% order_less_asym'
thf(fact_229_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_230_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_231_order__less__trans,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ( ord_le1188267648640031866_ereal @ Y @ Z )
=> ( ord_le1188267648640031866_ereal @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_232_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_233_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_234_ord__eq__less__subst,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_235_ord__eq__less__subst,axiom,
! [A: num,F: extended_ereal > num,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_236_ord__eq__less__subst,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_237_ord__eq__less__subst,axiom,
! [A: extended_ereal,F: num > extended_ereal,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_238_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_239_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_240_ord__eq__less__subst,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_241_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_242_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_243_ord__less__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_244_ord__less__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > num,C: num] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_245_ord__less__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_246_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > extended_ereal,C: extended_ereal] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_247_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_248_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_249_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_250_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_251_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_252_order__less__irrefl,axiom,
! [X: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ X @ X ) ).
% order_less_irrefl
thf(fact_253_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_254_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_255_order__less__subst1,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_256_order__less__subst1,axiom,
! [A: extended_ereal,F: num > extended_ereal,B: num,C: num] :
( ( ord_le1188267648640031866_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_257_order__less__subst1,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( ord_le1188267648640031866_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_258_order__less__subst1,axiom,
! [A: num,F: extended_ereal > num,B: extended_ereal,C: extended_ereal] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_259_order__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_260_order__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_261_order__less__subst1,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_262_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_263_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_264_order__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_265_order__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > num,C: num] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_266_order__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_267_order__less__subst2,axiom,
! [A: num,B: num,F: num > extended_ereal,C: extended_ereal] :
( ( ord_less_num @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_268_order__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_269_order__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_270_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_271_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_272_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_273_order__less__not__sym,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ~ ( ord_le1188267648640031866_ereal @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_274_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_275_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_276_order__less__imp__triv,axiom,
! [X: extended_ereal,Y: extended_ereal,P: $o] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ( ord_le1188267648640031866_ereal @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_277_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_278_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_279_linorder__less__linear,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
| ( X = Y )
| ( ord_le1188267648640031866_ereal @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_280_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_281_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_282_order__less__imp__not__eq,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_283_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_284_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_285_order__less__imp__not__eq2,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_286_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_287_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_288_order__less__imp__not__less,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ~ ( ord_le1188267648640031866_ereal @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_289_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_290_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_291_semigroup__def,axiom,
( semigroup_nat
= ( ^ [F2: nat > nat > nat] :
! [A3: nat,B3: nat,C3: nat] :
( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
= ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_292_semigroup__def,axiom,
( semigr3155017398426397084_ereal
= ( ^ [F2: extended_ereal > extended_ereal > extended_ereal] :
! [A3: extended_ereal,B3: extended_ereal,C3: extended_ereal] :
( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
= ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_293_mult_Oabel__semigroup__axioms,axiom,
abel_s1811588628328619620_ereal @ times_7703590493115627913_ereal ).
% mult.abel_semigroup_axioms
thf(fact_294_mult_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat @ times_times_nat ).
% mult.abel_semigroup_axioms
thf(fact_295_mult_Osemigroup__axioms,axiom,
semigr3155017398426397084_ereal @ times_7703590493115627913_ereal ).
% mult.semigroup_axioms
thf(fact_296_mult_Osemigroup__axioms,axiom,
semigroup_nat @ times_times_nat ).
% mult.semigroup_axioms
thf(fact_297_mult_Oleft__commute,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( times_7703590493115627913_ereal @ B @ ( times_7703590493115627913_ereal @ A @ C ) )
= ( times_7703590493115627913_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_298_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_299_mult_Ocommute,axiom,
( times_7703590493115627913_ereal
= ( ^ [A3: extended_ereal,B3: extended_ereal] : ( times_7703590493115627913_ereal @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_300_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_301_mult_Oassoc,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ C )
= ( times_7703590493115627913_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_302_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_303_abel__semigroup_Oleft__commute,axiom,
! [F: nat > nat > nat,B: nat,A: nat,C: nat] :
( ( abel_semigroup_nat @ F )
=> ( ( F @ B @ ( F @ A @ C ) )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_304_abel__semigroup_Oleft__commute,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( abel_s1811588628328619620_ereal @ F )
=> ( ( F @ B @ ( F @ A @ C ) )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_305_abel__semigroup_Ocommute,axiom,
! [F: nat > nat > nat,A: nat,B: nat] :
( ( abel_semigroup_nat @ F )
=> ( ( F @ A @ B )
= ( F @ B @ A ) ) ) ).
% abel_semigroup.commute
thf(fact_306_abel__semigroup_Ocommute,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( abel_s1811588628328619620_ereal @ F )
=> ( ( F @ A @ B )
= ( F @ B @ A ) ) ) ).
% abel_semigroup.commute
thf(fact_307_semigroup_Ointro,axiom,
! [F: nat > nat > nat] :
( ! [A4: nat,B4: nat,C2: nat] :
( ( F @ ( F @ A4 @ B4 ) @ C2 )
= ( F @ A4 @ ( F @ B4 @ C2 ) ) )
=> ( semigroup_nat @ F ) ) ).
% semigroup.intro
thf(fact_308_semigroup_Ointro,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal] :
( ! [A4: extended_ereal,B4: extended_ereal,C2: extended_ereal] :
( ( F @ ( F @ A4 @ B4 ) @ C2 )
= ( F @ A4 @ ( F @ B4 @ C2 ) ) )
=> ( semigr3155017398426397084_ereal @ F ) ) ).
% semigroup.intro
thf(fact_309_semigroup_Oassoc,axiom,
! [F: nat > nat > nat,A: nat,B: nat,C: nat] :
( ( semigroup_nat @ F )
=> ( ( F @ ( F @ A @ B ) @ C )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% semigroup.assoc
thf(fact_310_semigroup_Oassoc,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( semigr3155017398426397084_ereal @ F )
=> ( ( F @ ( F @ A @ B ) @ C )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% semigroup.assoc
thf(fact_311_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ C )
= ( times_7703590493115627913_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_312_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_313_abel__semigroup_Oaxioms_I1_J,axiom,
! [F: nat > nat > nat] :
( ( abel_semigroup_nat @ F )
=> ( semigroup_nat @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_314_abel__semigroup_Oaxioms_I1_J,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal] :
( ( abel_s1811588628328619620_ereal @ F )
=> ( semigr3155017398426397084_ereal @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_315_verit__comp__simplify1_I1_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_316_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_317_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_318_ereal__mult__mono__strict_H,axiom,
! [A: extended_ereal,C: extended_ereal,B: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C @ D )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono_strict'
thf(fact_319_ereal__mult__mono__strict,axiom,
! [B: extended_ereal,C: extended_ereal,A: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C @ D )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono_strict
thf(fact_320_ereal__zero__less__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A @ B ) )
= ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_zero_less_0_iff
thf(fact_321_ereal__mult__less__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ zero_z2744965634713055877_ereal )
= ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) ) ) ) ).
% ereal_mult_less_0_iff
thf(fact_322_leD,axiom,
! [Y: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X )
=> ~ ( ord_less_o_nat @ X @ Y ) ) ).
% leD
thf(fact_323_leD,axiom,
! [Y: $o > num,X: $o > num] :
( ( ord_less_eq_o_num @ Y @ X )
=> ~ ( ord_less_o_num @ X @ Y ) ) ).
% leD
thf(fact_324_leD,axiom,
! [Y: $o > extended_ereal,X: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ Y @ X )
=> ~ ( ord_le5465781672467912687_ereal @ X @ Y ) ) ).
% leD
thf(fact_325_leD,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ~ ( ord_le1188267648640031866_ereal @ X @ Y ) ) ).
% leD
thf(fact_326_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_327_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_328_leI,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ord_le1083603963089353582_ereal @ Y @ X ) ) ).
% leI
thf(fact_329_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_330_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_331_nless__le,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ~ ( ord_less_o_nat @ A @ B ) )
= ( ~ ( ord_less_eq_o_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_332_nless__le,axiom,
! [A: $o > num,B: $o > num] :
( ( ~ ( ord_less_o_num @ A @ B ) )
= ( ~ ( ord_less_eq_o_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_333_nless__le,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal] :
( ( ~ ( ord_le5465781672467912687_ereal @ A @ B ) )
= ( ~ ( ord_le318542340408939003_ereal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_334_nless__le,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ~ ( ord_le1188267648640031866_ereal @ A @ B ) )
= ( ~ ( ord_le1083603963089353582_ereal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_335_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_336_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_337_antisym__conv1,axiom,
! [X: $o > nat,Y: $o > nat] :
( ~ ( ord_less_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_338_antisym__conv1,axiom,
! [X: $o > num,Y: $o > num] :
( ~ ( ord_less_o_num @ X @ Y )
=> ( ( ord_less_eq_o_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_339_antisym__conv1,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal] :
( ~ ( ord_le5465781672467912687_ereal @ X @ Y )
=> ( ( ord_le318542340408939003_ereal @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_340_antisym__conv1,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_341_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_342_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_343_antisym__conv2,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ~ ( ord_less_o_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_344_antisym__conv2,axiom,
! [X: $o > num,Y: $o > num] :
( ( ord_less_eq_o_num @ X @ Y )
=> ( ( ~ ( ord_less_o_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_345_antisym__conv2,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X @ Y )
=> ( ( ~ ( ord_le5465781672467912687_ereal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_346_antisym__conv2,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ~ ( ord_le1188267648640031866_ereal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_347_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_348_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_349_dense__ge,axiom,
! [Z: extended_ereal,Y: extended_ereal] :
( ! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z @ X4 )
=> ( ord_le1083603963089353582_ereal @ Y @ X4 ) )
=> ( ord_le1083603963089353582_ereal @ Y @ Z ) ) ).
% dense_ge
thf(fact_350_dense__le,axiom,
! [Y: extended_ereal,Z: extended_ereal] :
( ! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y )
=> ( ord_le1083603963089353582_ereal @ X4 @ Z ) )
=> ( ord_le1083603963089353582_ereal @ Y @ Z ) ) ).
% dense_le
thf(fact_351_less__le__not__le,axiom,
( ord_less_o_nat
= ( ^ [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_o_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_352_less__le__not__le,axiom,
( ord_less_o_num
= ( ^ [X2: $o > num,Y2: $o > num] :
( ( ord_less_eq_o_num @ X2 @ Y2 )
& ~ ( ord_less_eq_o_num @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_353_less__le__not__le,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [X2: $o > extended_ereal,Y2: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X2 @ Y2 )
& ~ ( ord_le318542340408939003_ereal @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_354_less__le__not__le,axiom,
( ord_le1188267648640031866_ereal
= ( ^ [X2: extended_ereal,Y2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
& ~ ( ord_le1083603963089353582_ereal @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_355_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
& ~ ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_356_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_357_not__le__imp__less,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ~ ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( ord_le1188267648640031866_ereal @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_358_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_359_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_360_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_nat
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_o_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_361_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_num
= ( ^ [A3: $o > num,B3: $o > num] :
( ( ord_less_o_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_362_order_Oorder__iff__strict,axiom,
( ord_le318542340408939003_ereal
= ( ^ [A3: $o > extended_ereal,B3: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_363_order_Oorder__iff__strict,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [A3: extended_ereal,B3: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_364_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_365_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_366_order_Ostrict__iff__order,axiom,
( ord_less_o_nat
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_367_order_Ostrict__iff__order,axiom,
( ord_less_o_num
= ( ^ [A3: $o > num,B3: $o > num] :
( ( ord_less_eq_o_num @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_368_order_Ostrict__iff__order,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [A3: $o > extended_ereal,B3: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_369_order_Ostrict__iff__order,axiom,
( ord_le1188267648640031866_ereal
= ( ^ [A3: extended_ereal,B3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_370_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_371_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_372_order_Ostrict__trans1,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_o_nat @ B @ C )
=> ( ord_less_o_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_373_order_Ostrict__trans1,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( ord_less_o_num @ B @ C )
=> ( ord_less_o_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_374_order_Ostrict__trans1,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A @ B )
=> ( ( ord_le5465781672467912687_ereal @ B @ C )
=> ( ord_le5465781672467912687_ereal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_375_order_Ostrict__trans1,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ord_le1188267648640031866_ereal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_376_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_377_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_378_order_Ostrict__trans2,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_o_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_379_order_Ostrict__trans2,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( ord_less_o_num @ A @ B )
=> ( ( ord_less_eq_o_num @ B @ C )
=> ( ord_less_o_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_380_order_Ostrict__trans2,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ A @ B )
=> ( ( ord_le318542340408939003_ereal @ B @ C )
=> ( ord_le5465781672467912687_ereal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_381_order_Ostrict__trans2,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ord_le1188267648640031866_ereal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_382_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_383_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_384_order_Ostrict__iff__not,axiom,
( ord_less_o_nat
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A3 @ B3 )
& ~ ( ord_less_eq_o_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_385_order_Ostrict__iff__not,axiom,
( ord_less_o_num
= ( ^ [A3: $o > num,B3: $o > num] :
( ( ord_less_eq_o_num @ A3 @ B3 )
& ~ ( ord_less_eq_o_num @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_386_order_Ostrict__iff__not,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [A3: $o > extended_ereal,B3: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A3 @ B3 )
& ~ ( ord_le318542340408939003_ereal @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_387_order_Ostrict__iff__not,axiom,
( ord_le1188267648640031866_ereal
= ( ^ [A3: extended_ereal,B3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A3 @ B3 )
& ~ ( ord_le1083603963089353582_ereal @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_388_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_389_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_390_dense__ge__bounded,axiom,
! [Z: extended_ereal,X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z @ X )
=> ( ! [W: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z @ W )
=> ( ( ord_le1188267648640031866_ereal @ W @ X )
=> ( ord_le1083603963089353582_ereal @ Y @ W ) ) )
=> ( ord_le1083603963089353582_ereal @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_391_dense__le__bounded,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ! [W: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ W )
=> ( ( ord_le1188267648640031866_ereal @ W @ Y )
=> ( ord_le1083603963089353582_ereal @ W @ Z ) ) )
=> ( ord_le1083603963089353582_ereal @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_392_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_nat
= ( ^ [B3: $o > nat,A3: $o > nat] :
( ( ord_less_o_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_393_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_num
= ( ^ [B3: $o > num,A3: $o > num] :
( ( ord_less_o_num @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_394_dual__order_Oorder__iff__strict,axiom,
( ord_le318542340408939003_ereal
= ( ^ [B3: $o > extended_ereal,A3: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_395_dual__order_Oorder__iff__strict,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [B3: extended_ereal,A3: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_396_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_num @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_397_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_398_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_nat
= ( ^ [B3: $o > nat,A3: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_399_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_num
= ( ^ [B3: $o > num,A3: $o > num] :
( ( ord_less_eq_o_num @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_400_dual__order_Ostrict__iff__order,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [B3: $o > extended_ereal,A3: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_401_dual__order_Ostrict__iff__order,axiom,
( ord_le1188267648640031866_ereal
= ( ^ [B3: extended_ereal,A3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_402_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_403_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_404_dual__order_Ostrict__trans1,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_o_nat @ C @ B )
=> ( ord_less_o_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_405_dual__order_Ostrict__trans1,axiom,
! [B: $o > num,A: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ B @ A )
=> ( ( ord_less_o_num @ C @ B )
=> ( ord_less_o_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_406_dual__order_Ostrict__trans1,axiom,
! [B: $o > extended_ereal,A: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ B @ A )
=> ( ( ord_le5465781672467912687_ereal @ C @ B )
=> ( ord_le5465781672467912687_ereal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_407_dual__order_Ostrict__trans1,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B @ A )
=> ( ( ord_le1188267648640031866_ereal @ C @ B )
=> ( ord_le1188267648640031866_ereal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_408_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_409_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_410_dual__order_Ostrict__trans2,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_o_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_411_dual__order_Ostrict__trans2,axiom,
! [B: $o > num,A: $o > num,C: $o > num] :
( ( ord_less_o_num @ B @ A )
=> ( ( ord_less_eq_o_num @ C @ B )
=> ( ord_less_o_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_412_dual__order_Ostrict__trans2,axiom,
! [B: $o > extended_ereal,A: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ B @ A )
=> ( ( ord_le318542340408939003_ereal @ C @ B )
=> ( ord_le5465781672467912687_ereal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_413_dual__order_Ostrict__trans2,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ B @ A )
=> ( ( ord_le1083603963089353582_ereal @ C @ B )
=> ( ord_le1188267648640031866_ereal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_414_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_415_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_416_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_nat
= ( ^ [B3: $o > nat,A3: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A3 )
& ~ ( ord_less_eq_o_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_417_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_num
= ( ^ [B3: $o > num,A3: $o > num] :
( ( ord_less_eq_o_num @ B3 @ A3 )
& ~ ( ord_less_eq_o_num @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_418_dual__order_Ostrict__iff__not,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [B3: $o > extended_ereal,A3: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ B3 @ A3 )
& ~ ( ord_le318542340408939003_ereal @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_419_dual__order_Ostrict__iff__not,axiom,
( ord_le1188267648640031866_ereal
= ( ^ [B3: extended_ereal,A3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B3 @ A3 )
& ~ ( ord_le1083603963089353582_ereal @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_420_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_421_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_422_order_Ostrict__implies__order,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_o_nat @ A @ B )
=> ( ord_less_eq_o_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_423_order_Ostrict__implies__order,axiom,
! [A: $o > num,B: $o > num] :
( ( ord_less_o_num @ A @ B )
=> ( ord_less_eq_o_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_424_order_Ostrict__implies__order,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ A @ B )
=> ( ord_le318542340408939003_ereal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_425_order_Ostrict__implies__order,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_426_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_427_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_428_dual__order_Ostrict__implies__order,axiom,
! [B: $o > nat,A: $o > nat] :
( ( ord_less_o_nat @ B @ A )
=> ( ord_less_eq_o_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_429_dual__order_Ostrict__implies__order,axiom,
! [B: $o > num,A: $o > num] :
( ( ord_less_o_num @ B @ A )
=> ( ord_less_eq_o_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_430_dual__order_Ostrict__implies__order,axiom,
! [B: $o > extended_ereal,A: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ B @ A )
=> ( ord_le318542340408939003_ereal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_431_dual__order_Ostrict__implies__order,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ B @ A )
=> ( ord_le1083603963089353582_ereal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_432_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_433_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_434_order__le__less,axiom,
( ord_less_eq_o_nat
= ( ^ [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_o_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_435_order__le__less,axiom,
( ord_less_eq_o_num
= ( ^ [X2: $o > num,Y2: $o > num] :
( ( ord_less_o_num @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_436_order__le__less,axiom,
( ord_le318542340408939003_ereal
= ( ^ [X2: $o > extended_ereal,Y2: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_437_order__le__less,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X2: extended_ereal,Y2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_438_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_439_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_440_order__less__le,axiom,
( ord_less_o_nat
= ( ^ [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_441_order__less__le,axiom,
( ord_less_o_num
= ( ^ [X2: $o > num,Y2: $o > num] :
( ( ord_less_eq_o_num @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_442_order__less__le,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [X2: $o > extended_ereal,Y2: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_443_order__less__le,axiom,
( ord_le1188267648640031866_ereal
= ( ^ [X2: extended_ereal,Y2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_444_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_445_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_446_linorder__not__le,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ~ ( ord_le1083603963089353582_ereal @ X @ Y ) )
= ( ord_le1188267648640031866_ereal @ Y @ X ) ) ).
% linorder_not_le
thf(fact_447_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_448_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_449_linorder__not__less,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ~ ( ord_le1188267648640031866_ereal @ X @ Y ) )
= ( ord_le1083603963089353582_ereal @ Y @ X ) ) ).
% linorder_not_less
thf(fact_450_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_451_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_452_order__less__imp__le,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_o_nat @ X @ Y )
=> ( ord_less_eq_o_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_453_order__less__imp__le,axiom,
! [X: $o > num,Y: $o > num] :
( ( ord_less_o_num @ X @ Y )
=> ( ord_less_eq_o_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_454_order__less__imp__le,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ X @ Y )
=> ( ord_le318542340408939003_ereal @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_455_order__less__imp__le,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ord_le1083603963089353582_ereal @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_456_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_457_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_458_order__le__neq__trans,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_459_order__le__neq__trans,axiom,
! [A: $o > num,B: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_460_order__le__neq__trans,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A @ B )
=> ( ( A != B )
=> ( ord_le5465781672467912687_ereal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_461_order__le__neq__trans,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( A != B )
=> ( ord_le1188267648640031866_ereal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_462_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_463_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_464_order__neq__le__trans,axiom,
! [A: $o > nat,B: $o > nat] :
( ( A != B )
=> ( ( ord_less_eq_o_nat @ A @ B )
=> ( ord_less_o_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_465_order__neq__le__trans,axiom,
! [A: $o > num,B: $o > num] :
( ( A != B )
=> ( ( ord_less_eq_o_num @ A @ B )
=> ( ord_less_o_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_466_order__neq__le__trans,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal] :
( ( A != B )
=> ( ( ord_le318542340408939003_ereal @ A @ B )
=> ( ord_le5465781672467912687_ereal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_467_order__neq__le__trans,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( A != B )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1188267648640031866_ereal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_468_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_469_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_470_order__le__less__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_o_nat @ Y @ Z )
=> ( ord_less_o_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_471_order__le__less__trans,axiom,
! [X: $o > num,Y: $o > num,Z: $o > num] :
( ( ord_less_eq_o_num @ X @ Y )
=> ( ( ord_less_o_num @ Y @ Z )
=> ( ord_less_o_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_472_order__le__less__trans,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal,Z: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X @ Y )
=> ( ( ord_le5465781672467912687_ereal @ Y @ Z )
=> ( ord_le5465781672467912687_ereal @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_473_order__le__less__trans,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1188267648640031866_ereal @ Y @ Z )
=> ( ord_le1188267648640031866_ereal @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_474_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_475_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_476_order__less__le__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z )
=> ( ord_less_o_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_477_order__less__le__trans,axiom,
! [X: $o > num,Y: $o > num,Z: $o > num] :
( ( ord_less_o_num @ X @ Y )
=> ( ( ord_less_eq_o_num @ Y @ Z )
=> ( ord_less_o_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_478_order__less__le__trans,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal,Z: $o > extended_ereal] :
( ( ord_le5465781672467912687_ereal @ X @ Y )
=> ( ( ord_le318542340408939003_ereal @ Y @ Z )
=> ( ord_le5465781672467912687_ereal @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_479_order__less__le__trans,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ Y @ Z )
=> ( ord_le1188267648640031866_ereal @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_480_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_481_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_482_order__le__less__subst1,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_483_order__le__less__subst1,axiom,
! [A: extended_ereal,F: num > extended_ereal,B: num,C: num] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_484_order__le__less__subst1,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_485_order__le__less__subst1,axiom,
! [A: num,F: extended_ereal > num,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_486_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_487_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_488_order__le__less__subst1,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_489_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_490_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_491_order__le__less__subst1,axiom,
! [A: $o > nat,F: extended_ereal > $o > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_o_nat @ A @ ( F @ B ) )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_o_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_492_order__le__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_493_order__le__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > num,C: num] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_494_order__le__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_495_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_496_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_497_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_498_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_499_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_500_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_501_order__le__less__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > $o > nat,C: $o > nat] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_less_o_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_o_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_502_order__less__le__subst1,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_503_order__less__le__subst1,axiom,
! [A: num,F: extended_ereal > num,B: extended_ereal,C: extended_ereal] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_504_order__less__le__subst1,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_505_order__less__le__subst1,axiom,
! [A: extended_ereal,F: num > extended_ereal,B: num,C: num] :
( ( ord_le1188267648640031866_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_506_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_507_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_508_order__less__le__subst1,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( ord_le1188267648640031866_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_509_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_510_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_511_order__less__le__subst1,axiom,
! [A: $o > nat,F: extended_ereal > $o > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_o_nat @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_o_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_512_order__less__le__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_513_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > extended_ereal,C: extended_ereal] :
( ( ord_less_num @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_514_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_le1188267648640031866_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1188267648640031866_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_515_order__less__le__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > num,C: num] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_516_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_517_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_518_order__less__le__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_519_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_num @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_520_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_521_order__less__le__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > $o > nat,C: $o > nat] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_less_eq_o_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Y5 )
=> ( ord_less_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_o_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_522_linorder__le__less__linear,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
| ( ord_le1188267648640031866_ereal @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_523_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_524_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_525_order__le__imp__less__or__eq,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_o_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_526_order__le__imp__less__or__eq,axiom,
! [X: $o > num,Y: $o > num] :
( ( ord_less_eq_o_num @ X @ Y )
=> ( ( ord_less_o_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_527_order__le__imp__less__or__eq,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X @ Y )
=> ( ( ord_le5465781672467912687_ereal @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_528_order__le__imp__less__or__eq,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1188267648640031866_ereal @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_529_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_530_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_531_ereal__mult__less__right,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ B @ A ) @ ( times_7703590493115627913_ereal @ C @ A ) )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ A @ extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ B @ C ) ) ) ) ).
% ereal_mult_less_right
thf(fact_532_ereal__mult__strict__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ C @ A ) @ ( times_7703590493115627913_ereal @ C @ B ) ) ) ) ) ).
% ereal_mult_strict_left_mono
thf(fact_533_ereal__mult__strict__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ) ) ).
% ereal_mult_strict_right_mono
thf(fact_534_verit__comp__simplify1_I3_J,axiom,
! [B5: extended_ereal,A5: extended_ereal] :
( ( ~ ( ord_le1083603963089353582_ereal @ B5 @ A5 ) )
= ( ord_le1188267648640031866_ereal @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_535_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_536_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_537_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_538_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_539_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_540_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_541_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_542_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_543_crossproduct__eq,axiom,
! [W2: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W2 @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_544_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_545_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_546_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_547_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_548_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_549_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_550_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_551_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_552_times__ereal_Osimps_I6_J,axiom,
( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) ).
% times_ereal.simps(6)
thf(fact_553_less__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ extend1530274965995635425_ereal @ A ) ).
% less_ereal.simps(2)
thf(fact_554_ereal__right__mult__cong,axiom,
! [C: extended_ereal,D: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( C = D )
=> ( ( ( D != zero_z2744965634713055877_ereal )
=> ( A = B ) )
=> ( ( times_7703590493115627913_ereal @ C @ A )
= ( times_7703590493115627913_ereal @ D @ B ) ) ) ) ).
% ereal_right_mult_cong
thf(fact_555_ereal__mult__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ) ).
% ereal_mult_right_mono
thf(fact_556_ereal__mult__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C @ A ) @ ( times_7703590493115627913_ereal @ C @ B ) ) ) ) ).
% ereal_mult_left_mono
thf(fact_557_ereal__left__mult__cong,axiom,
! [C: extended_ereal,D: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( C = D )
=> ( ( ( D != zero_z2744965634713055877_ereal )
=> ( A = B ) )
=> ( ( times_7703590493115627913_ereal @ A @ C )
= ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ).
% ereal_left_mult_cong
thf(fact_558_ereal__zero__le__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A @ B ) )
= ( ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_zero_le_0_iff
thf(fact_559_ereal__mult__le__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ zero_z2744965634713055877_ereal )
= ( ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B ) ) ) ) ).
% ereal_mult_le_0_iff
thf(fact_560_ereal__mult__mono_H,axiom,
! [A: extended_ereal,C: extended_ereal,B: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ C @ D )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono'
thf(fact_561_ereal__mult__mono,axiom,
! [B: extended_ereal,C: extended_ereal,A: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ C @ D )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono
thf(fact_562_ereal__0__le__mult,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ) ).
% ereal_0_le_mult
thf(fact_563_ereal__le__distrib,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C @ ( plus_p7876563987511257093_ereal @ A @ B ) ) @ ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C @ A ) @ ( times_7703590493115627913_ereal @ C @ B ) ) ) ).
% ereal_le_distrib
thf(fact_564_ereal__add__strict__mono2,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ).
% ereal_add_strict_mono2
thf(fact_565_add_Oabel__semigroup__axioms,axiom,
abel_s1811588628328619620_ereal @ plus_p7876563987511257093_ereal ).
% add.abel_semigroup_axioms
thf(fact_566_add_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat @ plus_plus_nat ).
% add.abel_semigroup_axioms
thf(fact_567_add_Osemigroup__axioms,axiom,
semigr3155017398426397084_ereal @ plus_p7876563987511257093_ereal ).
% add.semigroup_axioms
thf(fact_568_add_Osemigroup__axioms,axiom,
semigroup_nat @ plus_plus_nat ).
% add.semigroup_axioms
thf(fact_569_comm__monoid_Oaxioms_I1_J,axiom,
! [F: nat > nat > nat,Z: nat] :
( ( comm_monoid_nat @ F @ Z )
=> ( abel_semigroup_nat @ F ) ) ).
% comm_monoid.axioms(1)
thf(fact_570_comm__monoid_Oaxioms_I1_J,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal] :
( ( comm_m179930729793757699_ereal @ F @ Z )
=> ( abel_s1811588628328619620_ereal @ F ) ) ).
% comm_monoid.axioms(1)
thf(fact_571_monoid_Oaxioms_I1_J,axiom,
! [F: nat > nat > nat,Z: nat] :
( ( monoid_nat @ F @ Z )
=> ( semigroup_nat @ F ) ) ).
% monoid.axioms(1)
thf(fact_572_monoid_Oaxioms_I1_J,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,Z: extended_ereal] :
( ( monoid425827787017695951_ereal @ F @ Z )
=> ( semigr3155017398426397084_ereal @ F ) ) ).
% monoid.axioms(1)
thf(fact_573_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_574_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_575_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_576_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_577_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_578_add__neg__neg,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ zero_z2744965634713055877_ereal ) ) ) ).
% add_neg_neg
thf(fact_579_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_580_add__pos__pos,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_581_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_582_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_583_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_584_ereal__less_I5_J,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ extend1530274965995635425_ereal ).
% ereal_less(5)
thf(fact_585_ereal__left__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,R: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( times_7703590493115627913_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ R )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ A @ R ) @ ( times_7703590493115627913_ereal @ B @ R ) ) ) ) ) ).
% ereal_left_distrib
thf(fact_586_ereal__right__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,R: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( times_7703590493115627913_ereal @ R @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ R @ A ) @ ( times_7703590493115627913_ereal @ R @ B ) ) ) ) ) ).
% ereal_right_distrib
thf(fact_587_ereal__le__epsilon,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ! [E: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ E )
=> ( ord_le1083603963089353582_ereal @ X @ ( plus_p7876563987511257093_ereal @ Y @ E ) ) )
=> ( ord_le1083603963089353582_ereal @ X @ Y ) ) ).
% ereal_le_epsilon
thf(fact_588_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_589_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_590_add__pos__nonneg,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_591_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_592_add__nonpos__neg,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ zero_z2744965634713055877_ereal ) ) ) ).
% add_nonpos_neg
thf(fact_593_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_594_add__nonneg__pos,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_595_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_596_add__neg__nonpos,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ zero_z2744965634713055877_ereal ) ) ) ).
% add_neg_nonpos
thf(fact_597_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_598_nle__le,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ~ ( ord_le1083603963089353582_ereal @ A @ B ) )
= ( ( ord_le1083603963089353582_ereal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_599_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_600_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_601_le__cases3,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ~ ( ord_le1083603963089353582_ereal @ Y @ Z ) )
=> ( ( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ~ ( ord_le1083603963089353582_ereal @ X @ Z ) )
=> ( ( ( ord_le1083603963089353582_ereal @ X @ Z )
=> ~ ( ord_le1083603963089353582_ereal @ Z @ Y ) )
=> ( ( ( ord_le1083603963089353582_ereal @ Z @ Y )
=> ~ ( ord_le1083603963089353582_ereal @ Y @ X ) )
=> ( ( ( ord_le1083603963089353582_ereal @ Y @ Z )
=> ~ ( ord_le1083603963089353582_ereal @ Z @ X ) )
=> ~ ( ( ord_le1083603963089353582_ereal @ Z @ X )
=> ~ ( ord_le1083603963089353582_ereal @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_602_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_603_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_604_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: $o > nat,Z5: $o > nat] : ( Y6 = Z5 ) )
= ( ^ [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
& ( ord_less_eq_o_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_605_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: $o > num,Z5: $o > num] : ( Y6 = Z5 ) )
= ( ^ [X2: $o > num,Y2: $o > num] :
( ( ord_less_eq_o_num @ X2 @ Y2 )
& ( ord_less_eq_o_num @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_606_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: $o > extended_ereal,Z5: $o > extended_ereal] : ( Y6 = Z5 ) )
= ( ^ [X2: $o > extended_ereal,Y2: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X2 @ Y2 )
& ( ord_le318542340408939003_ereal @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_607_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: extended_ereal,Z5: extended_ereal] : ( Y6 = Z5 ) )
= ( ^ [X2: extended_ereal,Y2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
& ( ord_le1083603963089353582_ereal @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_608_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
= ( ^ [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_609_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_610_ord__eq__le__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( A = B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_611_ord__eq__le__trans,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( A = B )
=> ( ( ord_less_eq_o_num @ B @ C )
=> ( ord_less_eq_o_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_612_ord__eq__le__trans,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal,C: $o > extended_ereal] :
( ( A = B )
=> ( ( ord_le318542340408939003_ereal @ B @ C )
=> ( ord_le318542340408939003_ereal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_613_ord__eq__le__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A = B )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ord_le1083603963089353582_ereal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_614_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_615_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_616_ord__le__eq__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_617_ord__le__eq__trans,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_618_ord__le__eq__trans,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A @ B )
=> ( ( B = C )
=> ( ord_le318542340408939003_ereal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_619_ord__le__eq__trans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( B = C )
=> ( ord_le1083603963089353582_ereal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_620_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_621_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_622_order__antisym,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_623_order__antisym,axiom,
! [X: $o > num,Y: $o > num] :
( ( ord_less_eq_o_num @ X @ Y )
=> ( ( ord_less_eq_o_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_624_order__antisym,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X @ Y )
=> ( ( ord_le318542340408939003_ereal @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_625_order__antisym,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_626_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_627_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_628_order_Otrans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_629_order_Otrans,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( ord_less_eq_o_num @ B @ C )
=> ( ord_less_eq_o_num @ A @ C ) ) ) ).
% order.trans
thf(fact_630_order_Otrans,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A @ B )
=> ( ( ord_le318542340408939003_ereal @ B @ C )
=> ( ord_le318542340408939003_ereal @ A @ C ) ) ) ).
% order.trans
thf(fact_631_order_Otrans,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ord_le1083603963089353582_ereal @ A @ C ) ) ) ).
% order.trans
thf(fact_632_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_633_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_634_order__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z )
=> ( ord_less_eq_o_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_635_order__trans,axiom,
! [X: $o > num,Y: $o > num,Z: $o > num] :
( ( ord_less_eq_o_num @ X @ Y )
=> ( ( ord_less_eq_o_num @ Y @ Z )
=> ( ord_less_eq_o_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_636_order__trans,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal,Z: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ X @ Y )
=> ( ( ord_le318542340408939003_ereal @ Y @ Z )
=> ( ord_le318542340408939003_ereal @ X @ Z ) ) ) ).
% order_trans
thf(fact_637_order__trans,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ Y @ Z )
=> ( ord_le1083603963089353582_ereal @ X @ Z ) ) ) ).
% order_trans
thf(fact_638_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_639_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_640_linorder__wlog,axiom,
! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
( ! [A4: extended_ereal,B4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: extended_ereal,B4: extended_ereal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_641_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_642_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_643_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: $o > nat,Z5: $o > nat] : ( Y6 = Z5 ) )
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A3 )
& ( ord_less_eq_o_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_644_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: $o > num,Z5: $o > num] : ( Y6 = Z5 ) )
= ( ^ [A3: $o > num,B3: $o > num] :
( ( ord_less_eq_o_num @ B3 @ A3 )
& ( ord_less_eq_o_num @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_645_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: $o > extended_ereal,Z5: $o > extended_ereal] : ( Y6 = Z5 ) )
= ( ^ [A3: $o > extended_ereal,B3: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ B3 @ A3 )
& ( ord_le318542340408939003_ereal @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_646_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: extended_ereal,Z5: extended_ereal] : ( Y6 = Z5 ) )
= ( ^ [A3: extended_ereal,B3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B3 @ A3 )
& ( ord_le1083603963089353582_ereal @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_647_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A3 )
& ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_648_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_649_dual__order_Oantisym,axiom,
! [B: $o > nat,A: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_650_dual__order_Oantisym,axiom,
! [B: $o > num,A: $o > num] :
( ( ord_less_eq_o_num @ B @ A )
=> ( ( ord_less_eq_o_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_651_dual__order_Oantisym,axiom,
! [B: $o > extended_ereal,A: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ B @ A )
=> ( ( ord_le318542340408939003_ereal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_652_dual__order_Oantisym,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B @ A )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_653_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_654_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_655_dual__order_Otrans,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_656_dual__order_Otrans,axiom,
! [B: $o > num,A: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ B @ A )
=> ( ( ord_less_eq_o_num @ C @ B )
=> ( ord_less_eq_o_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_657_dual__order_Otrans,axiom,
! [B: $o > extended_ereal,A: $o > extended_ereal,C: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ B @ A )
=> ( ( ord_le318542340408939003_ereal @ C @ B )
=> ( ord_le318542340408939003_ereal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_658_dual__order_Otrans,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B @ A )
=> ( ( ord_le1083603963089353582_ereal @ C @ B )
=> ( ord_le1083603963089353582_ereal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_659_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_660_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_661_antisym,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_662_antisym,axiom,
! [A: $o > num,B: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( ord_less_eq_o_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_663_antisym,axiom,
! [A: $o > extended_ereal,B: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A @ B )
=> ( ( ord_le318542340408939003_ereal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_664_antisym,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_665_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_666_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_667_le__funD,axiom,
! [F: $o > nat,G: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_668_le__funD,axiom,
! [F: $o > num,G: $o > num,X: $o] :
( ( ord_less_eq_o_num @ F @ G )
=> ( ord_less_eq_num @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_669_le__funD,axiom,
! [F: $o > extended_ereal,G: $o > extended_ereal,X: $o] :
( ( ord_le318542340408939003_ereal @ F @ G )
=> ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_670_le__funE,axiom,
! [F: $o > nat,G: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_671_le__funE,axiom,
! [F: $o > num,G: $o > num,X: $o] :
( ( ord_less_eq_o_num @ F @ G )
=> ( ord_less_eq_num @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_672_le__funE,axiom,
! [F: $o > extended_ereal,G: $o > extended_ereal,X: $o] :
( ( ord_le318542340408939003_ereal @ F @ G )
=> ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_673_le__funI,axiom,
! [F: $o > extended_ereal,G: $o > extended_ereal] :
( ! [X4: $o] : ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_le318542340408939003_ereal @ F @ G ) ) ).
% le_funI
thf(fact_674_le__funI,axiom,
! [F: $o > num,G: $o > num] :
( ! [X4: $o] : ( ord_less_eq_num @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_o_num @ F @ G ) ) ).
% le_funI
thf(fact_675_le__funI,axiom,
! [F: $o > nat,G: $o > nat] :
( ! [X4: $o] : ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_o_nat @ F @ G ) ) ).
% le_funI
thf(fact_676_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G2: $o > nat] :
! [X2: $o] : ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_677_le__fun__def,axiom,
( ord_less_eq_o_num
= ( ^ [F2: $o > num,G2: $o > num] :
! [X2: $o] : ( ord_less_eq_num @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_678_le__fun__def,axiom,
( ord_le318542340408939003_ereal
= ( ^ [F2: $o > extended_ereal,G2: $o > extended_ereal] :
! [X2: $o] : ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_679_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: $o > nat,Z5: $o > nat] : ( Y6 = Z5 ) )
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A3 @ B3 )
& ( ord_less_eq_o_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_680_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: $o > num,Z5: $o > num] : ( Y6 = Z5 ) )
= ( ^ [A3: $o > num,B3: $o > num] :
( ( ord_less_eq_o_num @ A3 @ B3 )
& ( ord_less_eq_o_num @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_681_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: $o > extended_ereal,Z5: $o > extended_ereal] : ( Y6 = Z5 ) )
= ( ^ [A3: $o > extended_ereal,B3: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ A3 @ B3 )
& ( ord_le318542340408939003_ereal @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_682_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: extended_ereal,Z5: extended_ereal] : ( Y6 = Z5 ) )
= ( ^ [A3: extended_ereal,B3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A3 @ B3 )
& ( ord_le1083603963089353582_ereal @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_683_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_684_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_685_order__subst1,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_686_order__subst1,axiom,
! [A: extended_ereal,F: num > extended_ereal,B: num,C: num] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst1,axiom,
! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_688_order__subst1,axiom,
! [A: num,F: extended_ereal > num,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_689_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_690_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_691_order__subst1,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_692_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_693_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_694_order__subst1,axiom,
! [A: extended_ereal,F: ( $o > nat ) > extended_ereal,B: $o > nat,C: $o > nat] :
( ( ord_le1083603963089353582_ereal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ! [X4: $o > nat,Y5: $o > nat] :
( ( ord_less_eq_o_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_695_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_696_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > num,C: num] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_697_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_698_order__subst2,axiom,
! [A: num,B: num,F: num > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_699_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_700_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_701_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_702_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_703_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_704_order__subst2,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > $o > nat,C: $o > nat] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_less_eq_o_nat @ ( F @ B ) @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_705_order__eq__refl,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( X = Y )
=> ( ord_less_eq_o_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_706_order__eq__refl,axiom,
! [X: $o > num,Y: $o > num] :
( ( X = Y )
=> ( ord_less_eq_o_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_707_order__eq__refl,axiom,
! [X: $o > extended_ereal,Y: $o > extended_ereal] :
( ( X = Y )
=> ( ord_le318542340408939003_ereal @ X @ Y ) ) ).
% order_eq_refl
thf(fact_708_order__eq__refl,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( X = Y )
=> ( ord_le1083603963089353582_ereal @ X @ Y ) ) ).
% order_eq_refl
thf(fact_709_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_710_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_711_linorder__linear,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
| ( ord_le1083603963089353582_ereal @ Y @ X ) ) ).
% linorder_linear
thf(fact_712_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_713_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_714_ord__eq__le__subst,axiom,
! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_715_ord__eq__le__subst,axiom,
! [A: num,F: extended_ereal > num,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_716_ord__eq__le__subst,axiom,
! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_717_ord__eq__le__subst,axiom,
! [A: extended_ereal,F: num > extended_ereal,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_718_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_719_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_720_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 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_721_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_722_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_723_ord__eq__le__subst,axiom,
! [A: $o > nat,F: extended_ereal > $o > nat,B: extended_ereal,C: extended_ereal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_724_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_725_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > num,C: num] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_726_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_727_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > extended_ereal,C: extended_ereal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_728_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_729_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: num,Y5: num] :
( ( ord_less_eq_num @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_730_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 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_731_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_732_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_733_ord__le__eq__subst,axiom,
! [A: extended_ereal,B: extended_ereal,F: extended_ereal > $o > nat,C: $o > nat] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: extended_ereal,Y5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X4 @ Y5 )
=> ( ord_less_eq_o_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_734_linorder__le__cases,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ~ ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ord_le1083603963089353582_ereal @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_735_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_736_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_737_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_738_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_739_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_740_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_741_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_742_less__fun__def,axiom,
( ord_less_o_nat
= ( ^ [F2: $o > nat,G2: $o > nat] :
( ( ord_less_eq_o_nat @ F2 @ G2 )
& ~ ( ord_less_eq_o_nat @ G2 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_743_less__fun__def,axiom,
( ord_less_o_num
= ( ^ [F2: $o > num,G2: $o > num] :
( ( ord_less_eq_o_num @ F2 @ G2 )
& ~ ( ord_less_eq_o_num @ G2 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_744_less__fun__def,axiom,
( ord_le5465781672467912687_ereal
= ( ^ [F2: $o > extended_ereal,G2: $o > extended_ereal] :
( ( ord_le318542340408939003_ereal @ F2 @ G2 )
& ~ ( ord_le318542340408939003_ereal @ G2 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_745_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_746_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_747_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_748_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_749_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_750_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_751_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_752_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_753_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_754_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_755_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_756_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_757_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_758_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_759_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_760_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_761_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_762_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_763_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_764_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_765_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_766_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_767_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_768_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_769_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_770_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_771_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_772_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_773_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_774_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_775_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_776_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_777_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_778_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_779_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_780_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_781_minf_I8_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ~ ( ord_le1083603963089353582_ereal @ T @ X6 ) ) ).
% minf(8)
thf(fact_782_minf_I8_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% minf(8)
thf(fact_783_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_784_minf_I6_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ( ord_le1083603963089353582_ereal @ X6 @ T ) ) ).
% minf(6)
thf(fact_785_minf_I6_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( ord_less_eq_num @ X6 @ T ) ) ).
% minf(6)
thf(fact_786_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_787_pinf_I8_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ( ord_le1083603963089353582_ereal @ T @ X6 ) ) ).
% pinf(8)
thf(fact_788_pinf_I8_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( ord_less_eq_num @ T @ X6 ) ) ).
% pinf(8)
thf(fact_789_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_790_pinf_I6_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ~ ( ord_le1083603963089353582_ereal @ X6 @ T ) ) ).
% pinf(6)
thf(fact_791_pinf_I6_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% pinf(6)
thf(fact_792_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_793_pinf_I1_J,axiom,
! [P: extended_ereal > $o,P4: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_794_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_795_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_796_pinf_I2_J,axiom,
! [P: extended_ereal > $o,P4: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_797_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_798_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_799_pinf_I3_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_800_pinf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_801_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_802_pinf_I4_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_803_pinf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_804_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_805_pinf_I5_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ~ ( ord_le1188267648640031866_ereal @ X6 @ T ) ) ).
% pinf(5)
thf(fact_806_pinf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ~ ( ord_less_num @ X6 @ T ) ) ).
% pinf(5)
thf(fact_807_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_808_pinf_I7_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ Z4 @ X6 )
=> ( ord_le1188267648640031866_ereal @ T @ X6 ) ) ).
% pinf(7)
thf(fact_809_pinf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ Z4 @ X6 )
=> ( ord_less_num @ T @ X6 ) ) ).
% pinf(7)
thf(fact_810_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_811_minf_I1_J,axiom,
! [P: extended_ereal > $o,P4: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_812_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_813_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_814_minf_I2_J,axiom,
! [P: extended_ereal > $o,P4: extended_ereal > $o,Q: extended_ereal > $o,Q2: extended_ereal > $o] :
( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: extended_ereal] :
! [X4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_815_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_816_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_817_minf_I3_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_818_minf_I3_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_819_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_820_minf_I4_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_821_minf_I4_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_822_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_823_minf_I5_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ( ord_le1188267648640031866_ereal @ X6 @ T ) ) ).
% minf(5)
thf(fact_824_minf_I5_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ( ord_less_num @ X6 @ T ) ) ).
% minf(5)
thf(fact_825_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_826_minf_I7_J,axiom,
! [T: extended_ereal] :
? [Z4: extended_ereal] :
! [X6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X6 @ Z4 )
=> ~ ( ord_le1188267648640031866_ereal @ T @ X6 ) ) ).
% minf(7)
thf(fact_827_minf_I7_J,axiom,
! [T: num] :
? [Z4: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z4 )
=> ~ ( ord_less_num @ T @ X6 ) ) ).
% minf(7)
thf(fact_828_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_829_complete__interval,axiom,
! [A: extended_ereal,B: extended_ereal,P: extended_ereal > $o] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ C2 )
& ( ord_le1083603963089353582_ereal @ C2 @ B )
& ! [X6: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ A @ X6 )
& ( ord_le1188267648640031866_ereal @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D2: extended_ereal] :
( ! [X4: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ A @ X4 )
& ( ord_le1188267648640031866_ereal @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_le1083603963089353582_ereal @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_830_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_831_abel__semigroup_Ointro,axiom,
! [F: nat > nat > nat] :
( ( semigroup_nat @ F )
=> ( ( abel_s2057502115565749341ms_nat @ F )
=> ( abel_semigroup_nat @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_832_abel__semigroup_Ointro,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal] :
( ( semigr3155017398426397084_ereal @ F )
=> ( ( abel_s6691961667147211777_ereal @ F )
=> ( abel_s1811588628328619620_ereal @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_833_abel__semigroup__def,axiom,
( abel_semigroup_nat
= ( ^ [F2: nat > nat > nat] :
( ( semigroup_nat @ F2 )
& ( abel_s2057502115565749341ms_nat @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_834_abel__semigroup__def,axiom,
( abel_s1811588628328619620_ereal
= ( ^ [F2: extended_ereal > extended_ereal > extended_ereal] :
( ( semigr3155017398426397084_ereal @ F2 )
& ( abel_s6691961667147211777_ereal @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_835_ereal__le__divide__pos,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( X != extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ Y @ ( divide8893690120176169980_ereal @ Z @ X ) )
= ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ X @ Y ) @ Z ) ) ) ) ).
% ereal_le_divide_pos
thf(fact_836_ereal__divide__le__pos,axiom,
! [X: extended_ereal,Z: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( X != extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ ( divide8893690120176169980_ereal @ Z @ X ) @ Y )
= ( ord_le1083603963089353582_ereal @ Z @ ( times_7703590493115627913_ereal @ X @ Y ) ) ) ) ) ).
% ereal_divide_le_pos
thf(fact_837_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_838_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_839_ereal__divide__zero__left,axiom,
! [A: extended_ereal] :
( ( divide8893690120176169980_ereal @ zero_z2744965634713055877_ereal @ A )
= zero_z2744965634713055877_ereal ) ).
% ereal_divide_zero_left
thf(fact_840_ereal__times__divide__eq__left,axiom,
! [B: extended_ereal,C: extended_ereal,A: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( divide8893690120176169980_ereal @ B @ C ) @ A )
= ( divide8893690120176169980_ereal @ ( times_7703590493115627913_ereal @ B @ A ) @ C ) ) ).
% ereal_times_divide_eq_left
thf(fact_841_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_842_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_843_ereal__divide__Infty_I1_J,axiom,
! [X: extended_ereal] :
( ( divide8893690120176169980_ereal @ X @ extend1530274965995635425_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_divide_Infty(1)
thf(fact_844_ereal__times__divide__eq,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ ( divide8893690120176169980_ereal @ B @ C ) )
= ( divide8893690120176169980_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ C ) ) ).
% ereal_times_divide_eq
thf(fact_845_zero__le__divide__ereal,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( divide8893690120176169980_ereal @ A @ B ) ) ) ) ).
% zero_le_divide_ereal
thf(fact_846_ex__gt__or__lt,axiom,
! [A: extended_ereal] :
? [B4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B4 )
| ( ord_le1188267648640031866_ereal @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_847_abel__semigroup_Oaxioms_I2_J,axiom,
! [F: nat > nat > nat] :
( ( abel_semigroup_nat @ F )
=> ( abel_s2057502115565749341ms_nat @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_848_abel__semigroup_Oaxioms_I2_J,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal] :
( ( abel_s1811588628328619620_ereal @ F )
=> ( abel_s6691961667147211777_ereal @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_849_ereal__divide__right__mono,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ Z )
=> ( ord_le1083603963089353582_ereal @ ( divide8893690120176169980_ereal @ X @ Z ) @ ( divide8893690120176169980_ereal @ Y @ Z ) ) ) ) ).
% ereal_divide_right_mono
thf(fact_850_ereal__mult__divide,axiom,
! [B: extended_ereal,A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ord_le1188267648640031866_ereal @ B @ extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ B @ ( divide8893690120176169980_ereal @ A @ B ) )
= A ) ) ) ).
% ereal_mult_divide
thf(fact_851_ereal__divide__less__iff,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ ( divide8893690120176169980_ereal @ A @ C ) @ B )
= ( ord_le1188267648640031866_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ) ) ).
% ereal_divide_less_iff
thf(fact_852_ereal__divide__less__pos,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( X != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ ( divide8893690120176169980_ereal @ Y @ X ) @ Z )
= ( ord_le1188267648640031866_ereal @ Y @ ( times_7703590493115627913_ereal @ X @ Z ) ) ) ) ) ).
% ereal_divide_less_pos
thf(fact_853_ereal__less__divide__iff,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ A @ ( divide8893690120176169980_ereal @ B @ C ) )
= ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ B ) ) ) ) ).
% ereal_less_divide_iff
thf(fact_854_ereal__less__divide__pos,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( X != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ Y @ ( divide8893690120176169980_ereal @ Z @ X ) )
= ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ X @ Y ) @ Z ) ) ) ) ).
% ereal_less_divide_pos
thf(fact_855_ereal__divide__left__mono,axiom,
! [Y: extended_ereal,X: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y @ X )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ Z )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ X @ Y ) )
=> ( ord_le1083603963089353582_ereal @ ( divide8893690120176169980_ereal @ Z @ X ) @ ( divide8893690120176169980_ereal @ Z @ Y ) ) ) ) ) ).
% ereal_divide_left_mono
thf(fact_856_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_857_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_858_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_859_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_860_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_861_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_862_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_863_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_864_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_865_ereal__divide__le__neg,axiom,
! [X: extended_ereal,Z: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ zero_z2744965634713055877_ereal )
=> ( ( X
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ord_le1083603963089353582_ereal @ ( divide8893690120176169980_ereal @ Z @ X ) @ Y )
= ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ X @ Y ) @ Z ) ) ) ) ).
% ereal_divide_le_neg
thf(fact_866_ereal__le__divide__neg,axiom,
! [X: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ zero_z2744965634713055877_ereal )
=> ( ( X
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ord_le1083603963089353582_ereal @ Y @ ( divide8893690120176169980_ereal @ Z @ X ) )
= ( ord_le1083603963089353582_ereal @ Z @ ( times_7703590493115627913_ereal @ X @ Y ) ) ) ) ) ).
% ereal_le_divide_neg
thf(fact_867_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus27091377158695749_ereal @ ( uminus27091377158695749_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_868_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= ( uminus27091377158695749_ereal @ B ) )
= ( A = B ) ) ).
% ereal_uminus_eq_iff
thf(fact_869_ereal__minus__le__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1083603963089353582_ereal @ B @ A ) ) ).
% ereal_minus_le_minus
thf(fact_870_ereal__uminus__zero__iff,axiom,
! [A: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= zero_z2744965634713055877_ereal )
= ( A = zero_z2744965634713055877_ereal ) ) ).
% ereal_uminus_zero_iff
thf(fact_871_ereal__uminus__zero,axiom,
( ( uminus27091377158695749_ereal @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_uminus_zero
thf(fact_872_ereal__minus__less__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ A ) ) ).
% ereal_minus_less_minus
thf(fact_873_ereal__mult__minus__right,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ ( uminus27091377158695749_ereal @ B ) )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_right
thf(fact_874_ereal__mult__minus__left,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_left
thf(fact_875_ereal__uminus__divide,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( divide8893690120176169980_ereal @ ( uminus27091377158695749_ereal @ X ) @ Y )
= ( uminus27091377158695749_ereal @ ( divide8893690120176169980_ereal @ X @ Y ) ) ) ).
% ereal_uminus_divide
thf(fact_876_ereal__infty__less__eq_I2_J,axiom,
! [X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( X
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_877_ereal__MInfty__lessI,axiom,
! [A: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ A ) ) ).
% ereal_MInfty_lessI
thf(fact_878_ereal__infty__less_I2_J,axiom,
! [X: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X )
= ( X
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less(2)
thf(fact_879_ereal__plus__eq__MInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ A @ B )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( A != extend1530274965995635425_ereal )
& ( B != extend1530274965995635425_ereal ) ) ) ).
% ereal_plus_eq_MInfty
thf(fact_880_ereal__MInfty__eq__plus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal )
= ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B != extend1530274965995635425_ereal ) )
| ( ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_MInfty_eq_plus
thf(fact_881_ereal__uminus__le__0__iff,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ zero_z2744965634713055877_ereal )
= ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A ) ) ).
% ereal_uminus_le_0_iff
thf(fact_882_ereal__0__le__uminus__iff,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% ereal_0_le_uminus_iff
thf(fact_883_neg__0__less__iff__less__erea,axiom,
! [A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% neg_0_less_iff_less_erea
thf(fact_884_ereal__divide__Infty_I2_J,axiom,
! [X: extended_ereal] :
( ( divide8893690120176169980_ereal @ X @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= zero_z2744965634713055877_ereal ) ).
% ereal_divide_Infty(2)
thf(fact_885_ereal__infty__mult,axiom,
! [A: extended_ereal] :
( ( ( A = zero_z2744965634713055877_ereal )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ A )
= zero_z2744965634713055877_ereal ) )
& ( ( A != zero_z2744965634713055877_ereal )
=> ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ A )
= extend1530274965995635425_ereal ) )
& ( ~ ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ A )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% ereal_infty_mult
thf(fact_886_ereal__mult__infty,axiom,
! [A: extended_ereal] :
( ( ( A = zero_z2744965634713055877_ereal )
=> ( ( times_7703590493115627913_ereal @ A @ extend1530274965995635425_ereal )
= zero_z2744965634713055877_ereal ) )
& ( ( A != zero_z2744965634713055877_ereal )
=> ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ A @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) )
& ( ~ ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ A @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% ereal_mult_infty
thf(fact_887_ereal__mult__eq__MInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( ( ( A = extend1530274965995635425_ereal )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( B = extend1530274965995635425_ereal ) )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_mult_eq_MInfty
thf(fact_888_ereal__mult__eq__PInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= extend1530274965995635425_ereal )
= ( ( ( A = extend1530274965995635425_ereal )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( B = extend1530274965995635425_ereal ) )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_mult_eq_PInfty
thf(fact_889_ereal__uminus__le__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ B ) @ A ) ) ).
% ereal_uminus_le_reorder
thf(fact_890_ereal__less__uminus__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ ( uminus27091377158695749_ereal @ A ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_891_ereal__uminus__less__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ B ) @ A ) ) ).
% ereal_uminus_less_reorder
thf(fact_892_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= B )
= ( A
= ( uminus27091377158695749_ereal @ B ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_893_MInfty__neq__PInfty_I1_J,axiom,
( extend1530274965995635425_ereal
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_894_ereal__infty__less__eq2_I2_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_895_ereal__less__eq_I2_J,axiom,
! [X: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X ) ).
% ereal_less_eq(2)
thf(fact_896_Infty__neq__0_I3_J,axiom,
( ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal )
!= zero_z2744965634713055877_ereal ) ).
% Infty_neq_0(3)
thf(fact_897_less__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% less_ereal.simps(3)
thf(fact_898_less__ereal_Osimps_I6_J,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal ).
% less_ereal.simps(6)
thf(fact_899_times__ereal_Osimps_I7_J,axiom,
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% times_ereal.simps(7)
thf(fact_900_times__ereal_Osimps_I8_J,axiom,
( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% times_ereal.simps(8)
thf(fact_901_times__ereal_Osimps_I9_J,axiom,
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= extend1530274965995635425_ereal ) ).
% times_ereal.simps(9)
thf(fact_902_ereal__add__cancel__right,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ( plus_p7876563987511257093_ereal @ B @ A )
= ( plus_p7876563987511257093_ereal @ C @ A ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B = C ) ) ) ) ).
% ereal_add_cancel_right
thf(fact_903_ereal__add__cancel__left,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ( plus_p7876563987511257093_ereal @ A @ B )
= ( plus_p7876563987511257093_ereal @ A @ C ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B = C ) ) ) ) ).
% ereal_add_cancel_left
thf(fact_904_plus__ereal_Osimps_I6_J,axiom,
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% plus_ereal.simps(6)
thf(fact_905_not__MInfty__nonneg,axiom,
! [X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( X
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% not_MInfty_nonneg
thf(fact_906_ereal__less_I6_J,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ zero_z2744965634713055877_ereal ).
% ereal_less(6)
thf(fact_907_ereal__add__le__add__iff2,axiom,
! [A: extended_ereal,C: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ ( plus_p7876563987511257093_ereal @ B @ C ) )
= ( ( ord_le1083603963089353582_ereal @ A @ B )
| ( C = extend1530274965995635425_ereal )
| ( ( C
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal )
& ( B != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_add_le_add_iff2
thf(fact_908_ereal__add__le__add__iff,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ C @ A ) @ ( plus_p7876563987511257093_ereal @ C @ B ) )
= ( ( ord_le1083603963089353582_ereal @ A @ B )
| ( C = extend1530274965995635425_ereal )
| ( ( C
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal )
& ( B != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_add_le_add_iff
thf(fact_909_ereal__divide__le__posI,axiom,
! [X: extended_ereal,Z: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( Z
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ord_le1083603963089353582_ereal @ Z @ ( times_7703590493115627913_ereal @ X @ Y ) )
=> ( ord_le1083603963089353582_ereal @ ( divide8893690120176169980_ereal @ Z @ X ) @ Y ) ) ) ) ).
% ereal_divide_le_posI
thf(fact_910_ereal__le__mult__one__interval,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [Z4: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ Z4 )
=> ( ( ord_le1188267648640031866_ereal @ Z4 @ one_on4623092294121504201_ereal )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ Z4 @ X ) @ Y ) ) )
=> ( ord_le1083603963089353582_ereal @ X @ Y ) ) ) ).
% ereal_le_mult_one_interval
thf(fact_911_mult_Oright__neutral,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ one_on4623092294121504201_ereal )
= A ) ).
% mult.right_neutral
thf(fact_912_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_913_mult__1,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal @ A )
= A ) ).
% mult_1
thf(fact_914_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_915_ereal__divide__one,axiom,
! [X: extended_ereal] :
( ( divide8893690120176169980_ereal @ X @ one_on4623092294121504201_ereal )
= X ) ).
% ereal_divide_one
thf(fact_916_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_917_ereal__plus__1_I3_J,axiom,
( ( plus_p7876563987511257093_ereal @ one_on4623092294121504201_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_plus_1(3)
thf(fact_918_ereal__plus__1_I4_J,axiom,
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ one_on4623092294121504201_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_plus_1(4)
thf(fact_919_inverse__eq__infinity__iff__eq__zero,axiom,
! [X: extended_ereal] :
( ( ( divide8893690120176169980_ereal @ one_on4623092294121504201_ereal @ X )
= extend1530274965995635425_ereal )
= ( X = zero_z2744965634713055877_ereal ) ) ).
% inverse_eq_infinity_iff_eq_zero
thf(fact_920_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_921_zero__neq__one,axiom,
zero_z2744965634713055877_ereal != one_on4623092294121504201_ereal ).
% zero_neq_one
thf(fact_922_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_923_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_924_one__reorient,axiom,
! [X: extended_ereal] :
( ( one_on4623092294121504201_ereal = X )
= ( X = one_on4623092294121504201_ereal ) ) ).
% one_reorient
thf(fact_925_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_926_ereal__times_I1_J,axiom,
one_on4623092294121504201_ereal != extend1530274965995635425_ereal ).
% ereal_times(1)
thf(fact_927_mult_Ocomm__neutral,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ one_on4623092294121504201_ereal )
= A ) ).
% mult.comm_neutral
thf(fact_928_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_929_comm__monoid__mult__class_Omult__1,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_930_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_931_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_932_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_933_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_934_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_935_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_936_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_937_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_938_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_939_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_940_ereal__times_I3_J,axiom,
( one_on4623092294121504201_ereal
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_times(3)
thf(fact_941_zero__less__one__ereal,axiom,
ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).
% zero_less_one_ereal
thf(fact_942_one__not__le__zero__ereal,axiom,
~ ( ord_le1083603963089353582_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).
% one_not_le_zero_ereal
thf(fact_943_ereal__0__less__1,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).
% ereal_0_less_1
thf(fact_944_ereal__one__not__less__zero__ereal,axiom,
~ ( ord_le1188267648640031866_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).
% ereal_one_not_less_zero_ereal
thf(fact_945_mult_Ocomm__monoid__axioms,axiom,
comm_m179930729793757699_ereal @ times_7703590493115627913_ereal @ one_on4623092294121504201_ereal ).
% mult.comm_monoid_axioms
thf(fact_946_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ times_times_nat @ one_one_nat ).
% mult.comm_monoid_axioms
thf(fact_947_mult_Omonoid__axioms,axiom,
monoid425827787017695951_ereal @ times_7703590493115627913_ereal @ one_on4623092294121504201_ereal ).
% mult.monoid_axioms
thf(fact_948_mult_Omonoid__axioms,axiom,
monoid_nat @ times_times_nat @ one_one_nat ).
% mult.monoid_axioms
thf(fact_949_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_950_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_951_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_952_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_953_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_954_ereal__m1__less__0,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ one_on4623092294121504201_ereal ) @ zero_z2744965634713055877_ereal ).
% ereal_m1_less_0
thf(fact_955_ereal__divide__same,axiom,
! [X: extended_ereal] :
( ( ( ( ( abs_ab7465543570706387889_ereal @ X )
= extend1530274965995635425_ereal )
| ( X = zero_z2744965634713055877_ereal ) )
=> ( ( divide8893690120176169980_ereal @ X @ X )
= zero_z2744965634713055877_ereal ) )
& ( ~ ( ( ( abs_ab7465543570706387889_ereal @ X )
= extend1530274965995635425_ereal )
| ( X = zero_z2744965634713055877_ereal ) )
=> ( ( divide8893690120176169980_ereal @ X @ X )
= one_on4623092294121504201_ereal ) ) ) ).
% ereal_divide_same
thf(fact_956_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_957_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_958_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_959_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_960_numeral__times__numeral,axiom,
! [M2: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).
% numeral_times_numeral
thf(fact_961_add__numeral__left,axiom,
! [V: num,W2: num,Z: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ V ) @ ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ W2 ) @ Z ) )
= ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_962_add__numeral__left,axiom,
! [V: num,W2: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_963_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ M2 ) @ ( numera1204434989813589363_ereal @ N ) )
= ( numera1204434989813589363_ereal @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_964_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_965_abs__ereal__uminus,axiom,
! [X: extended_ereal] :
( ( abs_ab7465543570706387889_ereal @ ( uminus27091377158695749_ereal @ X ) )
= ( abs_ab7465543570706387889_ereal @ X ) ) ).
% abs_ereal_uminus
thf(fact_966_abs__ereal__zero,axiom,
( ( abs_ab7465543570706387889_ereal @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% abs_ereal_zero
thf(fact_967_abs__ereal__one,axiom,
( ( abs_ab7465543570706387889_ereal @ one_on4623092294121504201_ereal )
= one_on4623092294121504201_ereal ) ).
% abs_ereal_one
thf(fact_968_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_969_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_970_abs__ereal__ge0,axiom,
! [X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X )
=> ( ( abs_ab7465543570706387889_ereal @ X )
= X ) ) ).
% abs_ereal_ge0
thf(fact_971_abs__ereal__less0,axiom,
! [X: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ zero_z2744965634713055877_ereal )
=> ( ( abs_ab7465543570706387889_ereal @ X )
= ( uminus27091377158695749_ereal @ X ) ) ) ).
% abs_ereal_less0
thf(fact_972_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_973_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_974_abs__ereal_Osimps_I2_J,axiom,
( ( abs_ab7465543570706387889_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= extend1530274965995635425_ereal ) ).
% abs_ereal.simps(2)
thf(fact_975_ereal__infinity__cases,axiom,
! [A: extended_ereal] :
( ( A != extend1530274965995635425_ereal )
=> ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( abs_ab7465543570706387889_ereal @ A )
!= extend1530274965995635425_ereal ) ) ) ).
% ereal_infinity_cases
thf(fact_976_abs__eq__infinity__cases,axiom,
! [X: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X )
= extend1530274965995635425_ereal )
=> ( ( X != extend1530274965995635425_ereal )
=> ( X
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% abs_eq_infinity_cases
thf(fact_977_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_978_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_979_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_980_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_981_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p7876563987511257093_ereal @ one_on4623092294121504201_ereal @ ( numera1204434989813589363_ereal @ X ) )
= ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ X ) @ one_on4623092294121504201_ereal ) ) ).
% one_plus_numeral_commute
thf(fact_982_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_983_not__inftyI,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ B @ C )
=> ( ( abs_ab7465543570706387889_ereal @ B )
!= extend1530274965995635425_ereal ) ) ) ).
% not_inftyI
thf(fact_984_ereal__abs__leI,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y )
=> ( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X ) @ Y )
=> ( ord_le1083603963089353582_ereal @ ( abs_ab7465543570706387889_ereal @ X ) @ Y ) ) ) ).
% ereal_abs_leI
thf(fact_985_abs__ereal__pos,axiom,
! [X: extended_ereal] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( abs_ab7465543570706387889_ereal @ X ) ) ).
% abs_ereal_pos
thf(fact_986_ereal__abs__add,axiom,
! [A: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( abs_ab7465543570706387889_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) @ ( plus_p7876563987511257093_ereal @ ( abs_ab7465543570706387889_ereal @ A ) @ ( abs_ab7465543570706387889_ereal @ B ) ) ) ).
% ereal_abs_add
thf(fact_987_ereal__abs__mult,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( abs_ab7465543570706387889_ereal @ ( times_7703590493115627913_ereal @ X @ Y ) )
= ( times_7703590493115627913_ereal @ ( abs_ab7465543570706387889_ereal @ X ) @ ( abs_ab7465543570706387889_ereal @ Y ) ) ) ).
% ereal_abs_mult
thf(fact_988_abs__ereal_Osimps_I3_J,axiom,
( ( abs_ab7465543570706387889_ereal @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) ).
% abs_ereal.simps(3)
thf(fact_989_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_990_ereal__less__add,axiom,
! [A: extended_ereal,C: extended_ereal,B: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ A )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ C @ B )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% ereal_less_add
thf(fact_991_ereal__divide__eq__0__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ( divide8893690120176169980_ereal @ X @ Y )
= zero_z2744965634713055877_ereal )
= ( ( X = zero_z2744965634713055877_ereal )
| ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal ) ) ) ).
% ereal_divide_eq_0_iff
thf(fact_992_ereal__distrib__left,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ( A != extend1530274965995635425_ereal )
| ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B != extend1530274965995635425_ereal ) )
=> ( ( ( abs_ab7465543570706387889_ereal @ C )
!= extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ C @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C @ A ) @ ( times_7703590493115627913_ereal @ C @ B ) ) ) ) ) ) ).
% ereal_distrib_left
thf(fact_993_ereal__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ( A != extend1530274965995635425_ereal )
| ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B != extend1530274965995635425_ereal ) )
=> ( ( ( abs_ab7465543570706387889_ereal @ C )
!= extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ C )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ A @ C ) @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ) ) ) ).
% ereal_distrib
thf(fact_994_ereal__mult__cancel__left,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= ( times_7703590493115627913_ereal @ A @ C ) )
= ( ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ B @ C ) ) )
| ( A = zero_z2744965634713055877_ereal )
| ( B = C ) ) ) ).
% ereal_mult_cancel_left
thf(fact_995_ereal__between_I2_J,axiom,
! [X: extended_ereal,E2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ E2 )
=> ( ord_le1188267648640031866_ereal @ X @ ( plus_p7876563987511257093_ereal @ X @ E2 ) ) ) ) ).
% ereal_between(2)
thf(fact_996_ereal__divide__eq,axiom,
! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
( ( B != zero_z2744965634713055877_ereal )
=> ( ( ( abs_ab7465543570706387889_ereal @ B )
!= extend1530274965995635425_ereal )
=> ( ( ( divide8893690120176169980_ereal @ A @ B )
= C )
= ( A
= ( times_7703590493115627913_ereal @ B @ C ) ) ) ) ) ).
% ereal_divide_eq
thf(fact_997_ereal__mult__le__mult__iff,axiom,
! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ C )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C @ A ) @ ( times_7703590493115627913_ereal @ C @ B ) )
= ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C )
=> ( ord_le1083603963089353582_ereal @ A @ B ) )
& ( ( ord_le1188267648640031866_ereal @ C @ zero_z2744965634713055877_ereal )
=> ( ord_le1083603963089353582_ereal @ B @ A ) ) ) ) ) ).
% ereal_mult_le_mult_iff
thf(fact_998_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_999_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_1000_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times_num @ X @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_1001_numeral__inc,axiom,
! [X: num] :
( ( numera1204434989813589363_ereal @ ( inc @ X ) )
= ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ X ) @ one_on4623092294121504201_ereal ) ) ).
% numeral_inc
thf(fact_1002_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_1003_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1004_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1005_semiring__norm_I11_J,axiom,
! [M2: num] :
( ( times_times_num @ M2 @ one )
= M2 ) ).
% semiring_norm(11)
thf(fact_1006_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1007_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1008_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1009_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1010_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1011_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p7876563987511257093_ereal @ one_on4623092294121504201_ereal @ ( numera1204434989813589363_ereal @ N ) )
= ( numera1204434989813589363_ereal @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1012_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1013_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p7876563987511257093_ereal @ ( numera1204434989813589363_ereal @ N ) @ one_on4623092294121504201_ereal )
= ( numera1204434989813589363_ereal @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1014_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1015_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_1016_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_1017_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1018_numeral__One,axiom,
( ( numera1204434989813589363_ereal @ one )
= one_on4623092294121504201_ereal ) ).
% numeral_One
thf(fact_1019_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1020_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1021_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1022_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_1023_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_1024_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1025_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1026_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1027_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1028_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1029_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1030_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1031_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1032_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1033_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1034_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1035_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1036_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1037_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1038_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1039_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1040_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1041_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1042_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1043_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1044_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1045_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1046_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1047_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1048_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1049_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1050_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1051_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1052_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1053_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1054_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1055_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1056_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1057_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1058_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1059_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1060_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1061_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1062_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1063_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1064_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1065_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1066_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1067_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1068_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1069_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1070_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1071_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1072_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1073_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1074_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1075_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1076_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1077_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1078_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1079_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1080_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1081_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1082_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1083_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1084_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1085_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1086_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1087_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1088_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1089_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1090_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_1091_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_1092_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1093_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1094_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1095_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1096_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1097_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1098_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1099_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1100_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1101_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1102_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1103_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1104_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
% Conjectures (1)
thf(conj_0,conjecture,
( ( prefix3213528784805800034_count @ ( prefix5314359684614007693append @ x @ y ) )
= ( plus_p7876563987511257093_ereal @ ( prefix3213528784805800034_count @ x ) @ ( prefix3213528784805800034_count @ y ) ) ) ).
%------------------------------------------------------------------------------