TPTP Problem File: SLH0296^1.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 : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_00037_001022__16095564_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1585 ( 645 unt; 316 typ; 0 def)
% Number of atoms : 3690 (1180 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 10419 ( 224 ~; 60 |; 373 &;8280 @)
% ( 0 <=>;1482 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 50 ( 49 usr)
% Number of type conns : 669 ( 669 >; 0 *; 0 +; 0 <<)
% Number of symbols : 270 ( 267 usr; 13 con; 0-3 aty)
% Number of variables : 3334 ( 371 ^;2782 !; 181 ?;3334 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:47:54.176
%------------------------------------------------------------------------------
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ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nonnegative____Real__Oennreal,type,
power_6007165696250533058nnreal: extend8495563244428889912nnreal > nat > extend8495563244428889912nnreal ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Real__Oereal,type,
power_1054015426188190660_ereal: extended_ereal > nat > extended_ereal ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
collec2260605976452661553d_enat: ( set_Extended_enat > $o ) > set_se7270636423289371942d_enat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
collec9191517506022579601d_enat: ( set_se7270636423289371942d_enat > $o ) > set_se6182022730614456710d_enat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
collec7971272359403355785_set_a: ( set_set_set_a > $o ) > set_set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
is_singleton_complex: set_complex > $o ).
thf(sy_c_Set_Ois__singleton_001t__Extended____Nat__Oenat,type,
is_sin1871519699599484762d_enat: set_Extended_enat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
is_singleton_int: set_int > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Complex__Ocomplex,type,
sunflower_complex: set_set_complex > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Extended____Nat__Oenat,type,
sunflo6578399698544687513d_enat: set_se7270636423289371942d_enat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Int__Oint,type,
sunflower_int: set_set_int > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Nat__Onat,type,
sunflower_nat: set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
sunflo1977439652906400249d_enat: set_se6182022730614456710d_enat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Int__Oint_J,type,
sunflower_set_int: set_set_set_int > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Nat__Onat_J,type,
sunflower_set_nat: set_set_set_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sunflower_set_set_a: set_set_set_set_a > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Set__Oset_Itf__a_J,type,
sunflower_set_a: set_set_set_a > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
sunflo5758272774145246570ex_nat: set_se2148935472528660553ex_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Extended____Nat__Oenat_Mt__Nat__Onat_J,type,
sunflo4400997697864936730at_nat: set_se4292523913457816871at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J,type,
sunflo2840805039424630504nt_nat: set_se7091771524856616391nt_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sunflo1841451327523575948at_nat: set_se3873067930692246379at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Extended____Nat__Oenat_J_Mt__Nat__Onat_J,type,
sunflo6204964536344185210at_nat: set_se1186208456680207751at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
sunflo8837005380533273758nt_nat: set_se3658823449513578877nt_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
sunflo6650083805840251970at_nat: set_se8003284279568041249at_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_Itf__a_J_J_Mt__Nat__Onat_J,type,
sunflo722587752449041010_a_nat: set_se6906472397742828927_a_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
sunflo8923959888101223698_a_nat: set_se4329446959406904351_a_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sunflo8619437272074122162_a_nat: set_se4904748513628223167_a_nat > $o ).
thf(sy_c_Sunflower_Osunflower_001tf__a,type,
sunflower_a: set_set_a > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Extended____Nat__Oenat,type,
member_Extended_enat: extended_enat > set_Extended_enat > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
member350739656593644271d_enat: set_Extended_enat > set_se7270636423289371942d_enat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
member111414017764802383d_enat: set_se7270636423289371942d_enat > set_se6182022730614456710d_enat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member_set_set_set_a: set_set_set_a > set_set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Complex__Ocomplex_Mt__Nat__Onat_J_J,type,
member3193062100113663146ex_nat: set_Su8911350359323359379ex_nat > set_se2148935472528660553ex_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Extended____Nat__Oenat_Mt__Nat__Onat_J_J,type,
member6581178323384882544at_nat: set_Su4628009649352508615at_nat > set_se4292523913457816871at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Int__Oint_Mt__Nat__Onat_J_J,type,
member4056137903419529512nt_nat: set_Sum_sum_int_nat > set_se7091771524856616391nt_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1869216328726507724at_nat: set_Sum_sum_nat_nat > set_se3873067930692246379at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Extended____Nat__Oenat_J_Mt__Nat__Onat_J_J,type,
member8457675980879877584at_nat: set_Su2351696092937267495at_nat > set_se1186208456680207751at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
member8593605234572697566nt_nat: set_Su2054411880199856583nt_nat > set_se3658823449513578877nt_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member5374901640408327554at_nat: set_Su8059080322890262379at_nat > set_se8003284279568041249at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_It__Set__Oset_Itf__a_J_J_Mt__Nat__Onat_J_J,type,
member2313308252142779848_a_nat: set_Su6024358463950899231_a_nat > set_se6906472397742828927_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J_J,type,
member8795406846318732392_a_nat: set_Su8246183859449809599_a_nat > set_se4329446959406904351_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8098812455498974984_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_X,type,
x: set_a ).
% Relevant facts (1264)
thf(fact_0_finite__Collect__subsets,axiom,
! [A: set_set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( finite4249678464180374575et_int
@ ( collect_set_set_int
@ ^ [B: set_set_int] : ( ord_le4403425263959731960et_int @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1_finite__Collect__subsets,axiom,
! [A: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ A )
=> ( finite7012531805024004687d_enat
@ ( collec9191517506022579601d_enat
@ ^ [B: set_se7270636423289371942d_enat] : ( ord_le8264872500312945862d_enat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2_finite__Collect__subsets,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3_finite__Collect__subsets,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( finite5318320746233006407_set_a
@ ( collec7971272359403355785_set_a
@ ^ [B: set_set_set_a] : ( ord_le5722252365846178494_set_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_4_finite__Collect__subsets,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B: set_complex] : ( ord_le211207098394363844omplex @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_5_finite__Collect__subsets,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_6_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B: set_nat] : ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_7_finite__Collect__subsets,axiom,
! [A: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A )
=> ( finite5468666774076196335d_enat
@ ( collec2260605976452661553d_enat
@ ^ [B: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_8_finite__Collect__subsets,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B: set_int] : ( ord_less_eq_set_int @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_9_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B: set_a] : ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_10_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_11_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_12_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_13_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_14_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_15_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_16_card__2__iff_H,axiom,
! [S: set_int] :
( ( ( finite_card_int @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: int] :
( ( member_int @ X @ S )
& ? [Y: int] :
( ( member_int @ Y @ S )
& ( X != Y )
& ! [Z: int] :
( ( member_int @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_17_card__2__iff_H,axiom,
! [S: set_Extended_enat] :
( ( ( finite121521170596916366d_enat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: extended_enat] :
( ( member_Extended_enat @ X @ S )
& ? [Y: extended_enat] :
( ( member_Extended_enat @ Y @ S )
& ( X != Y )
& ! [Z: extended_enat] :
( ( member_Extended_enat @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_18_card__2__iff_H,axiom,
! [S: set_set_a] :
( ( ( finite_card_set_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ S )
& ? [Y: set_a] :
( ( member_set_a @ Y @ S )
& ( X != Y )
& ! [Z: set_a] :
( ( member_set_a @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_19_card__2__iff_H,axiom,
! [S: set_a] :
( ( ( finite_card_a @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: a] :
( ( member_a @ X @ S )
& ? [Y: a] :
( ( member_a @ Y @ S )
& ( X != Y )
& ! [Z: a] :
( ( member_a @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_20_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ S )
& ? [Y: nat] :
( ( member_nat @ Y @ S )
& ( X != Y )
& ! [Z: nat] :
( ( member_nat @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_21_card__2__iff_H,axiom,
! [S: set_complex] :
( ( ( finite_card_complex @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X: complex] :
( ( member_complex @ X @ S )
& ? [Y: complex] :
( ( member_complex @ Y @ S )
& ( X != Y )
& ! [Z: complex] :
( ( member_complex @ Z @ S )
=> ( ( Z = X )
| ( Z = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_22_sameprod__altdef,axiom,
! [X2: set_Extended_enat] :
( ( clique5877592057054161012d_enat @ X2 @ X2 )
= ( collec2260605976452661553d_enat
@ ^ [Y2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ Y2 @ X2 )
& ( ( finite121521170596916366d_enat @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_23_sameprod__altdef,axiom,
! [X2: set_set_a] :
( ( clique4415459440104970860_set_a @ X2 @ X2 )
= ( collect_set_set_a
@ ^ [Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X2 )
& ( ( finite_card_set_a @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_24_sameprod__altdef,axiom,
! [X2: set_int] :
( ( clique6719711917653413022od_int @ X2 @ X2 )
= ( collect_set_int
@ ^ [Y2: set_int] :
( ( ord_less_eq_set_int @ Y2 @ X2 )
& ( ( finite_card_int @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_25_sameprod__altdef,axiom,
! [X2: set_nat] :
( ( clique6722202388162463298od_nat @ X2 @ X2 )
= ( collect_set_nat
@ ^ [Y2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
& ( ( finite_card_nat @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_26_sameprod__altdef,axiom,
! [X2: set_complex] :
( ( clique7858167266224639776omplex @ X2 @ X2 )
= ( collect_set_complex
@ ^ [Y2: set_complex] :
( ( ord_le211207098394363844omplex @ Y2 @ X2 )
& ( ( finite_card_complex @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_27_sameprod__altdef,axiom,
! [X2: set_a] :
( ( clique9072761800073521420prod_a @ X2 @ X2 )
= ( collect_set_a
@ ^ [Y2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
& ( ( finite_card_a @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_28_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_29_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_30_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_31_finite__Collect__conjI,axiom,
! [P: extended_enat > $o,Q: extended_enat > $o] :
( ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
| ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) )
=> ( finite4001608067531595151d_enat
@ ( collec4429806609662206161d_enat
@ ^ [X: extended_enat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_32_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_33_finite__Collect__conjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_34_finite__Collect__conjI,axiom,
! [P: set_complex > $o,Q: set_complex > $o] :
( ( ( finite6551019134538273531omplex @ ( collect_set_complex @ P ) )
| ( finite6551019134538273531omplex @ ( collect_set_complex @ Q ) ) )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [X: set_complex] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_35_finite__Collect__conjI,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( ( finite6197958912794628473et_int @ ( collect_set_int @ P ) )
| ( finite6197958912794628473et_int @ ( collect_set_int @ Q ) ) )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [X: set_int] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_36_finite__Collect__conjI,axiom,
! [P: set_Extended_enat > $o,Q: set_Extended_enat > $o] :
( ( ( finite5468666774076196335d_enat @ ( collec2260605976452661553d_enat @ P ) )
| ( finite5468666774076196335d_enat @ ( collec2260605976452661553d_enat @ Q ) ) )
=> ( finite5468666774076196335d_enat
@ ( collec2260605976452661553d_enat
@ ^ [X: set_Extended_enat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_37_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_38_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_39_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_40_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_41_finite__Collect__disjI,axiom,
! [P: extended_enat > $o,Q: extended_enat > $o] :
( ( finite4001608067531595151d_enat
@ ( collec4429806609662206161d_enat
@ ^ [X: extended_enat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
& ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_42_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X: int] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_43_finite__Collect__disjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_44_finite__Collect__disjI,axiom,
! [P: set_complex > $o,Q: set_complex > $o] :
( ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [X: set_complex] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite6551019134538273531omplex @ ( collect_set_complex @ P ) )
& ( finite6551019134538273531omplex @ ( collect_set_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_45_finite__Collect__disjI,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [X: set_int] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite6197958912794628473et_int @ ( collect_set_int @ P ) )
& ( finite6197958912794628473et_int @ ( collect_set_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_46_finite__Collect__disjI,axiom,
! [P: set_Extended_enat > $o,Q: set_Extended_enat > $o] :
( ( finite5468666774076196335d_enat
@ ( collec2260605976452661553d_enat
@ ^ [X: set_Extended_enat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite5468666774076196335d_enat @ ( collec2260605976452661553d_enat @ P ) )
& ( finite5468666774076196335d_enat @ ( collec2260605976452661553d_enat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_47_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_48_card__subset__eq,axiom,
! [B2: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A @ B2 )
=> ( ( ( finite_card_set_int @ A )
= ( finite_card_set_int @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_49_card__subset__eq,axiom,
! [B2: set_se7270636423289371942d_enat,A: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ B2 )
=> ( ( ord_le8264872500312945862d_enat @ A @ B2 )
=> ( ( ( finite3719263829065406702d_enat @ A )
= ( finite3719263829065406702d_enat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_50_card__subset__eq,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ( finite_card_set_nat @ A )
= ( finite_card_set_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_51_card__subset__eq,axiom,
! [B2: set_set_set_a,A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ A @ B2 )
=> ( ( ( finite6524359278146944486_set_a @ A )
= ( finite6524359278146944486_set_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_52_card__subset__eq,axiom,
! [B2: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
=> ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ( finite_card_complex @ A )
= ( finite_card_complex @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_53_card__subset__eq,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( finite_card_set_a @ A )
= ( finite_card_set_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_54_card__subset__eq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_55_card__subset__eq,axiom,
! [B2: set_Extended_enat,A: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ B2 )
=> ( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( ( finite121521170596916366d_enat @ A )
= ( finite121521170596916366d_enat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_56_card__subset__eq,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ( finite_card_int @ A )
= ( finite_card_int @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_57_card__subset__eq,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_58_infinite__arbitrarily__large,axiom,
! [A: set_set_int,N: nat] :
( ~ ( finite6197958912794628473et_int @ A )
=> ? [B3: set_set_int] :
( ( finite6197958912794628473et_int @ B3 )
& ( ( finite_card_set_int @ B3 )
= N )
& ( ord_le4403425263959731960et_int @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_59_infinite__arbitrarily__large,axiom,
! [A: set_se7270636423289371942d_enat,N: nat] :
( ~ ( finite5468666774076196335d_enat @ A )
=> ? [B3: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ B3 )
& ( ( finite3719263829065406702d_enat @ B3 )
= N )
& ( ord_le8264872500312945862d_enat @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_60_infinite__arbitrarily__large,axiom,
! [A: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ( finite_card_set_nat @ B3 )
= N )
& ( ord_le6893508408891458716et_nat @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_61_infinite__arbitrarily__large,axiom,
! [A: set_set_set_a,N: nat] :
( ~ ( finite7209287970140883943_set_a @ A )
=> ? [B3: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B3 )
& ( ( finite6524359278146944486_set_a @ B3 )
= N )
& ( ord_le5722252365846178494_set_a @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_62_infinite__arbitrarily__large,axiom,
! [A: set_complex,N: nat] :
( ~ ( finite3207457112153483333omplex @ A )
=> ? [B3: set_complex] :
( ( finite3207457112153483333omplex @ B3 )
& ( ( finite_card_complex @ B3 )
= N )
& ( ord_le211207098394363844omplex @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_63_infinite__arbitrarily__large,axiom,
! [A: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A )
=> ? [B3: set_set_a] :
( ( finite_finite_set_a @ B3 )
& ( ( finite_card_set_a @ B3 )
= N )
& ( ord_le3724670747650509150_set_a @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_64_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ( finite_card_nat @ B3 )
= N )
& ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_65_infinite__arbitrarily__large,axiom,
! [A: set_Extended_enat,N: nat] :
( ~ ( finite4001608067531595151d_enat @ A )
=> ? [B3: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ B3 )
& ( ( finite121521170596916366d_enat @ B3 )
= N )
& ( ord_le7203529160286727270d_enat @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_66_infinite__arbitrarily__large,axiom,
! [A: set_int,N: nat] :
( ~ ( finite_finite_int @ A )
=> ? [B3: set_int] :
( ( finite_finite_int @ B3 )
& ( ( finite_card_int @ B3 )
= N )
& ( ord_less_eq_set_int @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_67_infinite__arbitrarily__large,axiom,
! [A: set_a,N: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ( finite_card_a @ B3 )
= N )
& ( ord_less_eq_set_a @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_68_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_69_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera1916890842035813515d_enat @ M )
= ( numera1916890842035813515d_enat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_70_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_71_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_72_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_73_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_74_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_75_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_76_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_77_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_78_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A )
=> ? [B4: a] :
( ( member_a @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B5: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_79_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ? [B4: a] :
( ( member_a @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_80_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_complex,R: complex > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A )
=> ? [B4: a] :
( ( member_a @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: complex,A22: complex,B5: a] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_a @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_81_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_int,R: int > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ A )
=> ? [B4: a] :
( ( member_a @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: int,A22: int,B5: a] :
( ( member_int @ A1 @ A )
=> ( ( member_int @ A22 @ A )
=> ( ( member_a @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_82_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_Extended_enat,R: extended_enat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A2: extended_enat] :
( ( member_Extended_enat @ A2 @ A )
=> ? [B4: a] :
( ( member_a @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: extended_enat,A22: extended_enat,B5: a] :
( ( member_Extended_enat @ A1 @ A )
=> ( ( member_Extended_enat @ A22 @ A )
=> ( ( member_a @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_83_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B5: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_84_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_85_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_complex,R: complex > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A2: complex] :
( ( member_complex @ A2 @ A )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: complex,A22: complex,B5: nat] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_86_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_int,R: int > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A2: int] :
( ( member_int @ A2 @ A )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: int,A22: int,B5: nat] :
( ( member_int @ A1 @ A )
=> ( ( member_int @ A22 @ A )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_87_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_Extended_enat,R: extended_enat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A2: extended_enat] :
( ( member_Extended_enat @ A2 @ A )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A2 @ B4 ) ) )
=> ( ! [A1: extended_enat,A22: extended_enat,B5: nat] :
( ( member_Extended_enat @ A1 @ A )
=> ( ( member_Extended_enat @ A22 @ A )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_88_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_int,C: nat] :
( ! [G: set_set_int] :
( ( ord_le4403425263959731960et_int @ G @ F )
=> ( ( finite6197958912794628473et_int @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ G ) @ C ) ) )
=> ( ( finite6197958912794628473et_int @ F )
& ( ord_less_eq_nat @ ( finite_card_set_int @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_89_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_se7270636423289371942d_enat,C: nat] :
( ! [G: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ G @ F )
=> ( ( finite5468666774076196335d_enat @ G )
=> ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ G ) @ C ) ) )
=> ( ( finite5468666774076196335d_enat @ F )
& ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_90_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_nat,C: nat] :
( ! [G: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ F )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C ) ) )
=> ( ( finite1152437895449049373et_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_91_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_set_a,C: nat] :
( ! [G: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ G @ F )
=> ( ( finite7209287970140883943_set_a @ G )
=> ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ G ) @ C ) ) )
=> ( ( finite7209287970140883943_set_a @ F )
& ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_92_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_93_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_Extended_enat,C: nat] :
( ! [G: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ G @ F )
=> ( ( finite4001608067531595151d_enat @ G )
=> ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ G ) @ C ) ) )
=> ( ( finite4001608067531595151d_enat @ F )
& ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_94_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_complex,C: nat] :
( ! [G: set_complex] :
( ( ord_le211207098394363844omplex @ G @ F )
=> ( ( finite3207457112153483333omplex @ G )
=> ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C ) ) )
=> ( ( finite3207457112153483333omplex @ F )
& ( ord_less_eq_nat @ ( finite_card_complex @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_95_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_a,C: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C ) ) )
=> ( ( finite_finite_set_a @ F )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_96_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_97_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_int,C: nat] :
( ! [G: set_int] :
( ( ord_less_eq_set_int @ G @ F )
=> ( ( finite_finite_int @ G )
=> ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C ) ) )
=> ( ( finite_finite_int @ F )
& ( ord_less_eq_nat @ ( finite_card_int @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_98_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_int @ S ) )
=> ~ ! [T: set_set_int] :
( ( ord_le4403425263959731960et_int @ T @ S )
=> ( ( ( finite_card_set_int @ T )
= N )
=> ~ ( finite6197958912794628473et_int @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_99_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_se7270636423289371942d_enat] :
( ( ord_less_eq_nat @ N @ ( finite3719263829065406702d_enat @ S ) )
=> ~ ! [T: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ T @ S )
=> ( ( ( finite3719263829065406702d_enat @ T )
= N )
=> ~ ( finite5468666774076196335d_enat @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_100_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S ) )
=> ~ ! [T: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T @ S )
=> ( ( ( finite_card_set_nat @ T )
= N )
=> ~ ( finite1152437895449049373et_nat @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_101_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite6524359278146944486_set_a @ S ) )
=> ~ ! [T: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ T @ S )
=> ( ( ( finite6524359278146944486_set_a @ T )
= N )
=> ~ ( finite7209287970140883943_set_a @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_102_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T: set_a] :
( ( ord_less_eq_set_a @ T @ S )
=> ( ( ( finite_card_a @ T )
= N )
=> ~ ( finite_finite_a @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_103_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Extended_enat] :
( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ S ) )
=> ~ ! [T: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ T @ S )
=> ( ( ( finite121521170596916366d_enat @ T )
= N )
=> ~ ( finite4001608067531595151d_enat @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_104_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_complex] :
( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S ) )
=> ~ ! [T: set_complex] :
( ( ord_le211207098394363844omplex @ T @ S )
=> ( ( ( finite_card_complex @ T )
= N )
=> ~ ( finite3207457112153483333omplex @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_105_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T @ S )
=> ( ( ( finite_card_set_a @ T )
= N )
=> ~ ( finite_finite_set_a @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_106_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T: set_nat] :
( ( ord_less_eq_set_nat @ T @ S )
=> ( ( ( finite_card_nat @ T )
= N )
=> ~ ( finite_finite_nat @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_107_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
=> ~ ! [T: set_int] :
( ( ord_less_eq_set_int @ T @ S )
=> ( ( ( finite_card_int @ T )
= N )
=> ~ ( finite_finite_int @ T ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_108_exists__subset__between,axiom,
! [A: set_set_int,N: nat,C: set_set_int] :
( ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_int @ C ) )
=> ( ( ord_le4403425263959731960et_int @ A @ C )
=> ( ( finite6197958912794628473et_int @ C )
=> ? [B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B3 )
& ( ord_le4403425263959731960et_int @ B3 @ C )
& ( ( finite_card_set_int @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_109_exists__subset__between,axiom,
! [A: set_se7270636423289371942d_enat,N: nat,C: set_se7270636423289371942d_enat] :
( ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite3719263829065406702d_enat @ C ) )
=> ( ( ord_le8264872500312945862d_enat @ A @ C )
=> ( ( finite5468666774076196335d_enat @ C )
=> ? [B3: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ A @ B3 )
& ( ord_le8264872500312945862d_enat @ B3 @ C )
& ( ( finite3719263829065406702d_enat @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_110_exists__subset__between,axiom,
! [A: set_set_nat,N: nat,C: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( finite1152437895449049373et_nat @ C )
=> ? [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ C )
& ( ( finite_card_set_nat @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_111_exists__subset__between,axiom,
! [A: set_set_set_a,N: nat,C: set_set_set_a] :
( ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite6524359278146944486_set_a @ C ) )
=> ( ( ord_le5722252365846178494_set_a @ A @ C )
=> ( ( finite7209287970140883943_set_a @ C )
=> ? [B3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B3 )
& ( ord_le5722252365846178494_set_a @ B3 @ C )
& ( ( finite6524359278146944486_set_a @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_112_exists__subset__between,axiom,
! [A: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( finite_finite_a @ C )
=> ? [B3: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
& ( ord_less_eq_set_a @ B3 @ C )
& ( ( finite_card_a @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_113_exists__subset__between,axiom,
! [A: set_Extended_enat,N: nat,C: set_Extended_enat] :
( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ C ) )
=> ( ( ord_le7203529160286727270d_enat @ A @ C )
=> ( ( finite4001608067531595151d_enat @ C )
=> ? [B3: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B3 )
& ( ord_le7203529160286727270d_enat @ B3 @ C )
& ( ( finite121521170596916366d_enat @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_114_exists__subset__between,axiom,
! [A: set_complex,N: nat,C: set_complex] :
( ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ C ) )
=> ( ( ord_le211207098394363844omplex @ A @ C )
=> ( ( finite3207457112153483333omplex @ C )
=> ? [B3: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B3 )
& ( ord_le211207098394363844omplex @ B3 @ C )
& ( ( finite_card_complex @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_115_exists__subset__between,axiom,
! [A: set_set_a,N: nat,C: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( finite_finite_set_a @ C )
=> ? [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ C )
& ( ( finite_card_set_a @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_116_exists__subset__between,axiom,
! [A: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
& ( ord_less_eq_set_nat @ B3 @ C )
& ( ( finite_card_nat @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_117_exists__subset__between,axiom,
! [A: set_int,N: nat,C: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_int @ C ) )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( finite_finite_int @ C )
=> ? [B3: set_int] :
( ( ord_less_eq_set_int @ A @ B3 )
& ( ord_less_eq_set_int @ B3 @ C )
& ( ( finite_card_int @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_118_card__seteq,axiom,
! [B2: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ B2 ) @ ( finite_card_set_int @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_119_card__seteq,axiom,
! [B2: set_se7270636423289371942d_enat,A: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ B2 )
=> ( ( ord_le8264872500312945862d_enat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ B2 ) @ ( finite3719263829065406702d_enat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_120_card__seteq,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_121_card__seteq,axiom,
! [B2: set_set_set_a,A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ B2 ) @ ( finite6524359278146944486_set_a @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_122_card__seteq,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_123_card__seteq,axiom,
! [B2: set_Extended_enat,A: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ B2 )
=> ( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ B2 ) @ ( finite121521170596916366d_enat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_124_card__seteq,axiom,
! [B2: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
=> ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ B2 ) @ ( finite_card_complex @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_125_card__seteq,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B2 ) @ ( finite_card_set_a @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_126_card__seteq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_127_card__seteq,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_128_card__mono,axiom,
! [B2: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_int @ A ) @ ( finite_card_set_int @ B2 ) ) ) ) ).
% card_mono
thf(fact_129_card__mono,axiom,
! [B2: set_se7270636423289371942d_enat,A: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ B2 )
=> ( ( ord_le8264872500312945862d_enat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ A ) @ ( finite3719263829065406702d_enat @ B2 ) ) ) ) ).
% card_mono
thf(fact_130_card__mono,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_131_card__mono,axiom,
! [B2: set_set_set_a,A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ A ) @ ( finite6524359278146944486_set_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_132_card__mono,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_133_card__mono,axiom,
! [B2: set_Extended_enat,A: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ B2 )
=> ( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).
% card_mono
thf(fact_134_card__mono,axiom,
! [B2: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
=> ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B2 ) ) ) ) ).
% card_mono
thf(fact_135_card__mono,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_136_card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_137_card__mono,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) ) ) ) ).
% card_mono
thf(fact_138_sameprod__mono,axiom,
! [X2: set_Extended_enat,Y3: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ X2 @ Y3 )
=> ( ord_le8264872500312945862d_enat @ ( clique5877592057054161012d_enat @ X2 @ X2 ) @ ( clique5877592057054161012d_enat @ Y3 @ Y3 ) ) ) ).
% sameprod_mono
thf(fact_139_sameprod__mono,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ord_le5722252365846178494_set_a @ ( clique4415459440104970860_set_a @ X2 @ X2 ) @ ( clique4415459440104970860_set_a @ Y3 @ Y3 ) ) ) ).
% sameprod_mono
thf(fact_140_sameprod__mono,axiom,
! [X2: set_int,Y3: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y3 )
=> ( ord_le4403425263959731960et_int @ ( clique6719711917653413022od_int @ X2 @ X2 ) @ ( clique6719711917653413022od_int @ Y3 @ Y3 ) ) ) ).
% sameprod_mono
thf(fact_141_sameprod__mono,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( clique9072761800073521420prod_a @ X2 @ X2 ) @ ( clique9072761800073521420prod_a @ Y3 @ Y3 ) ) ) ).
% sameprod_mono
thf(fact_142_sameprod__mono,axiom,
! [X2: set_complex,Y3: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y3 )
=> ( ord_le4750530260501030778omplex @ ( clique7858167266224639776omplex @ X2 @ X2 ) @ ( clique7858167266224639776omplex @ Y3 @ Y3 ) ) ) ).
% sameprod_mono
thf(fact_143_sameprod__mono,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X2 @ X2 ) @ ( clique6722202388162463298od_nat @ Y3 @ Y3 ) ) ) ).
% sameprod_mono
thf(fact_144_verit__la__disequality,axiom,
! [A3: real,B6: real] :
( ( A3 = B6 )
| ~ ( ord_less_eq_real @ A3 @ B6 )
| ~ ( ord_less_eq_real @ B6 @ A3 ) ) ).
% verit_la_disequality
thf(fact_145_verit__la__disequality,axiom,
! [A3: num,B6: num] :
( ( A3 = B6 )
| ~ ( ord_less_eq_num @ A3 @ B6 )
| ~ ( ord_less_eq_num @ B6 @ A3 ) ) ).
% verit_la_disequality
thf(fact_146_verit__la__disequality,axiom,
! [A3: nat,B6: nat] :
( ( A3 = B6 )
| ~ ( ord_less_eq_nat @ A3 @ B6 )
| ~ ( ord_less_eq_nat @ B6 @ A3 ) ) ).
% verit_la_disequality
thf(fact_147_verit__la__disequality,axiom,
! [A3: int,B6: int] :
( ( A3 = B6 )
| ~ ( ord_less_eq_int @ A3 @ B6 )
| ~ ( ord_less_eq_int @ B6 @ A3 ) ) ).
% verit_la_disequality
thf(fact_148_verit__comp__simplify1_I2_J,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_149_verit__comp__simplify1_I2_J,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_150_verit__comp__simplify1_I2_J,axiom,
! [A3: num] : ( ord_less_eq_num @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_151_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_152_verit__comp__simplify1_I2_J,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_153_verit__comp__simplify1_I2_J,axiom,
! [A3: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_154_verit__comp__simplify1_I2_J,axiom,
! [A3: set_complex] : ( ord_le211207098394363844omplex @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_155_verit__comp__simplify1_I2_J,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_156_verit__comp__simplify1_I2_J,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_157_verit__comp__simplify1_I2_J,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_158_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_a,R2: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: a] :
( ( member_a @ X4 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_159_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_nat,R2: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_160_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_Extended_enat,R2: a > extended_enat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite4001608067531595151d_enat @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_161_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_int,R2: a > int > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_162_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_complex,R2: a > complex > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_163_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_a,R2: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: a] :
( ( member_a @ X4 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_164_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_165_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_Extended_enat,R2: nat > extended_enat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite4001608067531595151d_enat @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_166_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_int,R2: nat > int > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_167_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_complex,R2: nat > complex > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B2 )
& ( R2 @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_168_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_169_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_170_not__finite__existsD,axiom,
! [P: extended_enat > $o] :
( ~ ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
=> ? [X_1: extended_enat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_171_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_172_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_1: complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_173_not__finite__existsD,axiom,
! [P: set_complex > $o] :
( ~ ( finite6551019134538273531omplex @ ( collect_set_complex @ P ) )
=> ? [X_1: set_complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_174_not__finite__existsD,axiom,
! [P: set_a > $o] :
( ~ ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ? [X_1: set_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_175_not__finite__existsD,axiom,
! [P: set_int > $o] :
( ~ ( finite6197958912794628473et_int @ ( collect_set_int @ P ) )
=> ? [X_1: set_int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_176_not__finite__existsD,axiom,
! [P: set_Extended_enat > $o] :
( ~ ( finite5468666774076196335d_enat @ ( collec2260605976452661553d_enat @ P ) )
=> ? [X_1: set_Extended_enat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_177_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_178_finite__has__minimal2,axiom,
! [A: set_Extended_enat,A3: extended_enat] :
( ( finite4001608067531595151d_enat @ A )
=> ( ( member_Extended_enat @ A3 @ A )
=> ? [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ A )
& ( ord_le2932123472753598470d_enat @ X4 @ A3 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A )
=> ( ( ord_le2932123472753598470d_enat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_179_finite__has__minimal2,axiom,
! [A: set_complex,A3: complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( member_complex @ A3 @ A )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ord_less_eq_complex @ X4 @ A3 )
& ! [Xa: complex] :
( ( member_complex @ Xa @ A )
=> ( ( ord_less_eq_complex @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_180_finite__has__minimal2,axiom,
! [A: set_real,A3: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_real @ X4 @ A3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_181_finite__has__minimal2,axiom,
! [A: set_num,A3: num] :
( ( finite_finite_num @ A )
=> ( ( member_num @ A3 @ A )
=> ? [X4: num] :
( ( member_num @ X4 @ A )
& ( ord_less_eq_num @ X4 @ A3 )
& ! [Xa: num] :
( ( member_num @ Xa @ A )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_182_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ X4 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_183_finite__has__minimal2,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A3 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_eq_int @ X4 @ A3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_184_finite__has__minimal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_185_finite__has__minimal2,axiom,
! [A: set_se7270636423289371942d_enat,A3: set_Extended_enat] :
( ( finite5468666774076196335d_enat @ A )
=> ( ( member350739656593644271d_enat @ A3 @ A )
=> ? [X4: set_Extended_enat] :
( ( member350739656593644271d_enat @ X4 @ A )
& ( ord_le7203529160286727270d_enat @ X4 @ A3 )
& ! [Xa: set_Extended_enat] :
( ( member350739656593644271d_enat @ Xa @ A )
=> ( ( ord_le7203529160286727270d_enat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_186_finite__has__minimal2,axiom,
! [A: set_set_complex,A3: set_complex] :
( ( finite6551019134538273531omplex @ A )
=> ( ( member_set_complex @ A3 @ A )
=> ? [X4: set_complex] :
( ( member_set_complex @ X4 @ A )
& ( ord_le211207098394363844omplex @ X4 @ A3 )
& ! [Xa: set_complex] :
( ( member_set_complex @ Xa @ A )
=> ( ( ord_le211207098394363844omplex @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_187_finite__has__minimal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ X4 @ A3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_188_finite__has__maximal2,axiom,
! [A: set_Extended_enat,A3: extended_enat] :
( ( finite4001608067531595151d_enat @ A )
=> ( ( member_Extended_enat @ A3 @ A )
=> ? [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ A )
& ( ord_le2932123472753598470d_enat @ A3 @ X4 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A )
=> ( ( ord_le2932123472753598470d_enat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_189_finite__has__maximal2,axiom,
! [A: set_complex,A3: complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( member_complex @ A3 @ A )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A )
& ( ord_less_eq_complex @ A3 @ X4 )
& ! [Xa: complex] :
( ( member_complex @ Xa @ A )
=> ( ( ord_less_eq_complex @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_190_finite__has__maximal2,axiom,
! [A: set_real,A3: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A3 @ A )
=> ? [X4: real] :
( ( member_real @ X4 @ A )
& ( ord_less_eq_real @ A3 @ X4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_191_finite__has__maximal2,axiom,
! [A: set_num,A3: num] :
( ( finite_finite_num @ A )
=> ( ( member_num @ A3 @ A )
=> ? [X4: num] :
( ( member_num @ X4 @ A )
& ( ord_less_eq_num @ A3 @ X4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_192_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ A3 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_193_finite__has__maximal2,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A3 @ A )
=> ? [X4: int] :
( ( member_int @ X4 @ A )
& ( ord_less_eq_int @ A3 @ X4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_194_finite__has__maximal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ A3 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_195_finite__has__maximal2,axiom,
! [A: set_se7270636423289371942d_enat,A3: set_Extended_enat] :
( ( finite5468666774076196335d_enat @ A )
=> ( ( member350739656593644271d_enat @ A3 @ A )
=> ? [X4: set_Extended_enat] :
( ( member350739656593644271d_enat @ X4 @ A )
& ( ord_le7203529160286727270d_enat @ A3 @ X4 )
& ! [Xa: set_Extended_enat] :
( ( member350739656593644271d_enat @ Xa @ A )
=> ( ( ord_le7203529160286727270d_enat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_196_finite__has__maximal2,axiom,
! [A: set_set_complex,A3: set_complex] :
( ( finite6551019134538273531omplex @ A )
=> ( ( member_set_complex @ A3 @ A )
=> ? [X4: set_complex] :
( ( member_set_complex @ X4 @ A )
& ( ord_le211207098394363844omplex @ A3 @ X4 )
& ! [Xa: set_complex] :
( ( member_set_complex @ Xa @ A )
=> ( ( ord_le211207098394363844omplex @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_197_finite__has__maximal2,axiom,
! [A: set_set_nat,A3: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A3 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ A3 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_198_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_199_rev__finite__subset,axiom,
! [B2: set_set_int,A: set_set_int] :
( ( finite6197958912794628473et_int @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A @ B2 )
=> ( finite6197958912794628473et_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_200_rev__finite__subset,axiom,
! [B2: set_se7270636423289371942d_enat,A: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ B2 )
=> ( ( ord_le8264872500312945862d_enat @ A @ B2 )
=> ( finite5468666774076196335d_enat @ A ) ) ) ).
% rev_finite_subset
thf(fact_201_rev__finite__subset,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_202_rev__finite__subset,axiom,
! [B2: set_set_set_a,A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ A @ B2 )
=> ( finite7209287970140883943_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_203_rev__finite__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_204_rev__finite__subset,axiom,
! [B2: set_Extended_enat,A: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ B2 )
=> ( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( finite4001608067531595151d_enat @ A ) ) ) ).
% rev_finite_subset
thf(fact_205_rev__finite__subset,axiom,
! [B2: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
=> ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% rev_finite_subset
thf(fact_206_rev__finite__subset,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_207_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_208_rev__finite__subset,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( finite_finite_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_209_infinite__super,axiom,
! [S: set_set_int,T2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ T2 )
=> ( ~ ( finite6197958912794628473et_int @ S )
=> ~ ( finite6197958912794628473et_int @ T2 ) ) ) ).
% infinite_super
thf(fact_210_infinite__super,axiom,
! [S: set_se7270636423289371942d_enat,T2: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ S @ T2 )
=> ( ~ ( finite5468666774076196335d_enat @ S )
=> ~ ( finite5468666774076196335d_enat @ T2 ) ) ) ).
% infinite_super
thf(fact_211_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_212_infinite__super,axiom,
! [S: set_set_set_a,T2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ S @ T2 )
=> ( ~ ( finite7209287970140883943_set_a @ S )
=> ~ ( finite7209287970140883943_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_213_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_214_infinite__super,axiom,
! [S: set_Extended_enat,T2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ S @ T2 )
=> ( ~ ( finite4001608067531595151d_enat @ S )
=> ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).
% infinite_super
thf(fact_215_infinite__super,axiom,
! [S: set_complex,T2: set_complex] :
( ( ord_le211207098394363844omplex @ S @ T2 )
=> ( ~ ( finite3207457112153483333omplex @ S )
=> ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% infinite_super
thf(fact_216_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_217_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_218_infinite__super,axiom,
! [S: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S @ T2 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_219_finite__subset,axiom,
! [A: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B2 )
=> ( ( finite6197958912794628473et_int @ B2 )
=> ( finite6197958912794628473et_int @ A ) ) ) ).
% finite_subset
thf(fact_220_finite__subset,axiom,
! [A: set_se7270636423289371942d_enat,B2: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ A @ B2 )
=> ( ( finite5468666774076196335d_enat @ B2 )
=> ( finite5468666774076196335d_enat @ A ) ) ) ).
% finite_subset
thf(fact_221_finite__subset,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_subset
thf(fact_222_finite__subset,axiom,
! [A: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B2 )
=> ( ( finite7209287970140883943_set_a @ B2 )
=> ( finite7209287970140883943_set_a @ A ) ) ) ).
% finite_subset
thf(fact_223_finite__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_224_finite__subset,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( finite4001608067531595151d_enat @ B2 )
=> ( finite4001608067531595151d_enat @ A ) ) ) ).
% finite_subset
thf(fact_225_finite__subset,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( finite3207457112153483333omplex @ B2 )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_subset
thf(fact_226_finite__subset,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_227_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_228_finite__subset,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( finite_finite_int @ B2 )
=> ( finite_finite_int @ A ) ) ) ).
% finite_subset
thf(fact_229_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_230_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_231_subset__antisym,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( ord_le7203529160286727270d_enat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_232_subset__antisym,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_233_subset__antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_234_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_235_subset__antisym,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_236_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_237_subsetI,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ! [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ A )
=> ( member_Extended_enat @ X4 @ B2 ) )
=> ( ord_le7203529160286727270d_enat @ A @ B2 ) ) ).
% subsetI
thf(fact_238_subsetI,axiom,
! [A: set_complex,B2: set_complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ( member_complex @ X4 @ B2 ) )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% subsetI
thf(fact_239_subsetI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_240_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_241_subsetI,axiom,
! [A: set_int,B2: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( member_int @ X4 @ B2 ) )
=> ( ord_less_eq_set_int @ A @ B2 ) ) ).
% subsetI
thf(fact_242_dual__order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_243_dual__order_Orefl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_244_dual__order_Orefl,axiom,
! [A3: num] : ( ord_less_eq_num @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_245_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_246_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_247_dual__order_Orefl,axiom,
! [A3: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_248_dual__order_Orefl,axiom,
! [A3: set_complex] : ( ord_le211207098394363844omplex @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_249_dual__order_Orefl,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_250_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_251_dual__order_Orefl,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_252_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_253_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_254_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_255_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_256_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_257_order__refl,axiom,
! [X3: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ X3 @ X3 ) ).
% order_refl
thf(fact_258_order__refl,axiom,
! [X3: set_complex] : ( ord_le211207098394363844omplex @ X3 @ X3 ) ).
% order_refl
thf(fact_259_order__refl,axiom,
! [X3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_260_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_261_order__refl,axiom,
! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).
% order_refl
thf(fact_262_mem__Collect__eq,axiom,
! [A3: extended_enat,P: extended_enat > $o] :
( ( member_Extended_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_263_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_264_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_265_mem__Collect__eq,axiom,
! [A3: complex,P: complex > $o] :
( ( member_complex @ A3 @ ( collect_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_266_mem__Collect__eq,axiom,
! [A3: int,P: int > $o] :
( ( member_int @ A3 @ ( collect_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_267_mem__Collect__eq,axiom,
! [A3: set_complex,P: set_complex > $o] :
( ( member_set_complex @ A3 @ ( collect_set_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_268_mem__Collect__eq,axiom,
! [A3: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A3 @ ( collect_set_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_269_mem__Collect__eq,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A3 @ ( collect_set_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_270_mem__Collect__eq,axiom,
! [A3: set_int,P: set_int > $o] :
( ( member_set_int @ A3 @ ( collect_set_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_271_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_272_Collect__mem__eq,axiom,
! [A: set_Extended_enat] :
( ( collec4429806609662206161d_enat
@ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_273_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_274_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_275_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X: complex] : ( member_complex @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_276_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_277_Collect__mem__eq,axiom,
! [A: set_set_complex] :
( ( collect_set_complex
@ ^ [X: set_complex] : ( member_set_complex @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_278_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X: set_set_a] : ( member_set_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_279_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_280_Collect__mem__eq,axiom,
! [A: set_set_int] :
( ( collect_set_int
@ ^ [X: set_int] : ( member_set_int @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_281_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_282_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_283_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_284_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X4: complex] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_285_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_286_Collect__cong,axiom,
! [P: set_complex > $o,Q: set_complex > $o] :
( ! [X4: set_complex] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_complex @ P )
= ( collect_set_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_287_Collect__cong,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ! [X4: set_set_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_set_a @ P )
= ( collect_set_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_288_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_289_Collect__cong,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X4: set_int] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_int @ P )
= ( collect_set_int @ Q ) ) ) ).
% Collect_cong
thf(fact_290_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_291_ex__card,axiom,
! [N: nat,A: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ A ) )
=> ? [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ A )
& ( ( finite_card_a @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_292_ex__card,axiom,
! [N: nat,A: set_Extended_enat] :
( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ A ) )
=> ? [S2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ S2 @ A )
& ( ( finite121521170596916366d_enat @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_293_ex__card,axiom,
! [N: nat,A: set_complex] :
( ( ord_less_eq_nat @ N @ ( finite_card_complex @ A ) )
=> ? [S2: set_complex] :
( ( ord_le211207098394363844omplex @ S2 @ A )
& ( ( finite_card_complex @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_294_ex__card,axiom,
! [N: nat,A: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ A ) )
=> ? [S2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ A )
& ( ( finite_card_set_a @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_295_ex__card,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A ) )
=> ? [S2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ A )
& ( ( finite_card_nat @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_296_ex__card,axiom,
! [N: nat,A: set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_int @ A ) )
=> ? [S2: set_int] :
( ( ord_less_eq_set_int @ S2 @ A )
& ( ( finite_card_int @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_297_finite__indexed__bound,axiom,
! [A: set_a,P: a > real > $o] :
( ( finite_finite_a @ A )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [X_12: real] : ( P @ X4 @ X_12 ) )
=> ? [M2: real] :
! [X5: a] :
( ( member_a @ X5 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_298_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [X_12: real] : ( P @ X4 @ X_12 ) )
=> ? [M2: real] :
! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_299_finite__indexed__bound,axiom,
! [A: set_Extended_enat,P: extended_enat > real > $o] :
( ( finite4001608067531595151d_enat @ A )
=> ( ! [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ A )
=> ? [X_12: real] : ( P @ X4 @ X_12 ) )
=> ? [M2: real] :
! [X5: extended_enat] :
( ( member_Extended_enat @ X5 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_300_finite__indexed__bound,axiom,
! [A: set_int,P: int > real > $o] :
( ( finite_finite_int @ A )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A )
=> ? [X_12: real] : ( P @ X4 @ X_12 ) )
=> ? [M2: real] :
! [X5: int] :
( ( member_int @ X5 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_301_finite__indexed__bound,axiom,
! [A: set_complex,P: complex > real > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ? [X_12: real] : ( P @ X4 @ X_12 ) )
=> ? [M2: real] :
! [X5: complex] :
( ( member_complex @ X5 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_302_finite__indexed__bound,axiom,
! [A: set_a,P: a > num > $o] :
( ( finite_finite_a @ A )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [X_12: num] : ( P @ X4 @ X_12 ) )
=> ? [M2: num] :
! [X5: a] :
( ( member_a @ X5 @ A )
=> ? [K: num] :
( ( ord_less_eq_num @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_303_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > num > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [X_12: num] : ( P @ X4 @ X_12 ) )
=> ? [M2: num] :
! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ? [K: num] :
( ( ord_less_eq_num @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_304_finite__indexed__bound,axiom,
! [A: set_Extended_enat,P: extended_enat > num > $o] :
( ( finite4001608067531595151d_enat @ A )
=> ( ! [X4: extended_enat] :
( ( member_Extended_enat @ X4 @ A )
=> ? [X_12: num] : ( P @ X4 @ X_12 ) )
=> ? [M2: num] :
! [X5: extended_enat] :
( ( member_Extended_enat @ X5 @ A )
=> ? [K: num] :
( ( ord_less_eq_num @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_305_finite__indexed__bound,axiom,
! [A: set_int,P: int > num > $o] :
( ( finite_finite_int @ A )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A )
=> ? [X_12: num] : ( P @ X4 @ X_12 ) )
=> ? [M2: num] :
! [X5: int] :
( ( member_int @ X5 @ A )
=> ? [K: num] :
( ( ord_less_eq_num @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_306_finite__indexed__bound,axiom,
! [A: set_complex,P: complex > num > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ? [X_12: num] : ( P @ X4 @ X_12 ) )
=> ? [M2: num] :
! [X5: complex] :
( ( member_complex @ X5 @ A )
=> ? [K: num] :
( ( ord_less_eq_num @ K @ M2 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_307_card__sameprod,axiom,
! [X2: set_set_a] :
( ( finite_finite_set_a @ X2 )
=> ( ( finite6524359278146944486_set_a @ ( clique4415459440104970860_set_a @ X2 @ X2 ) )
= ( binomial @ ( finite_card_set_a @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_308_card__sameprod,axiom,
! [X2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ X2 )
=> ( ( finite3719263829065406702d_enat @ ( clique5877592057054161012d_enat @ X2 @ X2 ) )
= ( binomial @ ( finite121521170596916366d_enat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_309_card__sameprod,axiom,
! [X2: set_int] :
( ( finite_finite_int @ X2 )
=> ( ( finite_card_set_int @ ( clique6719711917653413022od_int @ X2 @ X2 ) )
= ( binomial @ ( finite_card_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_310_card__sameprod,axiom,
! [X2: set_set_int] :
( ( finite6197958912794628473et_int @ X2 )
=> ( ( finite7882580182802147440et_int @ ( clique4728665409795342932et_int @ X2 @ X2 ) )
= ( binomial @ ( finite_card_set_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_311_card__sameprod,axiom,
! [X2: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ X2 )
=> ( ( finite7663542970459504974d_enat @ ( clique7486815425688566996d_enat @ X2 @ X2 ) )
= ( binomial @ ( finite3719263829065406702d_enat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_312_card__sameprod,axiom,
! [X2: set_set_nat] :
( ( finite1152437895449049373et_nat @ X2 )
=> ( ( finite1149291290879098388et_nat @ ( clique8906516429304539640et_nat @ X2 @ X2 ) )
= ( binomial @ ( finite_card_set_nat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_313_card__sameprod,axiom,
! [X2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ X2 )
=> ( ( finite5538070369716010822_set_a @ ( clique4732396645418960844_set_a @ X2 @ X2 ) )
= ( binomial @ ( finite6524359278146944486_set_a @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_314_card__sameprod,axiom,
! [X2: set_a] :
( ( finite_finite_a @ X2 )
=> ( ( finite_card_set_a @ ( clique9072761800073521420prod_a @ X2 @ X2 ) )
= ( binomial @ ( finite_card_a @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_315_card__sameprod,axiom,
! [X2: set_complex] :
( ( finite3207457112153483333omplex @ X2 )
=> ( ( finite903997441450111292omplex @ ( clique7858167266224639776omplex @ X2 @ X2 ) )
= ( binomial @ ( finite_card_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_316_card__sameprod,axiom,
! [X2: set_nat] :
( ( finite_finite_nat @ X2 )
=> ( ( finite_card_set_nat @ ( clique6722202388162463298od_nat @ X2 @ X2 ) )
= ( binomial @ ( finite_card_nat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% card_sameprod
thf(fact_317_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_318_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_319_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_320_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_321_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R2 )
@ ^ [X: a] : ( member_a @ X @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_322_pred__subset__eq,axiom,
! [R2: set_Extended_enat,S: set_Extended_enat] :
( ( ord_le100613205991271927enat_o
@ ^ [X: extended_enat] : ( member_Extended_enat @ X @ R2 )
@ ^ [X: extended_enat] : ( member_Extended_enat @ X @ S ) )
= ( ord_le7203529160286727270d_enat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_323_pred__subset__eq,axiom,
! [R2: set_complex,S: set_complex] :
( ( ord_le4573692005234683329plex_o
@ ^ [X: complex] : ( member_complex @ X @ R2 )
@ ^ [X: complex] : ( member_complex @ X @ S ) )
= ( ord_le211207098394363844omplex @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_324_pred__subset__eq,axiom,
! [R2: set_set_a,S: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ R2 )
@ ^ [X: set_a] : ( member_set_a @ X @ S ) )
= ( ord_le3724670747650509150_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_325_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_326_pred__subset__eq,axiom,
! [R2: set_int,S: set_int] :
( ( ord_less_eq_int_o
@ ^ [X: int] : ( member_int @ X @ R2 )
@ ^ [X: int] : ( member_int @ X @ S ) )
= ( ord_less_eq_set_int @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_327_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_328_less__eq__set__def,axiom,
( ord_le7203529160286727270d_enat
= ( ^ [A5: set_Extended_enat,B: set_Extended_enat] :
( ord_le100613205991271927enat_o
@ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A5 )
@ ^ [X: extended_enat] : ( member_Extended_enat @ X @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_329_less__eq__set__def,axiom,
( ord_le211207098394363844omplex
= ( ^ [A5: set_complex,B: set_complex] :
( ord_le4573692005234683329plex_o
@ ^ [X: complex] : ( member_complex @ X @ A5 )
@ ^ [X: complex] : ( member_complex @ X @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_330_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A5 )
@ ^ [X: set_a] : ( member_set_a @ X @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_331_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_332_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B: set_int] :
( ord_less_eq_int_o
@ ^ [X: int] : ( member_int @ X @ A5 )
@ ^ [X: int] : ( member_int @ X @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_333_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_334_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6690914467698888265omplex @ N )
= one_one_complex )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_335_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera1916890842035813515d_enat @ N )
= one_on7984719198319812577d_enat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_336_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_337_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_338_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_339_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_340_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on7984719198319812577d_enat
= ( numera1916890842035813515d_enat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_341_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_342_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_343_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_344_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_345_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M3: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_346_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_347_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_348_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_349_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_350_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_351_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_352_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_353_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_354_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_355_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_356_numeral__One,axiom,
( ( numera7754357348821619680l_num1 @ one )
= one_on7795324986448017462l_num1 ) ).
% numeral_One
thf(fact_357_numeral__One,axiom,
( ( numera1916890842035813515d_enat @ one )
= one_on7984719198319812577d_enat ) ).
% numeral_One
thf(fact_358_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_359_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_360_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_361_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_362_nle__le,axiom,
! [A3: real,B6: real] :
( ( ~ ( ord_less_eq_real @ A3 @ B6 ) )
= ( ( ord_less_eq_real @ B6 @ A3 )
& ( B6 != A3 ) ) ) ).
% nle_le
thf(fact_363_nle__le,axiom,
! [A3: num,B6: num] :
( ( ~ ( ord_less_eq_num @ A3 @ B6 ) )
= ( ( ord_less_eq_num @ B6 @ A3 )
& ( B6 != A3 ) ) ) ).
% nle_le
thf(fact_364_nle__le,axiom,
! [A3: nat,B6: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B6 ) )
= ( ( ord_less_eq_nat @ B6 @ A3 )
& ( B6 != A3 ) ) ) ).
% nle_le
thf(fact_365_nle__le,axiom,
! [A3: int,B6: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B6 ) )
= ( ( ord_less_eq_int @ B6 @ A3 )
& ( B6 != A3 ) ) ) ).
% nle_le
thf(fact_366_le__cases3,axiom,
! [X3: real,Y4: real,Z2: real] :
( ( ( ord_less_eq_real @ X3 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_367_le__cases3,axiom,
! [X3: num,Y4: num,Z2: num] :
( ( ( ord_less_eq_num @ X3 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_368_le__cases3,axiom,
! [X3: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_369_le__cases3,axiom,
! [X3: int,Y4: int,Z2: int] :
( ( ( ord_less_eq_int @ X3 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_370_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_371_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_372_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
& ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_373_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_374_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_375_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Extended_enat,Z3: set_Extended_enat] : ( Y5 = Z3 ) )
= ( ^ [X: set_Extended_enat,Y: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ X @ Y )
& ( ord_le7203529160286727270d_enat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_376_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_complex,Z3: set_complex] : ( Y5 = Z3 ) )
= ( ^ [X: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y )
& ( ord_le211207098394363844omplex @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_377_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_378_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_379_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_int,Z3: set_int] : ( Y5 = Z3 ) )
= ( ^ [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
& ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_380_ord__eq__le__trans,axiom,
! [A3: real,B6: real,C2: real] :
( ( A3 = B6 )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_381_ord__eq__le__trans,axiom,
! [A3: set_a,B6: set_a,C2: set_a] :
( ( A3 = B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_382_ord__eq__le__trans,axiom,
! [A3: num,B6: num,C2: num] :
( ( A3 = B6 )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ord_less_eq_num @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_383_ord__eq__le__trans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( A3 = B6 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_384_ord__eq__le__trans,axiom,
! [A3: int,B6: int,C2: int] :
( ( A3 = B6 )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_385_ord__eq__le__trans,axiom,
! [A3: set_Extended_enat,B6: set_Extended_enat,C2: set_Extended_enat] :
( ( A3 = B6 )
=> ( ( ord_le7203529160286727270d_enat @ B6 @ C2 )
=> ( ord_le7203529160286727270d_enat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_386_ord__eq__le__trans,axiom,
! [A3: set_complex,B6: set_complex,C2: set_complex] :
( ( A3 = B6 )
=> ( ( ord_le211207098394363844omplex @ B6 @ C2 )
=> ( ord_le211207098394363844omplex @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_387_ord__eq__le__trans,axiom,
! [A3: set_set_a,B6: set_set_a,C2: set_set_a] :
( ( A3 = B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_388_ord__eq__le__trans,axiom,
! [A3: set_nat,B6: set_nat,C2: set_nat] :
( ( A3 = B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_389_ord__eq__le__trans,axiom,
! [A3: set_int,B6: set_int,C2: set_int] :
( ( A3 = B6 )
=> ( ( ord_less_eq_set_int @ B6 @ C2 )
=> ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_390_ord__le__eq__trans,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_391_ord__le__eq__trans,axiom,
! [A3: set_a,B6: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_392_ord__le__eq__trans,axiom,
! [A3: num,B6: num,C2: num] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_num @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_393_ord__le__eq__trans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_394_ord__le__eq__trans,axiom,
! [A3: int,B6: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_395_ord__le__eq__trans,axiom,
! [A3: set_Extended_enat,B6: set_Extended_enat,C2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_le7203529160286727270d_enat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_396_ord__le__eq__trans,axiom,
! [A3: set_complex,B6: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_le211207098394363844omplex @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_397_ord__le__eq__trans,axiom,
! [A3: set_set_a,B6: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_398_ord__le__eq__trans,axiom,
! [A3: set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_399_ord__le__eq__trans,axiom,
! [A3: set_int,B6: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_400_order__antisym,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_401_order__antisym,axiom,
! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_402_order__antisym,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_403_order__antisym,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_404_order__antisym,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_405_order__antisym,axiom,
! [X3: set_Extended_enat,Y4: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ X3 @ Y4 )
=> ( ( ord_le7203529160286727270d_enat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_406_order__antisym,axiom,
! [X3: set_complex,Y4: set_complex] :
( ( ord_le211207098394363844omplex @ X3 @ Y4 )
=> ( ( ord_le211207098394363844omplex @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_407_order__antisym,axiom,
! [X3: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_408_order__antisym,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_409_order__antisym,axiom,
! [X3: set_int,Y4: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y4 )
=> ( ( ord_less_eq_set_int @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_410_order_Otrans,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ord_less_eq_real @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_411_order_Otrans,axiom,
! [A3: set_a,B6: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_412_order_Otrans,axiom,
! [A3: num,B6: num,C2: num] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ord_less_eq_num @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_413_order_Otrans,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_414_order_Otrans,axiom,
! [A3: int,B6: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B6 )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_415_order_Otrans,axiom,
! [A3: set_Extended_enat,B6: set_Extended_enat,C2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A3 @ B6 )
=> ( ( ord_le7203529160286727270d_enat @ B6 @ C2 )
=> ( ord_le7203529160286727270d_enat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_416_order_Otrans,axiom,
! [A3: set_complex,B6: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B6 )
=> ( ( ord_le211207098394363844omplex @ B6 @ C2 )
=> ( ord_le211207098394363844omplex @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_417_order_Otrans,axiom,
! [A3: set_set_a,B6: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_418_order_Otrans,axiom,
! [A3: set_nat,B6: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_419_order_Otrans,axiom,
! [A3: set_int,B6: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B6 )
=> ( ( ord_less_eq_set_int @ B6 @ C2 )
=> ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_420_order__trans,axiom,
! [X3: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_421_order__trans,axiom,
! [X3: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_422_order__trans,axiom,
! [X3: num,Y4: num,Z2: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ( ord_less_eq_num @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_423_order__trans,axiom,
! [X3: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_424_order__trans,axiom,
! [X3: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_425_order__trans,axiom,
! [X3: set_Extended_enat,Y4: set_Extended_enat,Z2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ X3 @ Y4 )
=> ( ( ord_le7203529160286727270d_enat @ Y4 @ Z2 )
=> ( ord_le7203529160286727270d_enat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_426_order__trans,axiom,
! [X3: set_complex,Y4: set_complex,Z2: set_complex] :
( ( ord_le211207098394363844omplex @ X3 @ Y4 )
=> ( ( ord_le211207098394363844omplex @ Y4 @ Z2 )
=> ( ord_le211207098394363844omplex @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_427_order__trans,axiom,
! [X3: set_set_a,Y4: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ Y4 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_428_order__trans,axiom,
! [X3: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_429_order__trans,axiom,
! [X3: set_int,Y4: set_int,Z2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y4 )
=> ( ( ord_less_eq_set_int @ Y4 @ Z2 )
=> ( ord_less_eq_set_int @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_430_linorder__wlog,axiom,
! [P: real > real > $o,A3: real,B6: real] :
( ! [A2: real,B5: real] :
( ( ord_less_eq_real @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: real,B5: real] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A3 @ B6 ) ) ) ).
% linorder_wlog
thf(fact_431_linorder__wlog,axiom,
! [P: num > num > $o,A3: num,B6: num] :
( ! [A2: num,B5: num] :
( ( ord_less_eq_num @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: num,B5: num] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A3 @ B6 ) ) ) ).
% linorder_wlog
thf(fact_432_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B6: nat] :
( ! [A2: nat,B5: nat] :
( ( ord_less_eq_nat @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: nat,B5: nat] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A3 @ B6 ) ) ) ).
% linorder_wlog
thf(fact_433_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B6: int] :
( ! [A2: int,B5: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: int,B5: int] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A3 @ B6 ) ) ) ).
% linorder_wlog
thf(fact_434_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B7: real] :
( ( ord_less_eq_real @ B7 @ A4 )
& ( ord_less_eq_real @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_435_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_436_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [A4: num,B7: num] :
( ( ord_less_eq_num @ B7 @ A4 )
& ( ord_less_eq_num @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_437_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B7: nat] :
( ( ord_less_eq_nat @ B7 @ A4 )
& ( ord_less_eq_nat @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_438_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B7: int] :
( ( ord_less_eq_int @ B7 @ A4 )
& ( ord_less_eq_int @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_439_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Extended_enat,Z3: set_Extended_enat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_Extended_enat,B7: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ B7 @ A4 )
& ( ord_le7203529160286727270d_enat @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_440_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_complex,Z3: set_complex] : ( Y5 = Z3 ) )
= ( ^ [A4: set_complex,B7: set_complex] :
( ( ord_le211207098394363844omplex @ B7 @ A4 )
& ( ord_le211207098394363844omplex @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_441_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_442_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_443_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_int,Z3: set_int] : ( Y5 = Z3 ) )
= ( ^ [A4: set_int,B7: set_int] :
( ( ord_less_eq_set_int @ B7 @ A4 )
& ( ord_less_eq_set_int @ A4 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_444_dual__order_Oantisym,axiom,
! [B6: real,A3: real] :
( ( ord_less_eq_real @ B6 @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_445_dual__order_Oantisym,axiom,
! [B6: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_446_dual__order_Oantisym,axiom,
! [B6: num,A3: num] :
( ( ord_less_eq_num @ B6 @ A3 )
=> ( ( ord_less_eq_num @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_447_dual__order_Oantisym,axiom,
! [B6: nat,A3: nat] :
( ( ord_less_eq_nat @ B6 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_448_dual__order_Oantisym,axiom,
! [B6: int,A3: int] :
( ( ord_less_eq_int @ B6 @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_449_dual__order_Oantisym,axiom,
! [B6: set_Extended_enat,A3: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ B6 @ A3 )
=> ( ( ord_le7203529160286727270d_enat @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_450_dual__order_Oantisym,axiom,
! [B6: set_complex,A3: set_complex] :
( ( ord_le211207098394363844omplex @ B6 @ A3 )
=> ( ( ord_le211207098394363844omplex @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_451_dual__order_Oantisym,axiom,
! [B6: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_452_dual__order_Oantisym,axiom,
! [B6: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_453_dual__order_Oantisym,axiom,
! [B6: set_int,A3: set_int] :
( ( ord_less_eq_set_int @ B6 @ A3 )
=> ( ( ord_less_eq_set_int @ A3 @ B6 )
=> ( A3 = B6 ) ) ) ).
% dual_order.antisym
thf(fact_454_dual__order_Otrans,axiom,
! [B6: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B6 @ A3 )
=> ( ( ord_less_eq_real @ C2 @ B6 )
=> ( ord_less_eq_real @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_455_dual__order_Otrans,axiom,
! [B6: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B6 )
=> ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_456_dual__order_Otrans,axiom,
! [B6: num,A3: num,C2: num] :
( ( ord_less_eq_num @ B6 @ A3 )
=> ( ( ord_less_eq_num @ C2 @ B6 )
=> ( ord_less_eq_num @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_457_dual__order_Otrans,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B6 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B6 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_458_dual__order_Otrans,axiom,
! [B6: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B6 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B6 )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_459_dual__order_Otrans,axiom,
! [B6: set_Extended_enat,A3: set_Extended_enat,C2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ B6 @ A3 )
=> ( ( ord_le7203529160286727270d_enat @ C2 @ B6 )
=> ( ord_le7203529160286727270d_enat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_460_dual__order_Otrans,axiom,
! [B6: set_complex,A3: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ B6 @ A3 )
=> ( ( ord_le211207098394363844omplex @ C2 @ B6 )
=> ( ord_le211207098394363844omplex @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_461_dual__order_Otrans,axiom,
! [B6: set_set_a,A3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B6 )
=> ( ord_le3724670747650509150_set_a @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_462_dual__order_Otrans,axiom,
! [B6: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A3 )
=> ( ( ord_less_eq_set_nat @ C2 @ B6 )
=> ( ord_less_eq_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_463_dual__order_Otrans,axiom,
! [B6: set_int,A3: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B6 @ A3 )
=> ( ( ord_less_eq_set_int @ C2 @ B6 )
=> ( ord_less_eq_set_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_464_antisym,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_465_antisym,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_466_antisym,axiom,
! [A3: num,B6: num] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ord_less_eq_num @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_467_antisym,axiom,
! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_468_antisym,axiom,
! [A3: int,B6: int] :
( ( ord_less_eq_int @ A3 @ B6 )
=> ( ( ord_less_eq_int @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_469_antisym,axiom,
! [A3: set_Extended_enat,B6: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A3 @ B6 )
=> ( ( ord_le7203529160286727270d_enat @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_470_antisym,axiom,
! [A3: set_complex,B6: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B6 )
=> ( ( ord_le211207098394363844omplex @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_471_antisym,axiom,
! [A3: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_472_antisym,axiom,
! [A3: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_473_antisym,axiom,
! [A3: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A3 @ B6 )
=> ( ( ord_less_eq_set_int @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% antisym
thf(fact_474_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B7: real] :
( ( ord_less_eq_real @ A4 @ B7 )
& ( ord_less_eq_real @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_475_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A4 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_476_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [A4: num,B7: num] :
( ( ord_less_eq_num @ A4 @ B7 )
& ( ord_less_eq_num @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_477_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B7: nat] :
( ( ord_less_eq_nat @ A4 @ B7 )
& ( ord_less_eq_nat @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_478_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B7: int] :
( ( ord_less_eq_int @ A4 @ B7 )
& ( ord_less_eq_int @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_479_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Extended_enat,Z3: set_Extended_enat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_Extended_enat,B7: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A4 @ B7 )
& ( ord_le7203529160286727270d_enat @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_480_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_complex,Z3: set_complex] : ( Y5 = Z3 ) )
= ( ^ [A4: set_complex,B7: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B7 )
& ( ord_le211207098394363844omplex @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_481_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B7 )
& ( ord_le3724670747650509150_set_a @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_482_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_483_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_int,Z3: set_int] : ( Y5 = Z3 ) )
= ( ^ [A4: set_int,B7: set_int] :
( ( ord_less_eq_set_int @ A4 @ B7 )
& ( ord_less_eq_set_int @ B7 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_484_order__subst1,axiom,
! [A3: real,F2: real > real,B6: real,C2: real] :
( ( ord_less_eq_real @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_485_order__subst1,axiom,
! [A3: real,F2: num > real,B6: num,C2: num] :
( ( ord_less_eq_real @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_486_order__subst1,axiom,
! [A3: real,F2: nat > real,B6: nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_487_order__subst1,axiom,
! [A3: real,F2: int > real,B6: int,C2: int] :
( ( ord_less_eq_real @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_488_order__subst1,axiom,
! [A3: num,F2: real > num,B6: real,C2: real] :
( ( ord_less_eq_num @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_489_order__subst1,axiom,
! [A3: num,F2: num > num,B6: num,C2: num] :
( ( ord_less_eq_num @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_490_order__subst1,axiom,
! [A3: num,F2: nat > num,B6: nat,C2: nat] :
( ( ord_less_eq_num @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_491_order__subst1,axiom,
! [A3: num,F2: int > num,B6: int,C2: int] :
( ( ord_less_eq_num @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_int @ B6 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_492_order__subst1,axiom,
! [A3: nat,F2: real > nat,B6: real,C2: real] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_493_order__subst1,axiom,
! [A3: nat,F2: num > nat,B6: num,C2: num] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_494_order__subst2,axiom,
! [A3: real,B6: real,F2: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_495_order__subst2,axiom,
! [A3: real,B6: real,F2: real > num,C2: num] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_num @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_496_order__subst2,axiom,
! [A3: real,B6: real,F2: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_497_order__subst2,axiom,
! [A3: real,B6: real,F2: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_eq_int @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_498_order__subst2,axiom,
! [A3: num,B6: num,F2: num > real,C2: real] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ord_less_eq_real @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_499_order__subst2,axiom,
! [A3: num,B6: num,F2: num > num,C2: num] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ord_less_eq_num @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_500_order__subst2,axiom,
! [A3: num,B6: num,F2: num > nat,C2: nat] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_501_order__subst2,axiom,
! [A3: num,B6: num,F2: num > int,C2: int] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ord_less_eq_int @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_502_order__subst2,axiom,
! [A3: nat,B6: nat,F2: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_real @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_503_order__subst2,axiom,
! [A3: nat,B6: nat,F2: nat > num,C2: num] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_eq_num @ ( F2 @ B6 ) @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_504_order__eq__refl,axiom,
! [X3: real,Y4: real] :
( ( X3 = Y4 )
=> ( ord_less_eq_real @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_505_order__eq__refl,axiom,
! [X3: set_a,Y4: set_a] :
( ( X3 = Y4 )
=> ( ord_less_eq_set_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_506_order__eq__refl,axiom,
! [X3: num,Y4: num] :
( ( X3 = Y4 )
=> ( ord_less_eq_num @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_507_order__eq__refl,axiom,
! [X3: nat,Y4: nat] :
( ( X3 = Y4 )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_508_order__eq__refl,axiom,
! [X3: int,Y4: int] :
( ( X3 = Y4 )
=> ( ord_less_eq_int @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_509_order__eq__refl,axiom,
! [X3: set_Extended_enat,Y4: set_Extended_enat] :
( ( X3 = Y4 )
=> ( ord_le7203529160286727270d_enat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_510_order__eq__refl,axiom,
! [X3: set_complex,Y4: set_complex] :
( ( X3 = Y4 )
=> ( ord_le211207098394363844omplex @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_511_order__eq__refl,axiom,
! [X3: set_set_a,Y4: set_set_a] :
( ( X3 = Y4 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_512_order__eq__refl,axiom,
! [X3: set_nat,Y4: set_nat] :
( ( X3 = Y4 )
=> ( ord_less_eq_set_nat @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_513_order__eq__refl,axiom,
! [X3: set_int,Y4: set_int] :
( ( X3 = Y4 )
=> ( ord_less_eq_set_int @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_514_linorder__linear,axiom,
! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
| ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_515_linorder__linear,axiom,
! [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
| ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_516_linorder__linear,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_517_linorder__linear,axiom,
! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_linear
thf(fact_518_ord__eq__le__subst,axiom,
! [A3: real,F2: real > real,B6: real,C2: real] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_519_ord__eq__le__subst,axiom,
! [A3: num,F2: real > num,B6: real,C2: real] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_520_ord__eq__le__subst,axiom,
! [A3: nat,F2: real > nat,B6: real,C2: real] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_521_ord__eq__le__subst,axiom,
! [A3: int,F2: real > int,B6: real,C2: real] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_522_ord__eq__le__subst,axiom,
! [A3: real,F2: num > real,B6: num,C2: num] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_523_ord__eq__le__subst,axiom,
! [A3: num,F2: num > num,B6: num,C2: num] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_524_ord__eq__le__subst,axiom,
! [A3: nat,F2: num > nat,B6: num,C2: num] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_525_ord__eq__le__subst,axiom,
! [A3: int,F2: num > int,B6: num,C2: num] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_num @ B6 @ C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_526_ord__eq__le__subst,axiom,
! [A3: real,F2: nat > real,B6: nat,C2: nat] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_527_ord__eq__le__subst,axiom,
! [A3: num,F2: nat > num,B6: nat,C2: nat] :
( ( A3
= ( F2 @ B6 ) )
=> ( ( ord_less_eq_nat @ B6 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_528_ord__le__eq__subst,axiom,
! [A3: real,B6: real,F2: real > real,C2: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_529_ord__le__eq__subst,axiom,
! [A3: real,B6: real,F2: real > num,C2: num] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_530_ord__le__eq__subst,axiom,
! [A3: real,B6: real,F2: real > nat,C2: nat] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_531_ord__le__eq__subst,axiom,
! [A3: real,B6: real,F2: real > int,C2: int] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_532_ord__le__eq__subst,axiom,
! [A3: num,B6: num,F2: num > real,C2: real] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_533_ord__le__eq__subst,axiom,
! [A3: num,B6: num,F2: num > num,C2: num] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_534_ord__le__eq__subst,axiom,
! [A3: num,B6: num,F2: num > nat,C2: nat] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_535_ord__le__eq__subst,axiom,
! [A3: num,B6: num,F2: num > int,C2: int] :
( ( ord_less_eq_num @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: num,Y6: num] :
( ( ord_less_eq_num @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_536_ord__le__eq__subst,axiom,
! [A3: nat,B6: nat,F2: nat > real,C2: real] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_537_ord__le__eq__subst,axiom,
! [A3: nat,B6: nat,F2: nat > num,C2: num] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ( F2 @ B6 )
= C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_num @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_538_linorder__le__cases,axiom,
! [X3: real,Y4: real] :
( ~ ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_539_linorder__le__cases,axiom,
! [X3: num,Y4: num] :
( ~ ( ord_less_eq_num @ X3 @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_540_linorder__le__cases,axiom,
! [X3: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_541_linorder__le__cases,axiom,
! [X3: int,Y4: int] :
( ~ ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X3 ) ) ).
% linorder_le_cases
thf(fact_542_order__antisym__conv,axiom,
! [Y4: real,X3: real] :
( ( ord_less_eq_real @ Y4 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_543_order__antisym__conv,axiom,
! [Y4: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_544_order__antisym__conv,axiom,
! [Y4: num,X3: num] :
( ( ord_less_eq_num @ Y4 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_545_order__antisym__conv,axiom,
! [Y4: nat,X3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_546_order__antisym__conv,axiom,
! [Y4: int,X3: int] :
( ( ord_less_eq_int @ Y4 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_547_order__antisym__conv,axiom,
! [Y4: set_Extended_enat,X3: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ Y4 @ X3 )
=> ( ( ord_le7203529160286727270d_enat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_548_order__antisym__conv,axiom,
! [Y4: set_complex,X3: set_complex] :
( ( ord_le211207098394363844omplex @ Y4 @ X3 )
=> ( ( ord_le211207098394363844omplex @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_549_order__antisym__conv,axiom,
! [Y4: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_550_order__antisym__conv,axiom,
! [Y4: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_551_order__antisym__conv,axiom,
! [Y4: set_int,X3: set_int] :
( ( ord_less_eq_set_int @ Y4 @ X3 )
=> ( ( ord_less_eq_set_int @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_552_in__mono,axiom,
! [A: set_a,B2: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_553_in__mono,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat,X3: extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( member_Extended_enat @ X3 @ A )
=> ( member_Extended_enat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_554_in__mono,axiom,
! [A: set_complex,B2: set_complex,X3: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( member_complex @ X3 @ A )
=> ( member_complex @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_555_in__mono,axiom,
! [A: set_set_a,B2: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_556_in__mono,axiom,
! [A: set_nat,B2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_557_in__mono,axiom,
! [A: set_int,B2: set_int,X3: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ X3 @ A )
=> ( member_int @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_558_subsetD,axiom,
! [A: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_559_subsetD,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat,C2: extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( member_Extended_enat @ C2 @ A )
=> ( member_Extended_enat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_560_subsetD,axiom,
! [A: set_complex,B2: set_complex,C2: complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( member_complex @ C2 @ A )
=> ( member_complex @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_561_subsetD,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_562_subsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_563_subsetD,axiom,
! [A: set_int,B2: set_int,C2: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_564_equalityE,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_565_equalityE,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( A = B2 )
=> ~ ( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ~ ( ord_le7203529160286727270d_enat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_566_equalityE,axiom,
! [A: set_complex,B2: set_complex] :
( ( A = B2 )
=> ~ ( ( ord_le211207098394363844omplex @ A @ B2 )
=> ~ ( ord_le211207098394363844omplex @ B2 @ A ) ) ) ).
% equalityE
thf(fact_567_equalityE,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_568_equalityE,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_569_equalityE,axiom,
! [A: set_int,B2: set_int] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_int @ A @ B2 )
=> ~ ( ord_less_eq_set_int @ B2 @ A ) ) ) ).
% equalityE
thf(fact_570_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B ) ) ) ) ).
% subset_eq
thf(fact_571_subset__eq,axiom,
( ord_le7203529160286727270d_enat
= ( ^ [A5: set_Extended_enat,B: set_Extended_enat] :
! [X: extended_enat] :
( ( member_Extended_enat @ X @ A5 )
=> ( member_Extended_enat @ X @ B ) ) ) ) ).
% subset_eq
thf(fact_572_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A5: set_complex,B: set_complex] :
! [X: complex] :
( ( member_complex @ X @ A5 )
=> ( member_complex @ X @ B ) ) ) ) ).
% subset_eq
thf(fact_573_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B ) ) ) ) ).
% subset_eq
thf(fact_574_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_eq
thf(fact_575_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B: set_int] :
! [X: int] :
( ( member_int @ X @ A5 )
=> ( member_int @ X @ B ) ) ) ) ).
% subset_eq
thf(fact_576_equalityD1,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_577_equalityD1,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( A = B2 )
=> ( ord_le7203529160286727270d_enat @ A @ B2 ) ) ).
% equalityD1
thf(fact_578_equalityD1,axiom,
! [A: set_complex,B2: set_complex] :
( ( A = B2 )
=> ( ord_le211207098394363844omplex @ A @ B2 ) ) ).
% equalityD1
thf(fact_579_equalityD1,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A = B2 )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_580_equalityD1,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_581_equalityD1,axiom,
! [A: set_int,B2: set_int] :
( ( A = B2 )
=> ( ord_less_eq_set_int @ A @ B2 ) ) ).
% equalityD1
thf(fact_582_equalityD2,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_583_equalityD2,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( A = B2 )
=> ( ord_le7203529160286727270d_enat @ B2 @ A ) ) ).
% equalityD2
thf(fact_584_equalityD2,axiom,
! [A: set_complex,B2: set_complex] :
( ( A = B2 )
=> ( ord_le211207098394363844omplex @ B2 @ A ) ) ).
% equalityD2
thf(fact_585_equalityD2,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_586_equalityD2,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_587_equalityD2,axiom,
! [A: set_int,B2: set_int] :
( ( A = B2 )
=> ( ord_less_eq_set_int @ B2 @ A ) ) ).
% equalityD2
thf(fact_588_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A5 )
=> ( member_a @ T3 @ B ) ) ) ) ).
% subset_iff
thf(fact_589_subset__iff,axiom,
( ord_le7203529160286727270d_enat
= ( ^ [A5: set_Extended_enat,B: set_Extended_enat] :
! [T3: extended_enat] :
( ( member_Extended_enat @ T3 @ A5 )
=> ( member_Extended_enat @ T3 @ B ) ) ) ) ).
% subset_iff
thf(fact_590_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A5: set_complex,B: set_complex] :
! [T3: complex] :
( ( member_complex @ T3 @ A5 )
=> ( member_complex @ T3 @ B ) ) ) ) ).
% subset_iff
thf(fact_591_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A5 )
=> ( member_set_a @ T3 @ B ) ) ) ) ).
% subset_iff
thf(fact_592_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B ) ) ) ) ).
% subset_iff
thf(fact_593_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B: set_int] :
! [T3: int] :
( ( member_int @ T3 @ A5 )
=> ( member_int @ T3 @ B ) ) ) ) ).
% subset_iff
thf(fact_594_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_595_subset__refl,axiom,
! [A: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ A @ A ) ).
% subset_refl
thf(fact_596_subset__refl,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ A ) ).
% subset_refl
thf(fact_597_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_598_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_599_subset__refl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% subset_refl
thf(fact_600_Collect__mono,axiom,
! [P: set_complex > $o,Q: set_complex > $o] :
( ! [X4: set_complex] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le4750530260501030778omplex @ ( collect_set_complex @ P ) @ ( collect_set_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_601_Collect__mono,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ! [X4: set_set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_602_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_603_Collect__mono,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X4: set_int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_604_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_605_Collect__mono,axiom,
! [P: extended_enat > $o,Q: extended_enat > $o] :
( ! [X4: extended_enat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7203529160286727270d_enat @ ( collec4429806609662206161d_enat @ P ) @ ( collec4429806609662206161d_enat @ Q ) ) ) ).
% Collect_mono
thf(fact_606_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X4: complex] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_607_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_608_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_609_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_610_subset__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_611_subset__trans,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat,C: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ( ord_le7203529160286727270d_enat @ B2 @ C )
=> ( ord_le7203529160286727270d_enat @ A @ C ) ) ) ).
% subset_trans
thf(fact_612_subset__trans,axiom,
! [A: set_complex,B2: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C )
=> ( ord_le211207098394363844omplex @ A @ C ) ) ) ).
% subset_trans
thf(fact_613_subset__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_614_subset__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_615_subset__trans,axiom,
! [A: set_int,B2: set_int,C: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ C )
=> ( ord_less_eq_set_int @ A @ C ) ) ) ).
% subset_trans
thf(fact_616_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A5 @ B )
& ( ord_less_eq_set_a @ B @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_617_set__eq__subset,axiom,
( ( ^ [Y5: set_Extended_enat,Z3: set_Extended_enat] : ( Y5 = Z3 ) )
= ( ^ [A5: set_Extended_enat,B: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A5 @ B )
& ( ord_le7203529160286727270d_enat @ B @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_618_set__eq__subset,axiom,
( ( ^ [Y5: set_complex,Z3: set_complex] : ( Y5 = Z3 ) )
= ( ^ [A5: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A5 @ B )
& ( ord_le211207098394363844omplex @ B @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_619_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z3: set_set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B )
& ( ord_le3724670747650509150_set_a @ B @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_620_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A5: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B )
& ( ord_less_eq_set_nat @ B @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_621_set__eq__subset,axiom,
( ( ^ [Y5: set_int,Z3: set_int] : ( Y5 = Z3 ) )
= ( ^ [A5: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A5 @ B )
& ( ord_less_eq_set_int @ B @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_622_Collect__mono__iff,axiom,
! [P: set_complex > $o,Q: set_complex > $o] :
( ( ord_le4750530260501030778omplex @ ( collect_set_complex @ P ) @ ( collect_set_complex @ Q ) )
= ( ! [X: set_complex] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_623_Collect__mono__iff,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) )
= ( ! [X: set_set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_624_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_625_Collect__mono__iff,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) )
= ( ! [X: set_int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_626_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_627_Collect__mono__iff,axiom,
! [P: extended_enat > $o,Q: extended_enat > $o] :
( ( ord_le7203529160286727270d_enat @ ( collec4429806609662206161d_enat @ P ) @ ( collec4429806609662206161d_enat @ Q ) )
= ( ! [X: extended_enat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_628_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X: complex] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_629_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_630_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_631_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X: int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_632_Collect__subset,axiom,
! [A: set_set_complex,P: set_complex > $o] :
( ord_le4750530260501030778omplex
@ ( collect_set_complex
@ ^ [X: set_complex] :
( ( member_set_complex @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_633_Collect__subset,axiom,
! [A: set_set_set_a,P: set_set_a > $o] :
( ord_le5722252365846178494_set_a
@ ( collect_set_set_a
@ ^ [X: set_set_a] :
( ( member_set_set_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_634_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_635_Collect__subset,axiom,
! [A: set_set_int,P: set_int > $o] :
( ord_le4403425263959731960et_int
@ ( collect_set_int
@ ^ [X: set_int] :
( ( member_set_int @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_636_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_637_Collect__subset,axiom,
! [A: set_Extended_enat,P: extended_enat > $o] :
( ord_le7203529160286727270d_enat
@ ( collec4429806609662206161d_enat
@ ^ [X: extended_enat] :
( ( member_Extended_enat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_638_Collect__subset,axiom,
! [A: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X: complex] :
( ( member_complex @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_639_Collect__subset,axiom,
! [A: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_640_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_641_Collect__subset,axiom,
! [A: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_642_n__subsets,axiom,
! [A: set_set_int,K2: nat] :
( ( finite6197958912794628473et_int @ A )
=> ( ( finite7882580182802147440et_int
@ ( collect_set_set_int
@ ^ [B: set_set_int] :
( ( ord_le4403425263959731960et_int @ B @ A )
& ( ( finite_card_set_int @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_set_int @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_643_n__subsets,axiom,
! [A: set_se7270636423289371942d_enat,K2: nat] :
( ( finite5468666774076196335d_enat @ A )
=> ( ( finite7663542970459504974d_enat
@ ( collec9191517506022579601d_enat
@ ^ [B: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ B @ A )
& ( ( finite3719263829065406702d_enat @ B )
= K2 ) ) ) )
= ( binomial @ ( finite3719263829065406702d_enat @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_644_n__subsets,axiom,
! [A: set_set_nat,K2: nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite1149291290879098388et_nat
@ ( collect_set_set_nat
@ ^ [B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
& ( ( finite_card_set_nat @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_set_nat @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_645_n__subsets,axiom,
! [A: set_set_set_a,K2: nat] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( finite5538070369716010822_set_a
@ ( collec7971272359403355785_set_a
@ ^ [B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
& ( ( finite6524359278146944486_set_a @ B )
= K2 ) ) ) )
= ( binomial @ ( finite6524359278146944486_set_a @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_646_n__subsets,axiom,
! [A: set_a,K2: nat] :
( ( finite_finite_a @ A )
=> ( ( finite_card_set_a
@ ( collect_set_a
@ ^ [B: set_a] :
( ( ord_less_eq_set_a @ B @ A )
& ( ( finite_card_a @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_a @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_647_n__subsets,axiom,
! [A: set_Extended_enat,K2: nat] :
( ( finite4001608067531595151d_enat @ A )
=> ( ( finite3719263829065406702d_enat
@ ( collec2260605976452661553d_enat
@ ^ [B: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ B @ A )
& ( ( finite121521170596916366d_enat @ B )
= K2 ) ) ) )
= ( binomial @ ( finite121521170596916366d_enat @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_648_n__subsets,axiom,
! [A: set_complex,K2: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite903997441450111292omplex
@ ( collect_set_complex
@ ^ [B: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A )
& ( ( finite_card_complex @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_complex @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_649_n__subsets,axiom,
! [A: set_set_a,K2: nat] :
( ( finite_finite_set_a @ A )
=> ( ( finite6524359278146944486_set_a
@ ( collect_set_set_a
@ ^ [B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
& ( ( finite_card_set_a @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_set_a @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_650_n__subsets,axiom,
! [A: set_nat,K2: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_set_nat
@ ( collect_set_nat
@ ^ [B: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
& ( ( finite_card_nat @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_nat @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_651_n__subsets,axiom,
! [A: set_int,K2: nat] :
( ( finite_finite_int @ A )
=> ( ( finite_card_set_int
@ ( collect_set_int
@ ^ [B: set_int] :
( ( ord_less_eq_set_int @ B @ A )
& ( ( finite_card_int @ B )
= K2 ) ) ) )
= ( binomial @ ( finite_card_int @ A ) @ K2 ) ) ) ).
% n_subsets
thf(fact_652_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_653_choose__mono,axiom,
! [N: nat,M: nat,K2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ M @ K2 ) ) ) ).
% choose_mono
thf(fact_654_forall__2,axiom,
( ( ^ [P2: numera2417102609627094330l_num1 > $o] :
! [X6: numera2417102609627094330l_num1] : ( P2 @ X6 ) )
= ( ^ [P3: numera2417102609627094330l_num1 > $o] :
( ( P3 @ one_on3868389512446148991l_num1 )
& ( P3 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_655_exhaust__2,axiom,
! [X3: numera2417102609627094330l_num1] :
( ( X3 = one_on3868389512446148991l_num1 )
| ( X3
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_656_collinear__small,axiom,
! [S3: set_complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ S3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( linear2622894898690389117omplex @ S3 ) ) ) ).
% collinear_small
thf(fact_657_card2__sunflower,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ S ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( sunflower_a @ S ) ) ) ).
% card2_sunflower
thf(fact_658_card2__sunflower,axiom,
! [S: set_set_int] :
( ( finite6197958912794628473et_int @ S )
=> ( ( ord_less_eq_nat @ ( finite_card_set_int @ S ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( sunflower_int @ S ) ) ) ).
% card2_sunflower
thf(fact_659_card2__sunflower,axiom,
! [S: set_se7270636423289371942d_enat] :
( ( finite5468666774076196335d_enat @ S )
=> ( ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ S ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( sunflo6578399698544687513d_enat @ S ) ) ) ).
% card2_sunflower
thf(fact_660_card2__sunflower,axiom,
! [S: set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ S ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( sunflower_nat @ S ) ) ) ).
% card2_sunflower
thf(fact_661_card2__sunflower,axiom,
! [S: set_set_set_a] :
( ( finite7209287970140883943_set_a @ S )
=> ( ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ S ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( sunflower_set_a @ S ) ) ) ).
% card2_sunflower
thf(fact_662_landau__product__preprocess_I33_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% landau_product_preprocess(33)
thf(fact_663_landau__product__preprocess_I33_J,axiom,
( ( neg_nu5816564918971239084l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% landau_product_preprocess(33)
thf(fact_664_landau__product__preprocess_I33_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% landau_product_preprocess(33)
thf(fact_665_landau__product__preprocess_I33_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% landau_product_preprocess(33)
thf(fact_666_less__eq__enat__def,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [M4: extended_enat] :
( extended_case_enat_o
@ ^ [N1: nat] :
( extended_case_enat_o
@ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
@ $false
@ M4 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_667_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_668_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_669_enat_Osimps_I4_J,axiom,
! [F1: nat > $o,F22: $o,Nat: nat] :
( ( extended_case_enat_o @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(4)
thf(fact_670_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_671_landau__product__preprocess_I35_J,axiom,
! [K2: num] :
( ( neg_nu5816564918971239084l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) )
= ( numera7754357348821619680l_num1 @ ( bit0 @ K2 ) ) ) ).
% landau_product_preprocess(35)
thf(fact_672_landau__product__preprocess_I35_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) )
= ( numeral_numeral_real @ ( bit0 @ K2 ) ) ) ).
% landau_product_preprocess(35)
thf(fact_673_landau__product__preprocess_I35_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) ).
% landau_product_preprocess(35)
thf(fact_674_one__enat__def,axiom,
( one_on7984719198319812577d_enat
= ( extended_enat2 @ one_one_nat ) ) ).
% one_enat_def
thf(fact_675_enat__1__iff_I1_J,axiom,
! [X3: nat] :
( ( ( extended_enat2 @ X3 )
= one_on7984719198319812577d_enat )
= ( X3 = one_one_nat ) ) ).
% enat_1_iff(1)
thf(fact_676_enat__1__iff_I2_J,axiom,
! [X3: nat] :
( ( one_on7984719198319812577d_enat
= ( extended_enat2 @ X3 ) )
= ( X3 = one_one_nat ) ) ).
% enat_1_iff(2)
thf(fact_677_sunflower__def,axiom,
( sunflo6578399698544687513d_enat
= ( ^ [S4: set_se7270636423289371942d_enat] :
! [X: extended_enat] :
( ? [A5: set_Extended_enat,B: set_Extended_enat] :
( ( member350739656593644271d_enat @ A5 @ S4 )
& ( member350739656593644271d_enat @ B @ S4 )
& ( A5 != B )
& ( member_Extended_enat @ X @ A5 )
& ( member_Extended_enat @ X @ B ) )
=> ! [A5: set_Extended_enat] :
( ( member350739656593644271d_enat @ A5 @ S4 )
=> ( member_Extended_enat @ X @ A5 ) ) ) ) ) ).
% sunflower_def
thf(fact_678_sunflower__def,axiom,
( sunflower_nat
= ( ^ [S4: set_set_nat] :
! [X: nat] :
( ? [A5: set_nat,B: set_nat] :
( ( member_set_nat @ A5 @ S4 )
& ( member_set_nat @ B @ S4 )
& ( A5 != B )
& ( member_nat @ X @ A5 )
& ( member_nat @ X @ B ) )
=> ! [A5: set_nat] :
( ( member_set_nat @ A5 @ S4 )
=> ( member_nat @ X @ A5 ) ) ) ) ) ).
% sunflower_def
thf(fact_679_sunflower__subset,axiom,
! [F: set_set_a,G2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ F @ G2 )
=> ( ( sunflower_a @ G2 )
=> ( sunflower_a @ F ) ) ) ).
% sunflower_subset
thf(fact_680_collinear__subset,axiom,
! [T2: set_complex,S: set_complex] :
( ( linear2622894898690389117omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S @ T2 )
=> ( linear2622894898690389117omplex @ S ) ) ) ).
% collinear_subset
thf(fact_681_finite__enat__bounded,axiom,
! [A: set_Extended_enat,N: nat] :
( ! [Y6: extended_enat] :
( ( member_Extended_enat @ Y6 @ A )
=> ( ord_le2932123472753598470d_enat @ Y6 @ ( extended_enat2 @ N ) ) )
=> ( finite4001608067531595151d_enat @ A ) ) ).
% finite_enat_bounded
thf(fact_682_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_683_numeral__eq__enat,axiom,
( numera1916890842035813515d_enat
= ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K3 ) ) ) ) ).
% numeral_eq_enat
thf(fact_684_one__reorient,axiom,
! [X3: extended_enat] :
( ( one_on7984719198319812577d_enat = X3 )
= ( X3 = one_on7984719198319812577d_enat ) ) ).
% one_reorient
thf(fact_685_one__reorient,axiom,
! [X3: complex] :
( ( one_one_complex = X3 )
= ( X3 = one_one_complex ) ) ).
% one_reorient
thf(fact_686_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_687_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_688_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_689_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_690_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_691_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_692_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_set_a] :
( ! [G: set_se4329446959406904351_a_nat] :
( ! [X5: set_Su8246183859449809599_a_nat] :
( ( member8795406846318732392_a_nat @ X5 @ G )
=> ( ( finite7866281570360051208_a_nat @ X5 )
& ( ( finite1152564345524342151_a_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite6448316897771182823_a_nat @ G ) )
=> ? [S5: set_se4329446959406904351_a_nat] :
( ( ord_le7061570424243968959_a_nat @ S5 @ G )
& ( sunflo8923959888101223698_a_nat @ S5 )
& ( ( finite6448316897771182823_a_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ F )
=> ( ( finite_finite_set_a @ X4 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite6524359278146944486_set_a @ F ) )
=> ? [S2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ S2 @ F )
& ( sunflower_set_a @ S2 )
& ( ( finite6524359278146944486_set_a @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_693_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_nat] :
( ! [G: set_se3873067930692246379at_nat] :
( ! [X5: set_Sum_sum_nat_nat] :
( ( member1869216328726507724at_nat @ X5 @ G )
=> ( ( finite6187706683773761046at_nat @ X5 )
& ( ( finite8494011213269508311at_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite2024029949821234317at_nat @ G ) )
=> ? [S5: set_se3873067930692246379at_nat] :
( ( ord_le3495481059733392331at_nat @ S5 @ G )
& ( sunflo1841451327523575948at_nat @ S5 )
& ( ( finite2024029949821234317at_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ F )
=> ( ( finite_finite_nat @ X4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite_card_set_nat @ F ) )
=> ? [S2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ F )
& ( sunflower_nat @ S2 )
& ( ( finite_card_set_nat @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_694_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_se7270636423289371942d_enat] :
( ! [G: set_se4292523913457816871at_nat] :
( ! [X5: set_Su4628009649352508615at_nat] :
( ( member6581178323384882544at_nat @ X5 @ G )
=> ( ( finite6886502600733102096at_nat @ X5 )
& ( ( finite8792554634114447503at_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite2141207583037811951at_nat @ G ) )
=> ? [S5: set_se4292523913457816871at_nat] :
( ( ord_le5284647008997055687at_nat @ S5 @ G )
& ( sunflo4400997697864936730at_nat @ S5 )
& ( ( finite2141207583037811951at_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_Extended_enat] :
( ( member350739656593644271d_enat @ X4 @ F )
=> ( ( finite4001608067531595151d_enat @ X4 )
& ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite3719263829065406702d_enat @ F ) )
=> ? [S2: set_se7270636423289371942d_enat] :
( ( ord_le8264872500312945862d_enat @ S2 @ F )
& ( sunflo6578399698544687513d_enat @ S2 )
& ( ( finite3719263829065406702d_enat @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_695_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_int] :
( ! [G: set_se7091771524856616391nt_nat] :
( ! [X5: set_Sum_sum_int_nat] :
( ( member4056137903419529512nt_nat @ X5 @ G )
=> ( ( finite7187060395674815602nt_nat @ X5 )
& ( ( finite269992888315787059nt_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite4210951524514256105nt_nat @ G ) )
=> ? [S5: set_se7091771524856616391nt_nat] :
( ( ord_le6714184653897762343nt_nat @ S5 @ G )
& ( sunflo2840805039424630504nt_nat @ S5 )
& ( ( finite4210951524514256105nt_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ F )
=> ( ( finite_finite_int @ X4 )
& ( ord_less_eq_nat @ ( finite_card_int @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite_card_set_int @ F ) )
=> ? [S2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ F )
& ( sunflower_int @ S2 )
& ( ( finite_card_set_int @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_696_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_set_int] :
( ! [G: set_se3658823449513578877nt_nat] :
( ! [X5: set_Su2054411880199856583nt_nat] :
( ( member8593605234572697566nt_nat @ X5 @ G )
=> ( ( finite4678490111301253672nt_nat @ X5 )
& ( ( finite1376619864360116457nt_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite1753754214872078239nt_nat @ G ) )
=> ? [S5: set_se3658823449513578877nt_nat] :
( ( ord_le386859186808701405nt_nat @ S5 @ G )
& ( sunflo8837005380533273758nt_nat @ S5 )
& ( ( finite1753754214872078239nt_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ F )
=> ( ( finite6197958912794628473et_int @ X4 )
& ( ord_less_eq_nat @ ( finite_card_set_int @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite7882580182802147440et_int @ F ) )
=> ? [S2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ S2 @ F )
& ( sunflower_set_int @ S2 )
& ( ( finite7882580182802147440et_int @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_697_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_se6182022730614456710d_enat] :
( ! [G: set_se1186208456680207751at_nat] :
( ! [X5: set_Su2351696092937267495at_nat] :
( ( member8457675980879877584at_nat @ X5 @ G )
=> ( ( finite518351062452474992at_nat @ X5 )
& ( ( finite2731598656194920687at_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite1174306818983817551at_nat @ G ) )
=> ? [S5: set_se1186208456680207751at_nat] :
( ( ord_le5406584517758696743at_nat @ S5 @ G )
& ( sunflo6204964536344185210at_nat @ S5 )
& ( ( finite1174306818983817551at_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_se7270636423289371942d_enat] :
( ( member111414017764802383d_enat @ X4 @ F )
=> ( ( finite5468666774076196335d_enat @ X4 )
& ( ord_less_eq_nat @ ( finite3719263829065406702d_enat @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite7663542970459504974d_enat @ F ) )
=> ? [S2: set_se6182022730614456710d_enat] :
( ( ord_le2441283519433537830d_enat @ S2 @ F )
& ( sunflo1977439652906400249d_enat @ S2 )
& ( ( finite7663542970459504974d_enat @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_698_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_set_nat] :
( ! [G: set_se8003284279568041249at_nat] :
( ! [X5: set_Su8059080322890262379at_nat] :
( ( member5374901640408327554at_nat @ X5 @ G )
=> ( ( finite2491568536608231884at_nat @ X5 )
& ( ( finite8413070326521870477at_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite7758422657562484035at_nat @ G ) )
=> ? [S5: set_se8003284279568041249at_nat] :
( ( ord_le4731320016863163777at_nat @ S5 @ G )
& ( sunflo6650083805840251970at_nat @ S5 )
& ( ( finite7758422657562484035at_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ F )
=> ( ( finite1152437895449049373et_nat @ X4 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite1149291290879098388et_nat @ F ) )
=> ? [S2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ F )
& ( sunflower_set_nat @ S2 )
& ( ( finite1149291290879098388et_nat @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_699_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_set_set_a] :
( ! [G: set_se6906472397742828927_a_nat] :
( ! [X5: set_Su6024358463950899231_a_nat] :
( ( member2313308252142779848_a_nat @ X5 @ G )
=> ( ( finite2735258220151874408_a_nat @ X5 )
& ( ( finite5999300584321593063_a_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite2516327209276323911_a_nat @ G ) )
=> ? [S5: set_se6906472397742828927_a_nat] :
( ( ord_le2638365525317604639_a_nat @ S5 @ G )
& ( sunflo722587752449041010_a_nat @ S5 )
& ( ( finite2516327209276323911_a_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_set_set_a] :
( ( member_set_set_set_a @ X4 @ F )
=> ( ( finite7209287970140883943_set_a @ X4 )
& ( ord_less_eq_nat @ ( finite6524359278146944486_set_a @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite5538070369716010822_set_a @ F ) )
=> ? [S2: set_set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ S2 @ F )
& ( sunflower_set_set_a @ S2 )
& ( ( finite5538070369716010822_set_a @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_700_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_complex] :
( ! [G: set_se2148935472528660553ex_nat] :
( ! [X5: set_Su8911350359323359379ex_nat] :
( ( member3193062100113663146ex_nat @ X5 @ G )
=> ( ( finite975246697100726004ex_nat @ X5 )
& ( ( finite2444157462872061877ex_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite4339497693622181483ex_nat @ G ) )
=> ? [S5: set_se2148935472528660553ex_nat] :
( ( ord_le5086581535570348201ex_nat @ S5 @ G )
& ( sunflo5758272774145246570ex_nat @ S5 )
& ( ( finite4339497693622181483ex_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_complex] :
( ( member_set_complex @ X4 @ F )
=> ( ( finite3207457112153483333omplex @ X4 )
& ( ord_less_eq_nat @ ( finite_card_complex @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite903997441450111292omplex @ F ) )
=> ? [S2: set_set_complex] :
( ( ord_le4750530260501030778omplex @ S2 @ F )
& ( sunflower_complex @ S2 )
& ( ( finite903997441450111292omplex @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_701_sunflower__card__subset__lift,axiom,
! [K2: nat,C2: nat,R: nat,F: set_set_a] :
( ! [G: set_se4904748513628223167_a_nat] :
( ! [X5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X5 @ G )
=> ( ( finite502105017643426984_a_nat @ X5 )
& ( ( finite6080979521523705895_a_nat @ X5 )
= K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite7352162805081373063_a_nat @ G ) )
=> ? [S5: set_se4904748513628223167_a_nat] :
( ( ord_le7974500612278410847_a_nat @ S5 @ G )
& ( sunflo8619437272074122162_a_nat @ S5 )
& ( ( finite7352162805081373063_a_nat @ S5 )
= R ) ) ) )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ F )
=> ( ( finite_finite_a @ X4 )
& ( ord_less_eq_nat @ ( finite_card_a @ X4 ) @ K2 ) ) )
=> ( ( ord_less_nat @ C2 @ ( finite_card_set_a @ F ) )
=> ? [S2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ F )
& ( sunflower_a @ S2 )
& ( ( finite_card_set_a @ S2 )
= R ) ) ) ) ) ).
% sunflower_card_subset_lift
thf(fact_702_landau__product__preprocess_I34_J,axiom,
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% landau_product_preprocess(34)
thf(fact_703_landau__product__preprocess_I34_J,axiom,
( ( neg_nu5816564918971239084l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ) ).
% landau_product_preprocess(34)
thf(fact_704_landau__product__preprocess_I34_J,axiom,
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% landau_product_preprocess(34)
thf(fact_705_landau__product__preprocess_I34_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% landau_product_preprocess(34)
thf(fact_706_binomial__le__pow2,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_707_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_708_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_709_one__add__one,axiom,
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_710_one__add__one,axiom,
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_711_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_712_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_713_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_714_is__singleton__altdef,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
( ( finite_card_a @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_715_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( ( finite_card_nat @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_716_is__singleton__altdef,axiom,
( is_singleton_complex
= ( ^ [A5: set_complex] :
( ( finite_card_complex @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_717_is__singleton__altdef,axiom,
( is_singleton_int
= ( ^ [A5: set_int] :
( ( finite_card_int @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_718_is__singleton__altdef,axiom,
( is_sin1871519699599484762d_enat
= ( ^ [A5: set_Extended_enat] :
( ( finite121521170596916366d_enat @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_719_is__singleton__altdef,axiom,
( is_singleton_set_a
= ( ^ [A5: set_set_a] :
( ( finite_card_set_a @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_720_add__left__cancel,axiom,
! [A3: real,B6: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B6 )
= ( plus_plus_real @ A3 @ C2 ) )
= ( B6 = C2 ) ) ).
% add_left_cancel
thf(fact_721_add__left__cancel,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B6 )
= ( plus_plus_nat @ A3 @ C2 ) )
= ( B6 = C2 ) ) ).
% add_left_cancel
thf(fact_722_add__left__cancel,axiom,
! [A3: int,B6: int,C2: int] :
( ( ( plus_plus_int @ A3 @ B6 )
= ( plus_plus_int @ A3 @ C2 ) )
= ( B6 = C2 ) ) ).
% add_left_cancel
thf(fact_723_add__right__cancel,axiom,
! [B6: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B6 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
= ( B6 = C2 ) ) ).
% add_right_cancel
thf(fact_724_add__right__cancel,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B6 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
= ( B6 = C2 ) ) ).
% add_right_cancel
thf(fact_725_add__right__cancel,axiom,
! [B6: int,A3: int,C2: int] :
( ( ( plus_plus_int @ B6 @ A3 )
= ( plus_plus_int @ C2 @ A3 ) )
= ( B6 = C2 ) ) ).
% add_right_cancel
thf(fact_726_Compl__anti__mono,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B2 ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_727_Compl__anti__mono,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ A @ B2 )
=> ( ord_le7203529160286727270d_enat @ ( uminus417252749190364093d_enat @ B2 ) @ ( uminus417252749190364093d_enat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_728_Compl__anti__mono,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B2 )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ B2 ) @ ( uminus8566677241136511917omplex @ A ) ) ) ).
% Compl_anti_mono
thf(fact_729_Compl__anti__mono,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ B2 ) @ ( uminus6103902357914783669_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_730_Compl__anti__mono,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_731_Compl__anti__mono,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A ) ) ) ).
% Compl_anti_mono
thf(fact_732_Compl__subset__Compl__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B2 ) )
= ( ord_less_eq_set_a @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_733_Compl__subset__Compl__iff,axiom,
! [A: set_Extended_enat,B2: set_Extended_enat] :
( ( ord_le7203529160286727270d_enat @ ( uminus417252749190364093d_enat @ A ) @ ( uminus417252749190364093d_enat @ B2 ) )
= ( ord_le7203529160286727270d_enat @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_734_Compl__subset__Compl__iff,axiom,
! [A: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ A ) @ ( uminus8566677241136511917omplex @ B2 ) )
= ( ord_le211207098394363844omplex @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_735_Compl__subset__Compl__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ A ) @ ( uminus6103902357914783669_set_a @ B2 ) )
= ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_736_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B2 ) )
= ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_737_Compl__subset__Compl__iff,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( uminus1532241313380277803et_int @ B2 ) )
= ( ord_less_eq_set_int @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_738_landau__product__preprocess_I8_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% landau_product_preprocess(8)
thf(fact_739_add_Oinverse__inverse,axiom,
! [A3: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_740_add_Oinverse__inverse,axiom,
! [A3: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A3 ) )
= A3 ) ).
% add.inverse_inverse
thf(fact_741_neg__equal__iff__equal,axiom,
! [A3: real,B6: real] :
( ( ( uminus_uminus_real @ A3 )
= ( uminus_uminus_real @ B6 ) )
= ( A3 = B6 ) ) ).
% neg_equal_iff_equal
thf(fact_742_neg__equal__iff__equal,axiom,
! [A3: int,B6: int] :
( ( ( uminus_uminus_int @ A3 )
= ( uminus_uminus_int @ B6 ) )
= ( A3 = B6 ) ) ).
% neg_equal_iff_equal
thf(fact_743_verit__minus__simplify_I4_J,axiom,
! [B6: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B6 ) )
= B6 ) ).
% verit_minus_simplify(4)
thf(fact_744_verit__minus__simplify_I4_J,axiom,
! [B6: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B6 ) )
= B6 ) ).
% verit_minus_simplify(4)
thf(fact_745_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus_nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_746_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_747_add__le__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B6 ) )
= ( ord_less_eq_real @ A3 @ B6 ) ) ).
% add_le_cancel_left
thf(fact_748_add__le__cancel__left,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B6 ) )
= ( ord_less_eq_nat @ A3 @ B6 ) ) ).
% add_le_cancel_left
thf(fact_749_add__le__cancel__left,axiom,
! [C2: int,A3: int,B6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B6 ) )
= ( ord_less_eq_int @ A3 @ B6 ) ) ).
% add_le_cancel_left
thf(fact_750_add__le__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ C2 ) )
= ( ord_less_eq_real @ A3 @ B6 ) ) ).
% add_le_cancel_right
thf(fact_751_add__le__cancel__right,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ C2 ) )
= ( ord_less_eq_nat @ A3 @ B6 ) ) ).
% add_le_cancel_right
thf(fact_752_add__le__cancel__right,axiom,
! [A3: int,C2: int,B6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ C2 ) )
= ( ord_less_eq_int @ A3 @ B6 ) ) ).
% add_le_cancel_right
thf(fact_753_neg__le__iff__le,axiom,
! [B6: real,A3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B6 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_eq_real @ A3 @ B6 ) ) ).
% neg_le_iff_le
thf(fact_754_neg__le__iff__le,axiom,
! [B6: int,A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B6 ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_eq_int @ A3 @ B6 ) ) ).
% neg_le_iff_le
thf(fact_755_add__less__cancel__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B6 ) )
= ( ord_less_real @ A3 @ B6 ) ) ).
% add_less_cancel_left
thf(fact_756_add__less__cancel__left,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B6 ) )
= ( ord_less_nat @ A3 @ B6 ) ) ).
% add_less_cancel_left
thf(fact_757_add__less__cancel__left,axiom,
! [C2: int,A3: int,B6: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B6 ) )
= ( ord_less_int @ A3 @ B6 ) ) ).
% add_less_cancel_left
thf(fact_758_add__less__cancel__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ C2 ) )
= ( ord_less_real @ A3 @ B6 ) ) ).
% add_less_cancel_right
thf(fact_759_add__less__cancel__right,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ C2 ) )
= ( ord_less_nat @ A3 @ B6 ) ) ).
% add_less_cancel_right
thf(fact_760_add__less__cancel__right,axiom,
! [A3: int,C2: int,B6: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ C2 ) )
= ( ord_less_int @ A3 @ B6 ) ) ).
% add_less_cancel_right
thf(fact_761_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_762_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_763_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_764_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_765_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_766_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_767_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_768_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_769_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_770_add__numeral__left,axiom,
! [V: num,W: num,Z2: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ V ) @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ W ) @ Z2 ) )
= ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_771_add__numeral__left,axiom,
! [V: num,W: num,Z2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
= ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_772_add__numeral__left,axiom,
! [V: num,W: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_773_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_774_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_775_neg__less__iff__less,axiom,
! [B6: real,A3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B6 ) @ ( uminus_uminus_real @ A3 ) )
= ( ord_less_real @ A3 @ B6 ) ) ).
% neg_less_iff_less
thf(fact_776_neg__less__iff__less,axiom,
! [B6: int,A3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B6 ) @ ( uminus_uminus_int @ A3 ) )
= ( ord_less_int @ A3 @ B6 ) ) ).
% neg_less_iff_less
thf(fact_777_add__minus__cancel,axiom,
! [A3: real,B6: real] :
( ( plus_plus_real @ A3 @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ B6 ) )
= B6 ) ).
% add_minus_cancel
thf(fact_778_add__minus__cancel,axiom,
! [A3: int,B6: int] :
( ( plus_plus_int @ A3 @ ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ B6 ) )
= B6 ) ).
% add_minus_cancel
thf(fact_779_minus__add__cancel,axiom,
! [A3: real,B6: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( plus_plus_real @ A3 @ B6 ) )
= B6 ) ).
% minus_add_cancel
thf(fact_780_minus__add__cancel,axiom,
! [A3: int,B6: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ ( plus_plus_int @ A3 @ B6 ) )
= B6 ) ).
% minus_add_cancel
thf(fact_781_minus__add__distrib,axiom,
! [A3: real,B6: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B6 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B6 ) ) ) ).
% minus_add_distrib
thf(fact_782_minus__add__distrib,axiom,
! [A3: int,B6: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B6 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B6 ) ) ) ).
% minus_add_distrib
thf(fact_783_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_784_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_785_landau__product__preprocess_I4_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% landau_product_preprocess(4)
thf(fact_786_binomial__Suc__Suc,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( plus_plus_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_787_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_788_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_789_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_790_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_791_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_792_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y4: numera4273646738625120315l_num1] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ V ) ) @ ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ W ) ) @ Y4 ) )
= ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% semiring_norm(167)
thf(fact_793_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y4: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y4 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% semiring_norm(167)
thf(fact_794_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y4: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y4 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y4 ) ) ).
% semiring_norm(167)
thf(fact_795_landau__product__preprocess_I51_J,axiom,
! [M: num,N: num] :
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ M ) ) @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ N ) ) )
= ( uminus1336558196688952754l_num1 @ ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ M ) @ ( numera7754357348821619680l_num1 @ N ) ) ) ) ).
% landau_product_preprocess(51)
thf(fact_796_landau__product__preprocess_I51_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% landau_product_preprocess(51)
thf(fact_797_landau__product__preprocess_I51_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% landau_product_preprocess(51)
thf(fact_798_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_799_landau__product__preprocess_I31_J,axiom,
! [K2: num] :
( ( neg_nu5816564918971239084l_num1 @ ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) )
= ( uminus1336558196688952754l_num1 @ ( neg_nu5816564918971239084l_num1 @ ( numera7754357348821619680l_num1 @ K2 ) ) ) ) ).
% landau_product_preprocess(31)
thf(fact_800_landau__product__preprocess_I31_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
= ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).
% landau_product_preprocess(31)
thf(fact_801_landau__product__preprocess_I31_J,axiom,
! [K2: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).
% landau_product_preprocess(31)
thf(fact_802_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_803_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_804_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_805_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_806_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_807_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ ( numera7754357348821619680l_num1 @ N ) @ one_on7795324986448017462l_num1 )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_808_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_809_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_810_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_811_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_812_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_813_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p1441664204671982194l_num1 @ one_on7795324986448017462l_num1 @ ( numera7754357348821619680l_num1 @ N ) )
= ( numera7754357348821619680l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_814_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
= ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_815_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_816_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_817_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_818_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_819_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_820_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_821_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_822_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_823_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_824_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_825_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_826_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_827_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_828_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_829_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_830_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_831_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_832_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_833_add__neg__numeral__special_I9_J,axiom,
( ( plus_p1441664204671982194l_num1 @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) @ ( uminus1336558196688952754l_num1 @ one_on7795324986448017462l_num1 ) )
= ( uminus1336558196688952754l_num1 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_834_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_835_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_836_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_837_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_838_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_839_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_840_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_841_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_842_add__le__less__mono,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_eq_real @ A3 @ B6 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_843_add__le__less__mono,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_844_add__le__less__mono,axiom,
! [A3: int,B6: int,C2: int,D: int] :
( ( ord_less_eq_int @ A3 @ B6 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_845_add__less__le__mono,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_846_add__less__le__mono,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_847_add__less__le__mono,axiom,
! [A3: int,B6: int,C2: int,D: int] :
( ( ord_less_int @ A3 @ B6 )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_848_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: extended_enat,B6: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A3 @ B6 ) @ C2 )
= ( plus_p3455044024723400733d_enat @ A3 @ ( plus_p3455044024723400733d_enat @ B6 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_849_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: real,B6: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B6 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B6 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_850_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B6 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B6 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_851_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: int,B6: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B6 ) @ C2 )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B6 @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_852_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_853_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_854_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_855_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_856_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_857_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_858_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_859_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_860_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_861_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: extended_enat,J: extended_enat,K2: extended_enat,L: extended_enat] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_p3455044024723400733d_enat @ I2 @ K2 )
= ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_862_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K2: real,L: real] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_real @ I2 @ K2 )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_863_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I2 @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_864_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I2 @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_865_group__cancel_Oadd1,axiom,
! [A: extended_enat,K2: extended_enat,A3: extended_enat,B6: extended_enat] :
( ( A
= ( plus_p3455044024723400733d_enat @ K2 @ A3 ) )
=> ( ( plus_p3455044024723400733d_enat @ A @ B6 )
= ( plus_p3455044024723400733d_enat @ K2 @ ( plus_p3455044024723400733d_enat @ A3 @ B6 ) ) ) ) ).
% group_cancel.add1
thf(fact_866_group__cancel_Oadd1,axiom,
! [A: real,K2: real,A3: real,B6: real] :
( ( A
= ( plus_plus_real @ K2 @ A3 ) )
=> ( ( plus_plus_real @ A @ B6 )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A3 @ B6 ) ) ) ) ).
% group_cancel.add1
thf(fact_867_group__cancel_Oadd1,axiom,
! [A: nat,K2: nat,A3: nat,B6: nat] :
( ( A
= ( plus_plus_nat @ K2 @ A3 ) )
=> ( ( plus_plus_nat @ A @ B6 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A3 @ B6 ) ) ) ) ).
% group_cancel.add1
thf(fact_868_group__cancel_Oadd1,axiom,
! [A: int,K2: int,A3: int,B6: int] :
( ( A
= ( plus_plus_int @ K2 @ A3 ) )
=> ( ( plus_plus_int @ A @ B6 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A3 @ B6 ) ) ) ) ).
% group_cancel.add1
thf(fact_869_group__cancel_Oadd2,axiom,
! [B2: extended_enat,K2: extended_enat,B6: extended_enat,A3: extended_enat] :
( ( B2
= ( plus_p3455044024723400733d_enat @ K2 @ B6 ) )
=> ( ( plus_p3455044024723400733d_enat @ A3 @ B2 )
= ( plus_p3455044024723400733d_enat @ K2 @ ( plus_p3455044024723400733d_enat @ A3 @ B6 ) ) ) ) ).
% group_cancel.add2
thf(fact_870_group__cancel_Oadd2,axiom,
! [B2: real,K2: real,B6: real,A3: real] :
( ( B2
= ( plus_plus_real @ K2 @ B6 ) )
=> ( ( plus_plus_real @ A3 @ B2 )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A3 @ B6 ) ) ) ) ).
% group_cancel.add2
thf(fact_871_group__cancel_Oadd2,axiom,
! [B2: nat,K2: nat,B6: nat,A3: nat] :
( ( B2
= ( plus_plus_nat @ K2 @ B6 ) )
=> ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A3 @ B6 ) ) ) ) ).
% group_cancel.add2
thf(fact_872_group__cancel_Oadd2,axiom,
! [B2: int,K2: int,B6: int,A3: int] :
( ( B2
= ( plus_plus_int @ K2 @ B6 ) )
=> ( ( plus_plus_int @ A3 @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A3 @ B6 ) ) ) ) ).
% group_cancel.add2
thf(fact_873_group__cancel_Oneg1,axiom,
! [A: real,K2: real,A3: real] :
( ( A
= ( plus_plus_real @ K2 @ A3 ) )
=> ( ( uminus_uminus_real @ A )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( uminus_uminus_real @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_874_group__cancel_Oneg1,axiom,
! [A: int,K2: int,A3: int] :
( ( A
= ( plus_plus_int @ K2 @ A3 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A3 ) ) ) ) ).
% group_cancel.neg1
thf(fact_875_add_Oassoc,axiom,
! [A3: extended_enat,B6: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A3 @ B6 ) @ C2 )
= ( plus_p3455044024723400733d_enat @ A3 @ ( plus_p3455044024723400733d_enat @ B6 @ C2 ) ) ) ).
% add.assoc
thf(fact_876_add_Oassoc,axiom,
! [A3: real,B6: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A3 @ B6 ) @ C2 )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B6 @ C2 ) ) ) ).
% add.assoc
thf(fact_877_add_Oassoc,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B6 ) @ C2 )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B6 @ C2 ) ) ) ).
% add.assoc
thf(fact_878_add_Oassoc,axiom,
! [A3: int,B6: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A3 @ B6 ) @ C2 )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B6 @ C2 ) ) ) ).
% add.assoc
thf(fact_879_add_Oleft__cancel,axiom,
! [A3: real,B6: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B6 )
= ( plus_plus_real @ A3 @ C2 ) )
= ( B6 = C2 ) ) ).
% add.left_cancel
thf(fact_880_add_Oleft__cancel,axiom,
! [A3: int,B6: int,C2: int] :
( ( ( plus_plus_int @ A3 @ B6 )
= ( plus_plus_int @ A3 @ C2 ) )
= ( B6 = C2 ) ) ).
% add.left_cancel
thf(fact_881_add_Oright__cancel,axiom,
! [B6: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B6 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
= ( B6 = C2 ) ) ).
% add.right_cancel
thf(fact_882_add_Oright__cancel,axiom,
! [B6: int,A3: int,C2: int] :
( ( ( plus_plus_int @ B6 @ A3 )
= ( plus_plus_int @ C2 @ A3 ) )
= ( B6 = C2 ) ) ).
% add.right_cancel
thf(fact_883_add_Ocommute,axiom,
( plus_p3455044024723400733d_enat
= ( ^ [A4: extended_enat,B7: extended_enat] : ( plus_p3455044024723400733d_enat @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_884_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B7: real] : ( plus_plus_real @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_885_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B7: nat] : ( plus_plus_nat @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_886_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B7: int] : ( plus_plus_int @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_887_equation__minus__iff,axiom,
! [A3: real,B6: real] :
( ( A3
= ( uminus_uminus_real @ B6 ) )
= ( B6
= ( uminus_uminus_real @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_888_equation__minus__iff,axiom,
! [A3: int,B6: int] :
( ( A3
= ( uminus_uminus_int @ B6 ) )
= ( B6
= ( uminus_uminus_int @ A3 ) ) ) ).
% equation_minus_iff
thf(fact_889_minus__equation__iff,axiom,
! [A3: real,B6: real] :
( ( ( uminus_uminus_real @ A3 )
= B6 )
= ( ( uminus_uminus_real @ B6 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_890_minus__equation__iff,axiom,
! [A3: int,B6: int] :
( ( ( uminus_uminus_int @ A3 )
= B6 )
= ( ( uminus_uminus_int @ B6 )
= A3 ) ) ).
% minus_equation_iff
thf(fact_891_add_Oleft__commute,axiom,
! [B6: extended_enat,A3: extended_enat,C2: extended_enat] :
( ( plus_p3455044024723400733d_enat @ B6 @ ( plus_p3455044024723400733d_enat @ A3 @ C2 ) )
= ( plus_p3455044024723400733d_enat @ A3 @ ( plus_p3455044024723400733d_enat @ B6 @ C2 ) ) ) ).
% add.left_commute
thf(fact_892_add_Oleft__commute,axiom,
! [B6: real,A3: real,C2: real] :
( ( plus_plus_real @ B6 @ ( plus_plus_real @ A3 @ C2 ) )
= ( plus_plus_real @ A3 @ ( plus_plus_real @ B6 @ C2 ) ) ) ).
% add.left_commute
thf(fact_893_add_Oleft__commute,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( plus_plus_nat @ B6 @ ( plus_plus_nat @ A3 @ C2 ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B6 @ C2 ) ) ) ).
% add.left_commute
thf(fact_894_add_Oleft__commute,axiom,
! [B6: int,A3: int,C2: int] :
( ( plus_plus_int @ B6 @ ( plus_plus_int @ A3 @ C2 ) )
= ( plus_plus_int @ A3 @ ( plus_plus_int @ B6 @ C2 ) ) ) ).
% add.left_commute
thf(fact_895_add_Oinverse__distrib__swap,axiom,
! [A3: real,B6: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B6 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B6 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_896_add_Oinverse__distrib__swap,axiom,
! [A3: int,B6: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B6 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B6 ) @ ( uminus_uminus_int @ A3 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_897_less__minus__iff,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ A3 @ ( uminus_uminus_real @ B6 ) )
= ( ord_less_real @ B6 @ ( uminus_uminus_real @ A3 ) ) ) ).
% less_minus_iff
thf(fact_898_less__minus__iff,axiom,
! [A3: int,B6: int] :
( ( ord_less_int @ A3 @ ( uminus_uminus_int @ B6 ) )
= ( ord_less_int @ B6 @ ( uminus_uminus_int @ A3 ) ) ) ).
% less_minus_iff
thf(fact_899_minus__less__iff,axiom,
! [A3: real,B6: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B6 )
= ( ord_less_real @ ( uminus_uminus_real @ B6 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_900_minus__less__iff,axiom,
! [A3: int,B6: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B6 )
= ( ord_less_int @ ( uminus_uminus_int @ B6 ) @ A3 ) ) ).
% minus_less_iff
thf(fact_901_add__left__imp__eq,axiom,
! [A3: real,B6: real,C2: real] :
( ( ( plus_plus_real @ A3 @ B6 )
= ( plus_plus_real @ A3 @ C2 ) )
=> ( B6 = C2 ) ) ).
% add_left_imp_eq
thf(fact_902_add__left__imp__eq,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ( plus_plus_nat @ A3 @ B6 )
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( B6 = C2 ) ) ).
% add_left_imp_eq
thf(fact_903_add__left__imp__eq,axiom,
! [A3: int,B6: int,C2: int] :
( ( ( plus_plus_int @ A3 @ B6 )
= ( plus_plus_int @ A3 @ C2 ) )
=> ( B6 = C2 ) ) ).
% add_left_imp_eq
thf(fact_904_add__right__imp__eq,axiom,
! [B6: real,A3: real,C2: real] :
( ( ( plus_plus_real @ B6 @ A3 )
= ( plus_plus_real @ C2 @ A3 ) )
=> ( B6 = C2 ) ) ).
% add_right_imp_eq
thf(fact_905_add__right__imp__eq,axiom,
! [B6: nat,A3: nat,C2: nat] :
( ( ( plus_plus_nat @ B6 @ A3 )
= ( plus_plus_nat @ C2 @ A3 ) )
=> ( B6 = C2 ) ) ).
% add_right_imp_eq
thf(fact_906_add__right__imp__eq,axiom,
! [B6: int,A3: int,C2: int] :
( ( ( plus_plus_int @ B6 @ A3 )
= ( plus_plus_int @ C2 @ A3 ) )
=> ( B6 = C2 ) ) ).
% add_right_imp_eq
thf(fact_907_add__strict__mono,axiom,
! [A3: extended_enat,B6: extended_enat,C2: extended_enat,D: extended_enat] :
( ( ord_le72135733267957522d_enat @ A3 @ B6 )
=> ( ( ord_le72135733267957522d_enat @ C2 @ D )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A3 @ C2 ) @ ( plus_p3455044024723400733d_enat @ B6 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_908_add__strict__mono,axiom,
! [A3: real,B6: real,C2: real,D: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_909_add__strict__mono,axiom,
! [A3: nat,B6: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_910_add__strict__mono,axiom,
! [A3: int,B6: int,C2: int,D: int] :
( ( ord_less_int @ A3 @ B6 )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_911_add__strict__left__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B6 ) ) ) ).
% add_strict_left_mono
thf(fact_912_add__strict__left__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B6 ) ) ) ).
% add_strict_left_mono
thf(fact_913_add__strict__left__mono,axiom,
! [A3: int,B6: int,C2: int] :
( ( ord_less_int @ A3 @ B6 )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B6 ) ) ) ).
% add_strict_left_mono
thf(fact_914_add__strict__right__mono,axiom,
! [A3: real,B6: real,C2: real] :
( ( ord_less_real @ A3 @ B6 )
=> ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_915_add__strict__right__mono,axiom,
! [A3: nat,B6: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_916_add__strict__right__mono,axiom,
! [A3: int,B6: int,C2: int] :
( ( ord_less_int @ A3 @ B6 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_917_add__less__imp__less__left,axiom,
! [C2: real,A3: real,B6: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A3 ) @ ( plus_plus_real @ C2 @ B6 ) )
=> ( ord_less_real @ A3 @ B6 ) ) ).
% add_less_imp_less_left
thf(fact_918_add__less__imp__less__left,axiom,
! [C2: nat,A3: nat,B6: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B6 ) )
=> ( ord_less_nat @ A3 @ B6 ) ) ).
% add_less_imp_less_left
thf(fact_919_add__less__imp__less__left,axiom,
! [C2: int,A3: int,B6: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B6 ) )
=> ( ord_less_int @ A3 @ B6 ) ) ).
% add_less_imp_less_left
thf(fact_920_add__less__imp__less__right,axiom,
! [A3: real,C2: real,B6: real] :
( ( ord_less_real @ ( plus_plus_real @ A3 @ C2 ) @ ( plus_plus_real @ B6 @ C2 ) )
=> ( ord_less_real @ A3 @ B6 ) ) ).
% add_less_imp_less_right
thf(fact_921_add__less__imp__less__right,axiom,
! [A3: nat,C2: nat,B6: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B6 @ C2 ) )
=> ( ord_less_nat @ A3 @ B6 ) ) ).
% add_less_imp_less_right
thf(fact_922_add__less__imp__less__right,axiom,
! [A3: int,C2: int,B6: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B6 @ C2 ) )
=> ( ord_less_int @ A3 @ B6 ) ) ).
% add_less_imp_less_right
thf(fact_923_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_924_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_925_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_926_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_927_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_928_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_929_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_930_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_931_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_932_unbounded__k__infinite,axiom,
! [K2: nat,S: set_nat] :
( ! [M2: nat] :
( ( ord_less_nat @ K2 @ M2 )
=> ? [N5: nat] :
( ( ord_less_nat @ M2 @ N5 )
& ( member_nat @ N5 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_933_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_934_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_935_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_936_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_937_iadd__le__enat__iff,axiom,
! [X3: extended_enat,Y4: extended_enat,N: nat] :
( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y4 ) @ ( extended_enat2 @ N ) )
= ( ? [Y7: nat,X7: nat] :
( ( X3
= ( extended_enat2 @ X7 ) )
& ( Y4
= ( extended_enat2 @ Y7 ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ X7 @ Y7 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_938_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_939_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq_nat @ R @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% binomial_le_pow
thf(fact_940_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_941_ex__power__ivl1,axiom,
! [B6: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B6 @ N3 ) @ K2 )
& ( ord_less_nat @ K2 @ ( power_power_nat @ B6 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_942_ex__power__ivl2,axiom,
! [B6: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B6 @ N3 ) @ K2 )
& ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B6 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_943_sqrt__aux_I1_J,axiom,
! [N: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ) ).
% sqrt_aux(1)
thf(fact_944_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_945_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_946_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_947_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_948_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_949_numeral__less__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_950_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_iless
thf(fact_951_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K: nat] :
( ( N
= ( extended_enat2 @ K ) )
=> ~ ( ord_less_nat @ K @ M ) ) ) ).
% less_enatE
thf(fact_952_less__enat__def,axiom,
( ord_le72135733267957522d_enat
= ( ^ [M4: extended_enat,N2: extended_enat] :
( extended_case_enat_o
@ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
@ $false
@ M4 ) ) ) ).
% less_enat_def
thf(fact_953_Suc__ile__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
= ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% Suc_ile_eq
thf(fact_954_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_955_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_956_self__le__ge2__pow,axiom,
! [K2: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K2 @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_957_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_958_one__less__numeral,axiom,
! [N: num] :
( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( numera4658534427948366547nnreal @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral
thf(fact_959_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_960_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_961_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_962_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_963_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_964_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_965_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_966_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_967_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_968_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M5: extended_enat] :
( ( ord_le72135733267957522d_enat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_969_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_970_Suc__inject,axiom,
! [X3: nat,Y4: nat] :
( ( ( suc @ X3 )
= ( suc @ Y4 ) )
=> ( X3 = Y4 ) ) ).
% Suc_inject
thf(fact_971_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_972_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_973_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_974_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_975_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_976_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B6: nat] :
( ( P @ K2 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ B6 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y8: nat] :
( ( P @ Y8 )
=> ( ord_less_eq_nat @ Y8 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_977_linorder__neqE__nat,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ~ ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ Y4 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_978_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_979_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_980_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_981_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less_nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_982_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_983_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_984_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_985_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_986_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_987_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_988_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_989_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_990_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_991_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_992_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_993_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_994_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y6: nat,Z4: nat] :
( ( R2 @ X4 @ Y6 )
=> ( ( R2 @ Y6 @ Z4 )
=> ( R2 @ X4 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_995_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_996_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_997_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I3 @ K ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_998_less__trans__Suc,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_999_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1000_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1001_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1002_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1003_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1004_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1005_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1006_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1007_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1008_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1009_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1010_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1011_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1012_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1013_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1014_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1015_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1016_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1017_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F2 @ I3 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1018_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1019_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1020_nat__arith_Osuc1,axiom,
! [A: nat,K2: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K2 @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K2 @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1021_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1022_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1023_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1024_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1025_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1026_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1027_add__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1028_add__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1029_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1030_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1031_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1032_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1033_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1034_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1035_add__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1036_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1037_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1038_add__less__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1039_add__lessD1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
=> ( ord_less_nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_1040_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1041_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1042_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1043_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1044_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1045_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1046_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1047_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1048_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1049_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1050_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1051_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1052_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1053_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1054_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K2: nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( F2 @ M2 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K2 ) @ ( F2 @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1055_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1056_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1057_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1058_nat__add__1__add__1,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ one_one_nat )
= ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% nat_add_1_add_1
thf(fact_1059_realpow__square__minus__le,axiom,
! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_1060_sqrt__unique,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ( ord_less_nat @ N @ ( power_power_nat @ ( suc @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( sqrt @ N )
= M ) ) ) ).
% sqrt_unique
thf(fact_1061_sqrt__one,axiom,
( ( sqrt @ one_one_nat )
= one_one_nat ) ).
% sqrt_one
thf(fact_1062_sqrt__inverse__power2,axiom,
! [N: nat] :
( ( sqrt @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= N ) ).
% sqrt_inverse_power2
thf(fact_1063_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1064_real__arch__pow,axiom,
! [X3: real,Y4: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ? [N3: nat] : ( ord_less_real @ Y4 @ ( power_power_real @ X3 @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1065_sqrt__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( sqrt @ N ) @ N ) ).
% sqrt_le
thf(fact_1066_mono__sqrt_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( sqrt @ M ) @ ( sqrt @ N ) ) ) ).
% mono_sqrt'
thf(fact_1067_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1068_le__sqrtI,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ord_less_eq_nat @ X3 @ ( sqrt @ Y4 ) ) ) ).
% le_sqrtI
thf(fact_1069_sqrt__leI,axiom,
! [Y4: nat,X3: nat] :
( ! [Z4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ Z4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ord_less_eq_nat @ Z4 @ X3 ) )
=> ( ord_less_eq_nat @ ( sqrt @ Y4 ) @ X3 ) ) ).
% sqrt_leI
thf(fact_1070_le__sqrt__iff,axiom,
! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ ( sqrt @ Y4 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 ) ) ).
% le_sqrt_iff
thf(fact_1071_sqrt__le__iff,axiom,
! [Y4: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sqrt @ Y4 ) @ X3 )
= ( ! [Z: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ord_less_eq_nat @ Z @ X3 ) ) ) ) ).
% sqrt_le_iff
thf(fact_1072_sqrt__power2__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( power_power_nat @ ( sqrt @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ).
% sqrt_power2_le
thf(fact_1073_Suc__sqrt__power2__gt,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( suc @ ( sqrt @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_sqrt_power2_gt
thf(fact_1074_card__complex__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z: complex] :
( ( power_power_complex @ Z @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_complex_roots_unity
thf(fact_1075_pow_Osimps_I1_J,axiom,
! [X3: num] :
( ( pow @ X3 @ one )
= X3 ) ).
% pow.simps(1)
thf(fact_1076_seq__mono__lemma,axiom,
! [M: nat,D: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ord_less_real @ ( D @ N5 ) @ ( E @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1077_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1078_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1079_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1080_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1081_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_1082_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_1083_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_less_as_int
thf(fact_1084_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1085_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1086_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_1087_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1088_int__ops_I5_J,axiom,
! [A3: nat,B6: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A3 @ B6 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ).
% int_ops(5)
thf(fact_1089_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1090_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1091_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1092_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1093_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1094_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_1095_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1096_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1097_of__nat__eq__enat,axiom,
semiri4216267220026989637d_enat = extended_enat2 ).
% of_nat_eq_enat
thf(fact_1098_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1099_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1100_mult__le__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1101_mult__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1102_mult__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1103_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1104_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1105_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1106_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1107_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1108_Suc__times__binomial__eq,axiom,
! [N: nat,K2: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1109_Suc__times__binomial,axiom,
! [K2: nat,N: nat] :
( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).
% Suc_times_binomial
thf(fact_1110_choose__mult__lemma,axiom,
! [M: nat,R: nat,K2: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K2 ) @ ( plus_plus_nat @ M @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M @ K2 ) @ K2 ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_1111_numeral__eq__of__nat,axiom,
( numera4658534427948366547nnreal
= ( ^ [A4: num] : ( semiri6283507881447550617nnreal @ ( numeral_numeral_nat @ A4 ) ) ) ) ).
% numeral_eq_of_nat
thf(fact_1112_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M4: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M4 ) ) ) ) ).
% nat_less_real_le
thf(fact_1113_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M4: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M4 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1114_Suc__times__binomial__add,axiom,
! [A3: nat,B6: nat] :
( ( times_times_nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B6 ) ) @ ( suc @ A3 ) ) )
= ( times_times_nat @ ( suc @ B6 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B6 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_1115_binomial__mono,axiom,
! [K2: nat,K4: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ K4 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K4 ) ) ) ) ).
% binomial_mono
thf(fact_1116_binomial__maximum_H,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_1117_binomial__strict__mono,axiom,
! [K2: nat,K4: nat,N: nat] :
( ( ord_less_nat @ K2 @ K4 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K4 ) ) ) ) ).
% binomial_strict_mono
thf(fact_1118_binomial__strict__antimono,axiom,
! [K2: nat,K4: nat,N: nat] :
( ( ord_less_nat @ K2 @ K4 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) )
=> ( ( ord_less_eq_nat @ K4 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_1119_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_1120_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_1121_times__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( times_times_nat @ M @ N ) ) ) ).
% times_enat_simps(1)
thf(fact_1122_landau__product__preprocess_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% landau_product_preprocess(15)
thf(fact_1123_landau__product__preprocess_I13_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% landau_product_preprocess(13)
thf(fact_1124_landau__product__preprocess_I14_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% landau_product_preprocess(14)
thf(fact_1125_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1126_int__ops_I7_J,axiom,
! [A3: nat,B6: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A3 @ B6 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ).
% int_ops(7)
thf(fact_1127_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B7: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1128_int__if,axiom,
! [P: $o,A3: nat,B6: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A3 @ B6 ) )
= ( semiri1314217659103216013at_int @ A3 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A3 @ B6 ) )
= ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% int_if
thf(fact_1129_real__minus__mult__self__le,axiom,
! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X3 @ X3 ) ) ).
% real_minus_mult_self_le
thf(fact_1130_four__x__squared,axiom,
! [X3: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_1131_Bernoulli__inequality,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1132_L2__set__mult__ineq__lemma,axiom,
! [A3: real,C2: real,B6: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A3 @ C2 ) ) @ ( times_times_real @ B6 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1133_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1134_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1135_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_1136_finite__interval__int4,axiom,
! [A3: int,B6: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A3 @ I )
& ( ord_less_int @ I @ B6 ) ) ) ) ).
% finite_interval_int4
thf(fact_1137_finite__interval__int2,axiom,
! [A3: int,B6: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A3 @ I )
& ( ord_less_int @ I @ B6 ) ) ) ) ).
% finite_interval_int2
thf(fact_1138_finite__interval__int3,axiom,
! [A3: int,B6: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A3 @ I )
& ( ord_less_eq_int @ I @ B6 ) ) ) ) ).
% finite_interval_int3
thf(fact_1139_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1140_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1141_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1142_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1143_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1144_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1145_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1146_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_1147_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1148_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1149_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1150_int__ge__induct,axiom,
! [K2: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1151_int__gr__induct,axiom,
! [K2: int,I2: int,P: int > $o] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1152_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1153_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z: int] :
? [N2: nat] :
( Z
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1154_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z: int] :
? [N2: nat] :
( Z
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1155_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1156_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1157_real__of__nat__ge__one__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ one_one_nat @ N ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1158_sum__of__squares__ge__ennreal,axiom,
! [A3: extend8495563244428889912nnreal,B6: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ ( numera4658534427948366547nnreal @ ( bit0 @ one ) ) @ A3 ) @ B6 ) @ ( plus_p1859984266308609217nnreal @ ( power_6007165696250533058nnreal @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_6007165696250533058nnreal @ B6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_of_squares_ge_ennreal
thf(fact_1159_square__bound__lemma,axiom,
! [X3: real] : ( ord_less_real @ X3 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( plus_plus_real @ one_one_real @ X3 ) ) ) ).
% square_bound_lemma
thf(fact_1160_power__mono__ennreal,axiom,
! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,N: nat] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( power_6007165696250533058nnreal @ X3 @ N ) @ ( power_6007165696250533058nnreal @ Y4 @ N ) ) ) ).
% power_mono_ennreal
thf(fact_1161_sum__le__prod1,axiom,
! [A3: real,B6: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B6 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B6 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A3 @ B6 ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1162_Multiseries__Expansion_Ointyness__simps_I3_J,axiom,
! [A3: nat,N: nat] :
( ( power_power_real @ ( semiri5074537144036343181t_real @ A3 ) @ N )
= ( semiri5074537144036343181t_real @ ( power_power_nat @ A3 @ N ) ) ) ).
% Multiseries_Expansion.intyness_simps(3)
thf(fact_1163_Multiseries__Expansion_Ointyness__simps_I1_J,axiom,
! [A3: nat,B6: nat] :
( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A3 ) @ ( semiri5074537144036343181t_real @ B6 ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A3 @ B6 ) ) ) ).
% Multiseries_Expansion.intyness_simps(1)
thf(fact_1164_Multiseries__Expansion_Ointyness__uminus,axiom,
! [X3: real,N: nat] :
( ( X3
= ( semiri5074537144036343181t_real @ N ) )
=> ( ( uminus_uminus_real @ X3 )
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% Multiseries_Expansion.intyness_uminus
thf(fact_1165_Multiseries__Expansion_Ointyness__numeral,axiom,
! [Num: num] :
( ( Num = Num )
=> ( ( numeral_numeral_real @ Num )
= ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num ) ) ) ) ).
% Multiseries_Expansion.intyness_numeral
thf(fact_1166_Multiseries__Expansion_Ointyness__simps_I6_J,axiom,
( numeral_numeral_real
= ( ^ [Num2: num] : ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num2 ) ) ) ) ).
% Multiseries_Expansion.intyness_simps(6)
thf(fact_1167_Multiseries__Expansion_Ointyness__1,axiom,
( one_one_real
= ( semiri5074537144036343181t_real @ one_one_nat ) ) ).
% Multiseries_Expansion.intyness_1
thf(fact_1168_of__nat__less__ennreal__of__nat,axiom,
! [N: nat,X3: extended_enat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ N ) @ ( extend457580780380283419f_enat @ X3 ) )
= ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ N ) @ X3 ) ) ).
% of_nat_less_ennreal_of_nat
thf(fact_1169_ennreal__of__enat__le__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le3935885782089961368nnreal @ ( extend457580780380283419f_enat @ M ) @ ( extend457580780380283419f_enat @ N ) )
= ( ord_le2932123472753598470d_enat @ M @ N ) ) ).
% ennreal_of_enat_le_iff
thf(fact_1170_ennreal__of__enat__plus,axiom,
! [A3: extended_enat,B6: extended_enat] :
( ( extend457580780380283419f_enat @ ( plus_p3455044024723400733d_enat @ A3 @ B6 ) )
= ( plus_p1859984266308609217nnreal @ ( extend457580780380283419f_enat @ A3 ) @ ( extend457580780380283419f_enat @ B6 ) ) ) ).
% ennreal_of_enat_plus
thf(fact_1171_ennreal__of__enat__enat,axiom,
! [N: nat] :
( ( extend457580780380283419f_enat @ ( extended_enat2 @ N ) )
= ( semiri6283507881447550617nnreal @ N ) ) ).
% ennreal_of_enat_enat
thf(fact_1172_ennreal__of__enat__1,axiom,
( ( extend457580780380283419f_enat @ one_on7984719198319812577d_enat )
= one_on2969667320475766781nnreal ) ).
% ennreal_of_enat_1
thf(fact_1173_Bernoulli__inequality__even,axiom,
! [N: nat,X3: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_1174_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1175_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_1176_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_1177_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M4: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M4 ) ) ) ) ).
% int_less_real_le
thf(fact_1178_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M4: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M4 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1179_power__dvd__imp__le,axiom,
! [I2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1180_dvd__power__iff__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M ) @ ( power_power_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1181_ereal__power__uminus,axiom,
! [N: nat,X3: extended_ereal] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_1054015426188190660_ereal @ ( uminus27091377158695749_ereal @ X3 ) @ N )
= ( power_1054015426188190660_ereal @ X3 @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_1054015426188190660_ereal @ ( uminus27091377158695749_ereal @ X3 ) @ N )
= ( uminus27091377158695749_ereal @ ( power_1054015426188190660_ereal @ X3 @ N ) ) ) ) ) ).
% ereal_power_uminus
thf(fact_1182_ereal__mult__minus__left,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ A3 ) @ B6 )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A3 @ B6 ) ) ) ).
% ereal_mult_minus_left
thf(fact_1183_ereal__mult__minus__right,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( times_7703590493115627913_ereal @ A3 @ ( uminus27091377158695749_ereal @ B6 ) )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A3 @ B6 ) ) ) ).
% ereal_mult_minus_right
thf(fact_1184_ereal__minus__le__minus,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A3 ) @ ( uminus27091377158695749_ereal @ B6 ) )
= ( ord_le1083603963089353582_ereal @ B6 @ A3 ) ) ).
% ereal_minus_le_minus
thf(fact_1185_ereal__uminus__uminus,axiom,
! [A3: extended_ereal] :
( ( uminus27091377158695749_ereal @ ( uminus27091377158695749_ereal @ A3 ) )
= A3 ) ).
% ereal_uminus_uminus
thf(fact_1186_ereal__uminus__eq__iff,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A3 )
= ( uminus27091377158695749_ereal @ B6 ) )
= ( A3 = B6 ) ) ).
% ereal_uminus_eq_iff
thf(fact_1187_ereal__minus__less__minus,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A3 ) @ ( uminus27091377158695749_ereal @ B6 ) )
= ( ord_le1188267648640031866_ereal @ B6 @ A3 ) ) ).
% ereal_minus_less_minus
thf(fact_1188_zdvd__period,axiom,
! [A3: int,D: int,X3: int,T4: int,C2: int] :
( ( dvd_dvd_int @ A3 @ D )
=> ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ X3 @ T4 ) )
= ( dvd_dvd_int @ A3 @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C2 @ D ) ) @ T4 ) ) ) ) ).
% zdvd_period
thf(fact_1189_zdvd__reduce,axiom,
! [K2: int,N: int,M: int] :
( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N @ ( times_times_int @ K2 @ M ) ) )
= ( dvd_dvd_int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_1190_ereal__uminus__le__reorder,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A3 ) @ B6 )
= ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ B6 ) @ A3 ) ) ).
% ereal_uminus_le_reorder
thf(fact_1191_ereal__uminus__eq__reorder,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A3 )
= B6 )
= ( A3
= ( uminus27091377158695749_ereal @ B6 ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_1192_ereal__uminus__less__reorder,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A3 ) @ B6 )
= ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ B6 ) @ A3 ) ) ).
% ereal_uminus_less_reorder
thf(fact_1193_ereal__less__uminus__reorder,axiom,
! [A3: extended_ereal,B6: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A3 @ ( uminus27091377158695749_ereal @ B6 ) )
= ( ord_le1188267648640031866_ereal @ B6 @ ( uminus27091377158695749_ereal @ A3 ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_1194_ereal__le__distrib,axiom,
! [C2: extended_ereal,A3: extended_ereal,B6: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C2 @ ( plus_p7876563987511257093_ereal @ A3 @ B6 ) ) @ ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C2 @ A3 ) @ ( times_7703590493115627913_ereal @ C2 @ B6 ) ) ) ).
% ereal_le_distrib
thf(fact_1195_less__eq__ereal__def,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_ereal_def
thf(fact_1196_ereal__add__strict__mono2,axiom,
! [A3: extended_ereal,B6: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A3 @ B6 )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A3 @ C2 ) @ ( plus_p7876563987511257093_ereal @ B6 @ D ) ) ) ) ).
% ereal_add_strict_mono2
thf(fact_1197_bezout__lemma__nat,axiom,
! [D: nat,A3: nat,B6: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D @ A3 )
=> ( ( dvd_dvd_nat @ D @ B6 )
=> ( ( ( ( times_times_nat @ A3 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B6 @ Y4 ) @ D ) )
| ( ( times_times_nat @ B6 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ Y4 ) @ D ) ) )
=> ? [X4: nat,Y6: nat] :
( ( dvd_dvd_nat @ D @ A3 )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A3 @ B6 ) )
& ( ( ( times_times_nat @ A3 @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B6 ) @ Y6 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A3 @ B6 ) @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ Y6 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1198_bezout__add__nat,axiom,
! [A3: nat,B6: nat] :
? [D2: nat,X4: nat,Y6: nat] :
( ( dvd_dvd_nat @ D2 @ A3 )
& ( dvd_dvd_nat @ D2 @ B6 )
& ( ( ( times_times_nat @ A3 @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B6 @ Y6 ) @ D2 ) )
| ( ( times_times_nat @ B6 @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ Y6 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1199_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D3 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_1200_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int,T3: int] : ( dvd_dvd_int @ D3 @ ( uminus_uminus_int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1201_signed__take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_1202_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_1203_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_1204_signed__take__bit__minus,axiom,
! [N: nat,K2: int] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K2 ) ) ) ).
% signed_take_bit_minus
thf(fact_1205_signed__take__bit__add,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K2 @ L ) ) ) ).
% signed_take_bit_add
thf(fact_1206_push__bit__int__def,axiom,
( bit_se545348938243370406it_int
= ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_int_def
thf(fact_1207_push__bit__nat__def,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N2: nat,M4: nat] : ( times_times_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_nat_def
thf(fact_1208_signed__take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1209_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_1210_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_1211_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_1212_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_1213_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_1214_signed__take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_1215_signed__take__bit__int__eq__self,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
=> ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K2 )
= K2 ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_1216_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K2 )
= K2 )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
& ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_1217_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1218_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_1219_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_1220_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1221_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_1222_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_1223_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_1224_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_1225_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1226_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1227_landau__product__preprocess_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% landau_product_preprocess(9)
thf(fact_1228_landau__product__preprocess_I11_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% landau_product_preprocess(11)
thf(fact_1229_eq__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral_nat @ K2 )
= ( suc @ N ) )
= ( ( pred_numeral @ K2 )
= N ) ) ).
% eq_numeral_Suc
thf(fact_1230_Suc__eq__numeral,axiom,
! [N: nat,K2: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K2 ) )
= ( N
= ( pred_numeral @ K2 ) ) ) ).
% Suc_eq_numeral
thf(fact_1231_landau__product__preprocess_I17_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% landau_product_preprocess(17)
thf(fact_1232_landau__product__preprocess_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% landau_product_preprocess(16)
thf(fact_1233_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1234_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1235_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1236_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1237_landau__product__preprocess_I5_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% landau_product_preprocess(5)
thf(fact_1238_landau__product__preprocess_I6_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% landau_product_preprocess(6)
thf(fact_1239_landau__product__preprocess_I7_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% landau_product_preprocess(7)
thf(fact_1240_landau__product__preprocess_I10_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% landau_product_preprocess(10)
thf(fact_1241_landau__product__preprocess_I12_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% landau_product_preprocess(12)
thf(fact_1242_pred__numeral__simps_I3_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit1 @ K2 ) )
= ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ).
% pred_numeral_simps(3)
thf(fact_1243_le__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( ord_less_eq_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% le_Suc_numeral
thf(fact_1244_le__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% le_numeral_Suc
thf(fact_1245_less__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_1246_less__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( ord_less_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% less_Suc_numeral
thf(fact_1247_landau__product__preprocess_I18_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% landau_product_preprocess(18)
thf(fact_1248_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1249_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1250_signed__take__bit__numeral__bit0,axiom,
! [L: num,K2: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_1251_signed__take__bit__numeral__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_1252_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1253_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1254_num_Oexhaust,axiom,
! [Y4: num] :
( ( Y4 != one )
=> ( ! [X23: num] :
( Y4
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y4
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1255_forall__4,axiom,
( ( ^ [P2: numera4273646738625120315l_num1 > $o] :
! [X6: numera4273646738625120315l_num1] : ( P2 @ X6 ) )
= ( ^ [P3: numera4273646738625120315l_num1 > $o] :
( ( P3 @ one_on7795324986448017462l_num1 )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
& ( P3 @ ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).
% forall_4
thf(fact_1256_exhaust__4,axiom,
! [X3: numera4273646738625120315l_num1] :
( ( X3 = one_on7795324986448017462l_num1 )
| ( X3
= ( numera7754357348821619680l_num1 @ ( bit0 @ one ) ) )
| ( X3
= ( numera7754357348821619680l_num1 @ ( bit1 @ one ) ) )
| ( X3
= ( numera7754357348821619680l_num1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% exhaust_4
thf(fact_1257_forall__3,axiom,
( ( ^ [P2: numera6367994245245682809l_num1 > $o] :
! [X6: numera6367994245245682809l_num1] : ( P2 @ X6 ) )
= ( ^ [P3: numera6367994245245682809l_num1 > $o] :
( ( P3 @ one_on7819281148064737470l_num1 )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
& ( P3 @ ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ) ) ).
% forall_3
thf(fact_1258_exhaust__3,axiom,
! [X3: numera6367994245245682809l_num1] :
( ( X3 = one_on7819281148064737470l_num1 )
| ( X3
= ( numera6112219686443703444l_num1 @ ( bit0 @ one ) ) )
| ( X3
= ( numera6112219686443703444l_num1 @ ( bit1 @ one ) ) ) ) ).
% exhaust_3
thf(fact_1259_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_1260_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1261_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1262_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_1263_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y4: nat] :
( ( if_nat @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y4: nat] :
( ( if_nat @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
finite_finite_a @ x ).
thf(conj_1,conjecture,
( finite_finite_set_a
@ ( collect_set_a
@ ^ [Y2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ x )
& ( ( finite_card_a @ Y2 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
%------------------------------------------------------------------------------