TPTP Problem File: SLH0762^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Cotangent_PFD_Formula/0007_Cotangent_PFD_Formula/prob_00242_009268__14008316_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1368 ( 609 unt; 97 typ; 0 def)
% Number of atoms : 3685 (1196 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10361 ( 336 ~; 59 |; 198 &;8295 @)
% ( 0 <=>;1473 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 456 ( 456 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 90 usr; 22 con; 0-3 aty)
% Number of variables : 3426 ( 211 ^;3161 !; 54 ?;3426 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:57:34.280
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
set_complex_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (90)
thf(sy_c_Abstract__Topology__2_Oconstant__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
abstra1392856459910508790omplex: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Complex__Analysis__Basics_Oanalytic__on,type,
comple673786817313641009tic_on: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Complex__Analysis__Basics_Oholomorphic__on,type,
comple7700996537433184370hic_on: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Complex__Singularities_Oisolated__singularity__at,type,
comple1891072044276206784ity_at: ( complex > complex ) > complex > $o ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Complex__Ocomplex,type,
cotang8298477626502807258omplex: complex > complex ).
thf(sy_c_Derivative_Oderiv_001t__Complex__Ocomplex,type,
deriv_complex: ( complex > complex ) > complex > complex ).
thf(sy_c_Elementary__Metric__Spaces_Oball_001t__Complex__Ocomplex,type,
elemen509638587668912547omplex: complex > real > set_complex ).
thf(sy_c_Elementary__Metric__Spaces_Ocball_001t__Complex__Ocomplex,type,
elemen7827680097914048924omplex: complex > real > set_complex ).
thf(sy_c_Elementary__Metric__Spaces_Osphere_001t__Complex__Ocomplex,type,
elemen5844589719862857425omplex: complex > real > set_complex ).
thf(sy_c_Factorial__Ring_Ofactorial__semiring__class_OGcd__factorial_001t__Nat__Onat,type,
factor8539158941071730396al_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
minus_3522879524658371850omplex: set_complex_complex > set_complex_complex > set_complex_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Complex__Ocomplex_J,type,
one_one_set_complex: set_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
one_one_set_nat: set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Real__Oreal_J,type,
one_one_set_real: set_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
plus_p7052360327008956141omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
uminus4994531801924300922omplex: set_complex_complex > set_complex_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Complex__Ocomplex_J,type,
zero_z6614145512433583213omplex: set_complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
ring_1_Ints_complex: set_complex ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_M_Eo_J,type,
bot_bo466812550819818624plex_o: ( complex > complex ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Complex__Ocomplex_M_Eo_J,type,
bot_bot_complex_o: complex > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
bot_bo8693375350852365381omplex: set_complex_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
ord_le3207539288156484613omplex: set_complex_complex > set_complex_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
ord_le6271439605799870481omplex: set_complex_complex > set_complex_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $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__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_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Riemann__Mapping_OMoebius__function,type,
rieman142549540920964168nction: real > complex > complex > complex ).
thf(sy_c_Set_OCollect_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
collec7522758907530094160omplex: ( ( complex > complex ) > $o ) > set_complex_complex ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_Oinsert_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
insert2420511349202589739omplex: ( complex > complex ) > set_complex_complex > set_complex_complex ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Ois__empty_001t__Complex__Ocomplex,type,
is_empty_complex: set_complex > $o ).
thf(sy_c_Set_Ois__singleton_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
is_sin8098092515724343303omplex: set_complex_complex > $o ).
thf(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
is_singleton_complex: set_complex > $o ).
thf(sy_c_Set_Oremove_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
remove423721587600764758omplex: ( complex > complex ) > set_complex_complex > set_complex_complex ).
thf(sy_c_Set_Oremove_001t__Complex__Ocomplex,type,
remove_complex: complex > set_complex > set_complex ).
thf(sy_c_Set_Othe__elem_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
the_el7055940373909865542omplex: set_complex_complex > complex > complex ).
thf(sy_c_Set_Othe__elem_001t__Complex__Ocomplex,type,
the_elem_complex: set_complex > complex ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
topolo9015423870875150044omplex: set_complex > ( complex > complex ) > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
topolo7782564052695852148omplex: set_complex_complex > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
topolo4110288021797289639omplex: set_complex > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact_001t__Complex__Ocomplex,type,
topolo702044070747558609omplex: set_complex > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001t__Complex__Ocomplex,type,
topolo3972588530358341399omplex: set_complex > $o ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded_001t__Complex__Ocomplex,type,
topolo7663995495656592641omplex: set_complex > $o ).
thf(sy_c_member_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
member5128974058612258834omplex: ( complex > complex ) > set_complex_complex > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_A,type,
a: set_complex ).
% Relevant facts (1267)
thf(fact_0_assms,axiom,
ord_le211207098394363844omplex @ a @ ( uminus8566677241136511917omplex @ ( minus_811609699411566653omplex @ ring_1_Ints_complex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ) ).
% assms
thf(fact_1_insert__Diff__single,axiom,
! [A: complex > complex,A2: set_complex_complex] :
( ( insert2420511349202589739omplex @ A @ ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) ) )
= ( insert2420511349202589739omplex @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_2_insert__Diff__single,axiom,
! [A: complex,A2: set_complex] :
( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= ( insert_complex @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_3_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_5_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_6_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_7__C_K_C,axiom,
topolo4110288021797289639omplex @ ( uminus8566677241136511917omplex @ ( minus_811609699411566653omplex @ ring_1_Ints_complex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ) ).
% "*"
thf(fact_8_Diff__insert0,axiom,
! [X: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ~ ( member5128974058612258834omplex @ X @ A2 )
=> ( ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ X @ B2 ) )
= ( minus_3522879524658371850omplex @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_9_Diff__insert0,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ~ ( member_complex @ X @ A2 )
=> ( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ B2 ) )
= ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_10_insert__Diff1,axiom,
! [X: complex > complex,B2: set_complex_complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ X @ B2 )
=> ( ( minus_3522879524658371850omplex @ ( insert2420511349202589739omplex @ X @ A2 ) @ B2 )
= ( minus_3522879524658371850omplex @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_11_insert__Diff1,axiom,
! [X: complex,B2: set_complex,A2: set_complex] :
( ( member_complex @ X @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
= ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_12_Diff__empty,axiom,
! [A2: set_complex_complex] :
( ( minus_3522879524658371850omplex @ A2 @ bot_bo8693375350852365381omplex )
= A2 ) ).
% Diff_empty
thf(fact_13_Diff__empty,axiom,
! [A2: set_complex] :
( ( minus_811609699411566653omplex @ A2 @ bot_bot_set_complex )
= A2 ) ).
% Diff_empty
thf(fact_14_empty__Diff,axiom,
! [A2: set_complex_complex] :
( ( minus_3522879524658371850omplex @ bot_bo8693375350852365381omplex @ A2 )
= bot_bo8693375350852365381omplex ) ).
% empty_Diff
thf(fact_15_empty__Diff,axiom,
! [A2: set_complex] :
( ( minus_811609699411566653omplex @ bot_bot_set_complex @ A2 )
= bot_bot_set_complex ) ).
% empty_Diff
thf(fact_16_Diff__cancel,axiom,
! [A2: set_complex_complex] :
( ( minus_3522879524658371850omplex @ A2 @ A2 )
= bot_bo8693375350852365381omplex ) ).
% Diff_cancel
thf(fact_17_Diff__cancel,axiom,
! [A2: set_complex] :
( ( minus_811609699411566653omplex @ A2 @ A2 )
= bot_bot_set_complex ) ).
% Diff_cancel
thf(fact_18_singletonI,axiom,
! [A: complex > complex] : ( member5128974058612258834omplex @ A @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) ) ).
% singletonI
thf(fact_19_singletonI,axiom,
! [A: complex] : ( member_complex @ A @ ( insert_complex @ A @ bot_bot_set_complex ) ) ).
% singletonI
thf(fact_20_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_21_minus__diff__eq,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
= ( minus_minus_complex @ B @ A ) ) ).
% minus_diff_eq
thf(fact_22_subsetI,axiom,
! [A2: set_complex,B2: set_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_complex @ X2 @ B2 ) )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% subsetI
thf(fact_23_subsetI,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ! [X2: complex > complex] :
( ( member5128974058612258834omplex @ X2 @ A2 )
=> ( member5128974058612258834omplex @ X2 @ B2 ) )
=> ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ).
% subsetI
thf(fact_24_subset__antisym,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_25_subset__antisym,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( ord_le6271439605799870481omplex @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_26_empty__Collect__eq,axiom,
! [P: complex > $o] :
( ( bot_bot_set_complex
= ( collect_complex @ P ) )
= ( ! [X3: complex] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_27_Collect__empty__eq,axiom,
! [P: complex > $o] :
( ( ( collect_complex @ P )
= bot_bot_set_complex )
= ( ! [X3: complex] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_28_all__not__in__conv,axiom,
! [A2: set_complex_complex] :
( ( ! [X3: complex > complex] :
~ ( member5128974058612258834omplex @ X3 @ A2 ) )
= ( A2 = bot_bo8693375350852365381omplex ) ) ).
% all_not_in_conv
thf(fact_29_all__not__in__conv,axiom,
! [A2: set_complex] :
( ( ! [X3: complex] :
~ ( member_complex @ X3 @ A2 ) )
= ( A2 = bot_bot_set_complex ) ) ).
% all_not_in_conv
thf(fact_30_empty__iff,axiom,
! [C: complex > complex] :
~ ( member5128974058612258834omplex @ C @ bot_bo8693375350852365381omplex ) ).
% empty_iff
thf(fact_31_empty__iff,axiom,
! [C: complex] :
~ ( member_complex @ C @ bot_bot_set_complex ) ).
% empty_iff
thf(fact_32_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_33_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_34_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_35_add_Oinverse__inverse,axiom,
! [A: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_36_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_37_verit__minus__simplify_I4_J,axiom,
! [B: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_38_insert__absorb2,axiom,
! [X: complex,A2: set_complex] :
( ( insert_complex @ X @ ( insert_complex @ X @ A2 ) )
= ( insert_complex @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_39_insert__absorb2,axiom,
! [X: complex > complex,A2: set_complex_complex] :
( ( insert2420511349202589739omplex @ X @ ( insert2420511349202589739omplex @ X @ A2 ) )
= ( insert2420511349202589739omplex @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_40_insert__iff,axiom,
! [A: complex,B: complex,A2: set_complex] :
( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
= ( ( A = B )
| ( member_complex @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_41_insert__iff,axiom,
! [A: complex > complex,B: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ A @ ( insert2420511349202589739omplex @ B @ A2 ) )
= ( ( A = B )
| ( member5128974058612258834omplex @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_42_insertCI,axiom,
! [A: complex,B2: set_complex,B: complex] :
( ( ~ ( member_complex @ A @ B2 )
=> ( A = B ) )
=> ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% insertCI
thf(fact_43_insertCI,axiom,
! [A: complex > complex,B2: set_complex_complex,B: complex > complex] :
( ( ~ ( member5128974058612258834omplex @ A @ B2 )
=> ( A = B ) )
=> ( member5128974058612258834omplex @ A @ ( insert2420511349202589739omplex @ B @ B2 ) ) ) ).
% insertCI
thf(fact_44_Diff__idemp,axiom,
! [A2: set_complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) @ B2 )
= ( minus_811609699411566653omplex @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_45_Diff__idemp,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( minus_3522879524658371850omplex @ ( minus_3522879524658371850omplex @ A2 @ B2 ) @ B2 )
= ( minus_3522879524658371850omplex @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_46_Diff__iff,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
= ( ( member_complex @ C @ A2 )
& ~ ( member_complex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_47_Diff__iff,axiom,
! [C: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ ( minus_3522879524658371850omplex @ A2 @ B2 ) )
= ( ( member5128974058612258834omplex @ C @ A2 )
& ~ ( member5128974058612258834omplex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_48_DiffI,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ A2 )
=> ( ~ ( member_complex @ C @ B2 )
=> ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_49_DiffI,axiom,
! [C: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ A2 )
=> ( ~ ( member5128974058612258834omplex @ C @ B2 )
=> ( member5128974058612258834omplex @ C @ ( minus_3522879524658371850omplex @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_50_Compl__eq__Compl__iff,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ( uminus8566677241136511917omplex @ A2 )
= ( uminus8566677241136511917omplex @ B2 ) )
= ( A2 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_51_Compl__iff,axiom,
! [C: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ ( uminus4994531801924300922omplex @ A2 ) )
= ( ~ ( member5128974058612258834omplex @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_52_Compl__iff,axiom,
! [C: complex,A2: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
= ( ~ ( member_complex @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_53_ComplI,axiom,
! [C: complex > complex,A2: set_complex_complex] :
( ~ ( member5128974058612258834omplex @ C @ A2 )
=> ( member5128974058612258834omplex @ C @ ( uminus4994531801924300922omplex @ A2 ) ) ) ).
% ComplI
thf(fact_54_ComplI,axiom,
! [C: complex,A2: set_complex] :
( ~ ( member_complex @ C @ A2 )
=> ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% ComplI
thf(fact_55_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_56_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_57_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_58_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_59_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_60_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_61_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_62_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_63_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_64_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_65_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_66_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_67_neg__le__iff__le,axiom,
! [B: complex,A: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_68_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_69_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_70_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_71_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_72_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_73_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_74_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_75_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_76_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_77_empty__subsetI,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ bot_bot_set_complex @ A2 ) ).
% empty_subsetI
thf(fact_78_empty__subsetI,axiom,
! [A2: set_complex_complex] : ( ord_le6271439605799870481omplex @ bot_bo8693375350852365381omplex @ A2 ) ).
% empty_subsetI
thf(fact_79_subset__empty,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ bot_bot_set_complex )
= ( A2 = bot_bot_set_complex ) ) ).
% subset_empty
thf(fact_80_subset__empty,axiom,
! [A2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ bot_bo8693375350852365381omplex )
= ( A2 = bot_bo8693375350852365381omplex ) ) ).
% subset_empty
thf(fact_81_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A: complex > complex,P: ( complex > complex ) > $o] :
( ( member5128974058612258834omplex @ A @ ( collec7522758907530094160omplex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_complex_complex] :
( ( collec7522758907530094160omplex
@ ^ [X3: complex > complex] : ( member5128974058612258834omplex @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_insert__subset,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A2 ) @ B2 )
= ( ( member_complex @ X @ B2 )
& ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_86_insert__subset,axiom,
! [X: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ ( insert2420511349202589739omplex @ X @ A2 ) @ B2 )
= ( ( member5128974058612258834omplex @ X @ B2 )
& ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_87_Compl__subset__Compl__iff,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ A2 ) @ ( uminus8566677241136511917omplex @ B2 ) )
= ( ord_le211207098394363844omplex @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_88_Compl__subset__Compl__iff,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ ( uminus4994531801924300922omplex @ A2 ) @ ( uminus4994531801924300922omplex @ B2 ) )
= ( ord_le6271439605799870481omplex @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_89_Compl__anti__mono,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ B2 ) @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_90_Compl__anti__mono,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ord_le6271439605799870481omplex @ ( uminus4994531801924300922omplex @ B2 ) @ ( uminus4994531801924300922omplex @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_91_diff__ge__0__iff__ge,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( minus_minus_complex @ A @ B ) )
= ( ord_less_eq_complex @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_92_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_93_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_94_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_95_neg__le__0__iff__le,axiom,
! [A: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A ) @ zero_zero_complex )
= ( ord_less_eq_complex @ zero_zero_complex @ A ) ) ).
% neg_le_0_iff_le
thf(fact_96_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_97_neg__0__le__iff__le,axiom,
! [A: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% neg_0_le_iff_le
thf(fact_98_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_99_singleton__insert__inj__eq_H,axiom,
! [A: complex,A2: set_complex,B: complex] :
( ( ( insert_complex @ A @ A2 )
= ( insert_complex @ B @ bot_bot_set_complex ) )
= ( ( A = B )
& ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_100_singleton__insert__inj__eq_H,axiom,
! [A: complex > complex,A2: set_complex_complex,B: complex > complex] :
( ( ( insert2420511349202589739omplex @ A @ A2 )
= ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex ) )
= ( ( A = B )
& ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_101_singleton__insert__inj__eq,axiom,
! [B: complex,A: complex,A2: set_complex] :
( ( ( insert_complex @ B @ bot_bot_set_complex )
= ( insert_complex @ A @ A2 ) )
= ( ( A = B )
& ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ B @ bot_bot_set_complex ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_102_singleton__insert__inj__eq,axiom,
! [B: complex > complex,A: complex > complex,A2: set_complex_complex] :
( ( ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex )
= ( insert2420511349202589739omplex @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_103_Diff__eq__empty__iff,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ( minus_811609699411566653omplex @ A2 @ B2 )
= bot_bot_set_complex )
= ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_104_Diff__eq__empty__iff,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ( minus_3522879524658371850omplex @ A2 @ B2 )
= bot_bo8693375350852365381omplex )
= ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_105_subset__Compl__singleton,axiom,
! [A2: set_complex,B: complex] :
( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
= ( ~ ( member_complex @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_106_subset__Compl__singleton,axiom,
! [A2: set_complex_complex,B: complex > complex] :
( ( ord_le6271439605799870481omplex @ A2 @ ( uminus4994531801924300922omplex @ ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex ) ) )
= ( ~ ( member5128974058612258834omplex @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_107_verit__comp__simplify1_I2_J,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_108_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_109_verit__comp__simplify1_I2_J,axiom,
! [A: set_complex_complex] : ( ord_le6271439605799870481omplex @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_110_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_111_in__mono,axiom,
! [A2: set_complex,B2: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ X @ A2 )
=> ( member_complex @ X @ B2 ) ) ) ).
% in_mono
thf(fact_112_in__mono,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,X: complex > complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( member5128974058612258834omplex @ X @ A2 )
=> ( member5128974058612258834omplex @ X @ B2 ) ) ) ).
% in_mono
thf(fact_113_subsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% subsetD
thf(fact_114_subsetD,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,C: complex > complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( member5128974058612258834omplex @ C @ A2 )
=> ( member5128974058612258834omplex @ C @ B2 ) ) ) ).
% subsetD
thf(fact_115_equalityE,axiom,
! [A2: set_complex,B2: set_complex] :
( ( A2 = B2 )
=> ~ ( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ~ ( ord_le211207098394363844omplex @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_116_equalityE,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( A2 = B2 )
=> ~ ( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ~ ( ord_le6271439605799870481omplex @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_117_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
! [X3: complex] :
( ( member_complex @ X3 @ A3 )
=> ( member_complex @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_118_subset__eq,axiom,
( ord_le6271439605799870481omplex
= ( ^ [A3: set_complex_complex,B3: set_complex_complex] :
! [X3: complex > complex] :
( ( member5128974058612258834omplex @ X3 @ A3 )
=> ( member5128974058612258834omplex @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_119_equalityD1,axiom,
! [A2: set_complex,B2: set_complex] :
( ( A2 = B2 )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_120_equalityD1,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( A2 = B2 )
=> ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_121_equalityD2,axiom,
! [A2: set_complex,B2: set_complex] :
( ( A2 = B2 )
=> ( ord_le211207098394363844omplex @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_122_equalityD2,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( A2 = B2 )
=> ( ord_le6271439605799870481omplex @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_123_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
! [T: complex] :
( ( member_complex @ T @ A3 )
=> ( member_complex @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_124_subset__iff,axiom,
( ord_le6271439605799870481omplex
= ( ^ [A3: set_complex_complex,B3: set_complex_complex] :
! [T: complex > complex] :
( ( member5128974058612258834omplex @ T @ A3 )
=> ( member5128974058612258834omplex @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_125_subset__refl,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ A2 ) ).
% subset_refl
thf(fact_126_subset__refl,axiom,
! [A2: set_complex_complex] : ( ord_le6271439605799870481omplex @ A2 @ A2 ) ).
% subset_refl
thf(fact_127_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_128_Collect__mono,axiom,
! [P: ( complex > complex ) > $o,Q: ( complex > complex ) > $o] :
( ! [X2: complex > complex] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6271439605799870481omplex @ ( collec7522758907530094160omplex @ P ) @ ( collec7522758907530094160omplex @ Q ) ) ) ).
% Collect_mono
thf(fact_129_subset__trans,axiom,
! [A2: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C2 )
=> ( ord_le211207098394363844omplex @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_130_subset__trans,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,C2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( ord_le6271439605799870481omplex @ B2 @ C2 )
=> ( ord_le6271439605799870481omplex @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_131_set__eq__subset,axiom,
( ( ^ [Y: set_complex,Z: set_complex] : ( Y = Z ) )
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B3 )
& ( ord_le211207098394363844omplex @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_132_set__eq__subset,axiom,
( ( ^ [Y: set_complex_complex,Z: set_complex_complex] : ( Y = Z ) )
= ( ^ [A3: set_complex_complex,B3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A3 @ B3 )
& ( ord_le6271439605799870481omplex @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_133_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_134_Collect__mono__iff,axiom,
! [P: ( complex > complex ) > $o,Q: ( complex > complex ) > $o] :
( ( ord_le6271439605799870481omplex @ ( collec7522758907530094160omplex @ P ) @ ( collec7522758907530094160omplex @ Q ) )
= ( ! [X3: complex > complex] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_135_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_136_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_137_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_138_diff__eq__diff__less__eq,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( ord_less_eq_complex @ A @ B )
= ( ord_less_eq_complex @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_139_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_140_diff__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ A @ C ) @ ( minus_minus_complex @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_141_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_142_diff__left__mono,axiom,
! [B: complex,A: complex,C: complex] :
( ( ord_less_eq_complex @ B @ A )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ C @ A ) @ ( minus_minus_complex @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_143_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_144_diff__mono,axiom,
! [A: complex,B: complex,D: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ( ord_less_eq_complex @ D @ C )
=> ( ord_less_eq_complex @ ( minus_minus_complex @ A @ C ) @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_145_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_146_le__imp__neg__le,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% le_imp_neg_le
thf(fact_147_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_148_minus__le__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_149_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_150_le__minus__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( ord_less_eq_complex @ B @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% le_minus_iff
thf(fact_151_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_152_subset__insertI2,axiom,
! [A2: set_complex,B2: set_complex,B: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_153_subset__insertI2,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,B: complex > complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_154_subset__insertI,axiom,
! [B2: set_complex,A: complex] : ( ord_le211207098394363844omplex @ B2 @ ( insert_complex @ A @ B2 ) ) ).
% subset_insertI
thf(fact_155_subset__insertI,axiom,
! [B2: set_complex_complex,A: complex > complex] : ( ord_le6271439605799870481omplex @ B2 @ ( insert2420511349202589739omplex @ A @ B2 ) ) ).
% subset_insertI
thf(fact_156_subset__insert,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ~ ( member_complex @ X @ A2 )
=> ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
= ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_157_subset__insert,axiom,
! [X: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ~ ( member5128974058612258834omplex @ X @ A2 )
=> ( ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ X @ B2 ) )
= ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_158_insert__mono,axiom,
! [C2: set_complex,D2: set_complex,A: complex] :
( ( ord_le211207098394363844omplex @ C2 @ D2 )
=> ( ord_le211207098394363844omplex @ ( insert_complex @ A @ C2 ) @ ( insert_complex @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_159_insert__mono,axiom,
! [C2: set_complex_complex,D2: set_complex_complex,A: complex > complex] :
( ( ord_le6271439605799870481omplex @ C2 @ D2 )
=> ( ord_le6271439605799870481omplex @ ( insert2420511349202589739omplex @ A @ C2 ) @ ( insert2420511349202589739omplex @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_160_double__diff,axiom,
! [A2: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C2 )
=> ( ( minus_811609699411566653omplex @ B2 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_161_double__diff,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,C2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( ord_le6271439605799870481omplex @ B2 @ C2 )
=> ( ( minus_3522879524658371850omplex @ B2 @ ( minus_3522879524658371850omplex @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_162_Diff__subset,axiom,
! [A2: set_complex,B2: set_complex] : ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_163_Diff__subset,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] : ( ord_le6271439605799870481omplex @ ( minus_3522879524658371850omplex @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_164_Diff__mono,axiom,
! [A2: set_complex,C2: set_complex,D2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ C2 )
=> ( ( ord_le211207098394363844omplex @ D2 @ B2 )
=> ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) @ ( minus_811609699411566653omplex @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_165_Diff__mono,axiom,
! [A2: set_complex_complex,C2: set_complex_complex,D2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ C2 )
=> ( ( ord_le6271439605799870481omplex @ D2 @ B2 )
=> ( ord_le6271439605799870481omplex @ ( minus_3522879524658371850omplex @ A2 @ B2 ) @ ( minus_3522879524658371850omplex @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_166_le__iff__diff__le__0,axiom,
( ord_less_eq_complex
= ( ^ [A4: complex,B4: complex] : ( ord_less_eq_complex @ ( minus_minus_complex @ A4 @ B4 ) @ zero_zero_complex ) ) ) ).
% le_iff_diff_le_0
thf(fact_167_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_168_subset__singleton__iff,axiom,
! [X4: set_complex,A: complex] :
( ( ord_le211207098394363844omplex @ X4 @ ( insert_complex @ A @ bot_bot_set_complex ) )
= ( ( X4 = bot_bot_set_complex )
| ( X4
= ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).
% subset_singleton_iff
thf(fact_169_subset__singleton__iff,axiom,
! [X4: set_complex_complex,A: complex > complex] :
( ( ord_le6271439605799870481omplex @ X4 @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) )
= ( ( X4 = bot_bo8693375350852365381omplex )
| ( X4
= ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) ) ) ) ).
% subset_singleton_iff
thf(fact_170_subset__singletonD,axiom,
! [A2: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) )
=> ( ( A2 = bot_bot_set_complex )
| ( A2
= ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ).
% subset_singletonD
thf(fact_171_subset__singletonD,axiom,
! [A2: set_complex_complex,X: complex > complex] :
( ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) )
=> ( ( A2 = bot_bo8693375350852365381omplex )
| ( A2
= ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) ) ) ) ).
% subset_singletonD
thf(fact_172_subset__Diff__insert,axiom,
! [A2: set_complex,B2: set_complex,X: complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ ( insert_complex @ X @ C2 ) ) )
= ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ C2 ) )
& ~ ( member_complex @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_173_subset__Diff__insert,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,X: complex > complex,C2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ ( minus_3522879524658371850omplex @ B2 @ ( insert2420511349202589739omplex @ X @ C2 ) ) )
= ( ( ord_le6271439605799870481omplex @ A2 @ ( minus_3522879524658371850omplex @ B2 @ C2 ) )
& ~ ( member5128974058612258834omplex @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_174_subset__Compl__self__eq,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ A2 ) )
= ( A2 = bot_bot_set_complex ) ) ).
% subset_Compl_self_eq
thf(fact_175_subset__Compl__self__eq,axiom,
! [A2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ ( uminus4994531801924300922omplex @ A2 ) )
= ( A2 = bot_bo8693375350852365381omplex ) ) ).
% subset_Compl_self_eq
thf(fact_176_Diff__single__insert,axiom,
! [A2: set_complex,X: complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 )
=> ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_177_Diff__single__insert,axiom,
! [A2: set_complex_complex,X: complex > complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) ) @ B2 )
=> ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_178_subset__insert__iff,axiom,
! [A2: set_complex,X: complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
= ( ( ( member_complex @ X @ A2 )
=> ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
& ( ~ ( member_complex @ X @ A2 )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_179_subset__insert__iff,axiom,
! [A2: set_complex_complex,X: complex > complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ ( insert2420511349202589739omplex @ X @ B2 ) )
= ( ( ( member5128974058612258834omplex @ X @ A2 )
=> ( ord_le6271439605799870481omplex @ ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) ) @ B2 ) )
& ( ~ ( member5128974058612258834omplex @ X @ A2 )
=> ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_180_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_181_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_182_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_183_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_184_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_185_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B )
= ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_186_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_187_diff__eq__diff__eq,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_188_ex__in__conv,axiom,
! [A2: set_complex_complex] :
( ( ? [X3: complex > complex] : ( member5128974058612258834omplex @ X3 @ A2 ) )
= ( A2 != bot_bo8693375350852365381omplex ) ) ).
% ex_in_conv
thf(fact_189_ex__in__conv,axiom,
! [A2: set_complex] :
( ( ? [X3: complex] : ( member_complex @ X3 @ A2 ) )
= ( A2 != bot_bot_set_complex ) ) ).
% ex_in_conv
thf(fact_190_equals0I,axiom,
! [A2: set_complex_complex] :
( ! [Y2: complex > complex] :
~ ( member5128974058612258834omplex @ Y2 @ A2 )
=> ( A2 = bot_bo8693375350852365381omplex ) ) ).
% equals0I
thf(fact_191_equals0I,axiom,
! [A2: set_complex] :
( ! [Y2: complex] :
~ ( member_complex @ Y2 @ A2 )
=> ( A2 = bot_bot_set_complex ) ) ).
% equals0I
thf(fact_192_equals0D,axiom,
! [A2: set_complex_complex,A: complex > complex] :
( ( A2 = bot_bo8693375350852365381omplex )
=> ~ ( member5128974058612258834omplex @ A @ A2 ) ) ).
% equals0D
thf(fact_193_equals0D,axiom,
! [A2: set_complex,A: complex] :
( ( A2 = bot_bot_set_complex )
=> ~ ( member_complex @ A @ A2 ) ) ).
% equals0D
thf(fact_194_emptyE,axiom,
! [A: complex > complex] :
~ ( member5128974058612258834omplex @ A @ bot_bo8693375350852365381omplex ) ).
% emptyE
thf(fact_195_emptyE,axiom,
! [A: complex] :
~ ( member_complex @ A @ bot_bot_set_complex ) ).
% emptyE
thf(fact_196_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_197_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_198_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_199_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% equation_minus_iff
thf(fact_200_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_201_verit__negate__coefficient_I3_J,axiom,
! [A: complex,B: complex] :
( ( A = B )
=> ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_202_mk__disjoint__insert,axiom,
! [A: complex,A2: set_complex] :
( ( member_complex @ A @ A2 )
=> ? [B5: set_complex] :
( ( A2
= ( insert_complex @ A @ B5 ) )
& ~ ( member_complex @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_203_mk__disjoint__insert,axiom,
! [A: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ A @ A2 )
=> ? [B5: set_complex_complex] :
( ( A2
= ( insert2420511349202589739omplex @ A @ B5 ) )
& ~ ( member5128974058612258834omplex @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_204_insert__commute,axiom,
! [X: complex,Y3: complex,A2: set_complex] :
( ( insert_complex @ X @ ( insert_complex @ Y3 @ A2 ) )
= ( insert_complex @ Y3 @ ( insert_complex @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_205_insert__commute,axiom,
! [X: complex > complex,Y3: complex > complex,A2: set_complex_complex] :
( ( insert2420511349202589739omplex @ X @ ( insert2420511349202589739omplex @ Y3 @ A2 ) )
= ( insert2420511349202589739omplex @ Y3 @ ( insert2420511349202589739omplex @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_206_insert__eq__iff,axiom,
! [A: complex,A2: set_complex,B: complex,B2: set_complex] :
( ~ ( member_complex @ A @ A2 )
=> ( ~ ( member_complex @ B @ B2 )
=> ( ( ( insert_complex @ A @ A2 )
= ( insert_complex @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_complex] :
( ( A2
= ( insert_complex @ B @ C3 ) )
& ~ ( member_complex @ B @ C3 )
& ( B2
= ( insert_complex @ A @ C3 ) )
& ~ ( member_complex @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_207_insert__eq__iff,axiom,
! [A: complex > complex,A2: set_complex_complex,B: complex > complex,B2: set_complex_complex] :
( ~ ( member5128974058612258834omplex @ A @ A2 )
=> ( ~ ( member5128974058612258834omplex @ B @ B2 )
=> ( ( ( insert2420511349202589739omplex @ A @ A2 )
= ( insert2420511349202589739omplex @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_complex_complex] :
( ( A2
= ( insert2420511349202589739omplex @ B @ C3 ) )
& ~ ( member5128974058612258834omplex @ B @ C3 )
& ( B2
= ( insert2420511349202589739omplex @ A @ C3 ) )
& ~ ( member5128974058612258834omplex @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_208_insert__absorb,axiom,
! [A: complex,A2: set_complex] :
( ( member_complex @ A @ A2 )
=> ( ( insert_complex @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_209_insert__absorb,axiom,
! [A: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ A @ A2 )
=> ( ( insert2420511349202589739omplex @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_210_insert__ident,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ~ ( member_complex @ X @ A2 )
=> ( ~ ( member_complex @ X @ B2 )
=> ( ( ( insert_complex @ X @ A2 )
= ( insert_complex @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_211_insert__ident,axiom,
! [X: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ~ ( member5128974058612258834omplex @ X @ A2 )
=> ( ~ ( member5128974058612258834omplex @ X @ B2 )
=> ( ( ( insert2420511349202589739omplex @ X @ A2 )
= ( insert2420511349202589739omplex @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_212_Set_Oset__insert,axiom,
! [X: complex,A2: set_complex] :
( ( member_complex @ X @ A2 )
=> ~ ! [B5: set_complex] :
( ( A2
= ( insert_complex @ X @ B5 ) )
=> ( member_complex @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_213_Set_Oset__insert,axiom,
! [X: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ X @ A2 )
=> ~ ! [B5: set_complex_complex] :
( ( A2
= ( insert2420511349202589739omplex @ X @ B5 ) )
=> ( member5128974058612258834omplex @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_214_insertI2,axiom,
! [A: complex,B2: set_complex,B: complex] :
( ( member_complex @ A @ B2 )
=> ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).
% insertI2
thf(fact_215_insertI2,axiom,
! [A: complex > complex,B2: set_complex_complex,B: complex > complex] :
( ( member5128974058612258834omplex @ A @ B2 )
=> ( member5128974058612258834omplex @ A @ ( insert2420511349202589739omplex @ B @ B2 ) ) ) ).
% insertI2
thf(fact_216_insertI1,axiom,
! [A: complex,B2: set_complex] : ( member_complex @ A @ ( insert_complex @ A @ B2 ) ) ).
% insertI1
thf(fact_217_insertI1,axiom,
! [A: complex > complex,B2: set_complex_complex] : ( member5128974058612258834omplex @ A @ ( insert2420511349202589739omplex @ A @ B2 ) ) ).
% insertI1
thf(fact_218_insertE,axiom,
! [A: complex,B: complex,A2: set_complex] :
( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
=> ( ( A != B )
=> ( member_complex @ A @ A2 ) ) ) ).
% insertE
thf(fact_219_insertE,axiom,
! [A: complex > complex,B: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ A @ ( insert2420511349202589739omplex @ B @ A2 ) )
=> ( ( A != B )
=> ( member5128974058612258834omplex @ A @ A2 ) ) ) ).
% insertE
thf(fact_220_DiffD2,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
=> ~ ( member_complex @ C @ B2 ) ) ).
% DiffD2
thf(fact_221_DiffD2,axiom,
! [C: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ ( minus_3522879524658371850omplex @ A2 @ B2 ) )
=> ~ ( member5128974058612258834omplex @ C @ B2 ) ) ).
% DiffD2
thf(fact_222_DiffD1,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
=> ( member_complex @ C @ A2 ) ) ).
% DiffD1
thf(fact_223_DiffD1,axiom,
! [C: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ ( minus_3522879524658371850omplex @ A2 @ B2 ) )
=> ( member5128974058612258834omplex @ C @ A2 ) ) ).
% DiffD1
thf(fact_224_DiffE,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
=> ~ ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_225_DiffE,axiom,
! [C: complex > complex,A2: set_complex_complex,B2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ ( minus_3522879524658371850omplex @ A2 @ B2 ) )
=> ~ ( ( member5128974058612258834omplex @ C @ A2 )
=> ( member5128974058612258834omplex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_226_double__complement,axiom,
! [A2: set_complex] :
( ( uminus8566677241136511917omplex @ ( uminus8566677241136511917omplex @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_227_ComplD,axiom,
! [C: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ C @ ( uminus4994531801924300922omplex @ A2 ) )
=> ~ ( member5128974058612258834omplex @ C @ A2 ) ) ).
% ComplD
thf(fact_228_ComplD,axiom,
! [C: complex,A2: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
=> ~ ( member_complex @ C @ A2 ) ) ).
% ComplD
thf(fact_229_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_230_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: complex,Z: complex] : ( Y = Z ) )
= ( ^ [A4: complex,B4: complex] :
( ( minus_minus_complex @ A4 @ B4 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_231_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_232_minus__diff__commute,axiom,
! [B: complex,A: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_233_singleton__inject,axiom,
! [A: complex > complex,B: complex > complex] :
( ( ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex )
= ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_234_singleton__inject,axiom,
! [A: complex,B: complex] :
( ( ( insert_complex @ A @ bot_bot_set_complex )
= ( insert_complex @ B @ bot_bot_set_complex ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_235_insert__not__empty,axiom,
! [A: complex > complex,A2: set_complex_complex] :
( ( insert2420511349202589739omplex @ A @ A2 )
!= bot_bo8693375350852365381omplex ) ).
% insert_not_empty
thf(fact_236_insert__not__empty,axiom,
! [A: complex,A2: set_complex] :
( ( insert_complex @ A @ A2 )
!= bot_bot_set_complex ) ).
% insert_not_empty
thf(fact_237_doubleton__eq__iff,axiom,
! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex] :
( ( ( insert2420511349202589739omplex @ A @ ( insert2420511349202589739omplex @ B @ bot_bo8693375350852365381omplex ) )
= ( insert2420511349202589739omplex @ C @ ( insert2420511349202589739omplex @ D @ bot_bo8693375350852365381omplex ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_238_doubleton__eq__iff,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( insert_complex @ A @ ( insert_complex @ B @ bot_bot_set_complex ) )
= ( insert_complex @ C @ ( insert_complex @ D @ bot_bot_set_complex ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_239_singleton__iff,axiom,
! [B: complex > complex,A: complex > complex] :
( ( member5128974058612258834omplex @ B @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_240_singleton__iff,axiom,
! [B: complex,A: complex] :
( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_241_singletonD,axiom,
! [B: complex > complex,A: complex > complex] :
( ( member5128974058612258834omplex @ B @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_242_singletonD,axiom,
! [B: complex,A: complex] :
( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_243_insert__Diff__if,axiom,
! [X: complex,B2: set_complex,A2: set_complex] :
( ( ( member_complex @ X @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
= ( minus_811609699411566653omplex @ A2 @ B2 ) ) )
& ( ~ ( member_complex @ X @ B2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
= ( insert_complex @ X @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_244_insert__Diff__if,axiom,
! [X: complex > complex,B2: set_complex_complex,A2: set_complex_complex] :
( ( ( member5128974058612258834omplex @ X @ B2 )
=> ( ( minus_3522879524658371850omplex @ ( insert2420511349202589739omplex @ X @ A2 ) @ B2 )
= ( minus_3522879524658371850omplex @ A2 @ B2 ) ) )
& ( ~ ( member5128974058612258834omplex @ X @ B2 )
=> ( ( minus_3522879524658371850omplex @ ( insert2420511349202589739omplex @ X @ A2 ) @ B2 )
= ( insert2420511349202589739omplex @ X @ ( minus_3522879524658371850omplex @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_245_Diff__insert__absorb,axiom,
! [X: complex,A2: set_complex] :
( ~ ( member_complex @ X @ A2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_246_Diff__insert__absorb,axiom,
! [X: complex > complex,A2: set_complex_complex] :
( ~ ( member5128974058612258834omplex @ X @ A2 )
=> ( ( minus_3522879524658371850omplex @ ( insert2420511349202589739omplex @ X @ A2 ) @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_247_Diff__insert2,axiom,
! [A2: set_complex,A: complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) )
= ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_248_Diff__insert2,axiom,
! [A2: set_complex_complex,A: complex > complex,B2: set_complex_complex] :
( ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ A @ B2 ) )
= ( minus_3522879524658371850omplex @ ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_249_insert__Diff,axiom,
! [A: complex,A2: set_complex] :
( ( member_complex @ A @ A2 )
=> ( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_250_insert__Diff,axiom,
! [A: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ A @ A2 )
=> ( ( insert2420511349202589739omplex @ A @ ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_251_Diff__insert,axiom,
! [A2: set_complex,A: complex,B2: set_complex] :
( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) )
= ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ).
% Diff_insert
thf(fact_252_Diff__insert,axiom,
! [A2: set_complex_complex,A: complex > complex,B2: set_complex_complex] :
( ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ A @ B2 ) )
= ( minus_3522879524658371850omplex @ ( minus_3522879524658371850omplex @ A2 @ B2 ) @ ( insert2420511349202589739omplex @ A @ bot_bo8693375350852365381omplex ) ) ) ).
% Diff_insert
thf(fact_253_Compl__insert,axiom,
! [X: complex > complex,A2: set_complex_complex] :
( ( uminus4994531801924300922omplex @ ( insert2420511349202589739omplex @ X @ A2 ) )
= ( minus_3522879524658371850omplex @ ( uminus4994531801924300922omplex @ A2 ) @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) ) ) ).
% Compl_insert
thf(fact_254_Compl__insert,axiom,
! [X: complex,A2: set_complex] :
( ( uminus8566677241136511917omplex @ ( insert_complex @ X @ A2 ) )
= ( minus_811609699411566653omplex @ ( uminus8566677241136511917omplex @ A2 ) @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ).
% Compl_insert
thf(fact_255_open__empty,axiom,
topolo4110288021797289639omplex @ bot_bot_set_complex ).
% open_empty
thf(fact_256_compl__le__compl__iff,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ X ) @ ( uminus8566677241136511917omplex @ Y3 ) )
= ( ord_le211207098394363844omplex @ Y3 @ X ) ) ).
% compl_le_compl_iff
thf(fact_257_compl__le__compl__iff,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ ( uminus4994531801924300922omplex @ X ) @ ( uminus4994531801924300922omplex @ Y3 ) )
= ( ord_le6271439605799870481omplex @ Y3 @ X ) ) ).
% compl_le_compl_iff
thf(fact_258_open__delete,axiom,
! [S: set_complex,X: complex] :
( ( topolo4110288021797289639omplex @ S )
=> ( topolo4110288021797289639omplex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ).
% open_delete
thf(fact_259_not__open__singleton,axiom,
! [X: complex] :
~ ( topolo4110288021797289639omplex @ ( insert_complex @ X @ bot_bot_set_complex ) ) ).
% not_open_singleton
thf(fact_260_set__zero,axiom,
( zero_zero_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_261_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_262_set__zero,axiom,
( zero_z6614145512433583213omplex
= ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ).
% set_zero
thf(fact_263_diff__shunt__var,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ( minus_811609699411566653omplex @ X @ Y3 )
= bot_bot_set_complex )
= ( ord_le211207098394363844omplex @ X @ Y3 ) ) ).
% diff_shunt_var
thf(fact_264_diff__shunt__var,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ( minus_3522879524658371850omplex @ X @ Y3 )
= bot_bo8693375350852365381omplex )
= ( ord_le6271439605799870481omplex @ X @ Y3 ) ) ).
% diff_shunt_var
thf(fact_265_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A4 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_266_ball__insert,axiom,
! [A: complex,B2: set_complex,P: complex > $o] :
( ( ! [X3: complex] :
( ( member_complex @ X3 @ ( insert_complex @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
& ! [X3: complex] :
( ( member_complex @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_267_ball__insert,axiom,
! [A: complex > complex,B2: set_complex_complex,P: ( complex > complex ) > $o] :
( ( ! [X3: complex > complex] :
( ( member5128974058612258834omplex @ X3 @ ( insert2420511349202589739omplex @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
& ! [X3: complex > complex] :
( ( member5128974058612258834omplex @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_268_the__elem__eq,axiom,
! [X: complex > complex] :
( ( the_el7055940373909865542omplex @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) )
= X ) ).
% the_elem_eq
thf(fact_269_the__elem__eq,axiom,
! [X: complex] :
( ( the_elem_complex @ ( insert_complex @ X @ bot_bot_set_complex ) )
= X ) ).
% the_elem_eq
thf(fact_270_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ( uminus8566677241136511917omplex @ X )
= ( uminus8566677241136511917omplex @ Y3 ) )
= ( X = Y3 ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_271_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_complex] :
( ( uminus8566677241136511917omplex @ ( uminus8566677241136511917omplex @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_272_bot__set__def,axiom,
( bot_bot_set_complex
= ( collect_complex @ bot_bot_complex_o ) ) ).
% bot_set_def
thf(fact_273_separation__t1,axiom,
! [X: complex,Y3: complex] :
( ( X != Y3 )
= ( ? [U: set_complex] :
( ( topolo4110288021797289639omplex @ U )
& ( member_complex @ X @ U )
& ~ ( member_complex @ Y3 @ U ) ) ) ) ).
% separation_t1
thf(fact_274_separation__t0,axiom,
! [X: complex,Y3: complex] :
( ( X != Y3 )
= ( ? [U: set_complex] :
( ( topolo4110288021797289639omplex @ U )
& ( ( member_complex @ X @ U )
!= ( member_complex @ Y3 @ U ) ) ) ) ) ).
% separation_t0
thf(fact_275_t1__space,axiom,
! [X: complex,Y3: complex] :
( ( X != Y3 )
=> ? [U2: set_complex] :
( ( topolo4110288021797289639omplex @ U2 )
& ( member_complex @ X @ U2 )
& ~ ( member_complex @ Y3 @ U2 ) ) ) ).
% t1_space
thf(fact_276_t0__space,axiom,
! [X: complex,Y3: complex] :
( ( X != Y3 )
=> ? [U2: set_complex] :
( ( topolo4110288021797289639omplex @ U2 )
& ( ( member_complex @ X @ U2 )
!= ( member_complex @ Y3 @ U2 ) ) ) ) ).
% t0_space
thf(fact_277_compl__mono,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y3 )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ Y3 ) @ ( uminus8566677241136511917omplex @ X ) ) ) ).
% compl_mono
thf(fact_278_compl__mono,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X @ Y3 )
=> ( ord_le6271439605799870481omplex @ ( uminus4994531801924300922omplex @ Y3 ) @ ( uminus4994531801924300922omplex @ X ) ) ) ).
% compl_mono
thf(fact_279_compl__le__swap1,axiom,
! [Y3: set_complex,X: set_complex] :
( ( ord_le211207098394363844omplex @ Y3 @ ( uminus8566677241136511917omplex @ X ) )
=> ( ord_le211207098394363844omplex @ X @ ( uminus8566677241136511917omplex @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_280_compl__le__swap1,axiom,
! [Y3: set_complex_complex,X: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ Y3 @ ( uminus4994531801924300922omplex @ X ) )
=> ( ord_le6271439605799870481omplex @ X @ ( uminus4994531801924300922omplex @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_281_compl__le__swap2,axiom,
! [Y3: set_complex,X: set_complex] :
( ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ Y3 ) @ X )
=> ( ord_le211207098394363844omplex @ ( uminus8566677241136511917omplex @ X ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_282_compl__le__swap2,axiom,
! [Y3: set_complex_complex,X: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ ( uminus4994531801924300922omplex @ Y3 ) @ X )
=> ( ord_le6271439605799870481omplex @ ( uminus4994531801924300922omplex @ X ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_283_first__countable__basis,axiom,
! [X: complex] :
? [A5: nat > set_complex] :
( ! [I: nat] :
( ( member_complex @ X @ ( A5 @ I ) )
& ( topolo4110288021797289639omplex @ ( A5 @ I ) ) )
& ! [S2: set_complex] :
( ( ( topolo4110288021797289639omplex @ S2 )
& ( member_complex @ X @ S2 ) )
=> ? [I2: nat] : ( ord_le211207098394363844omplex @ ( A5 @ I2 ) @ S2 ) ) ) ).
% first_countable_basis
thf(fact_284_open__subopen,axiom,
( topolo7782564052695852148omplex
= ( ^ [S3: set_complex_complex] :
! [X3: complex > complex] :
( ( member5128974058612258834omplex @ X3 @ S3 )
=> ? [T2: set_complex_complex] :
( ( topolo7782564052695852148omplex @ T2 )
& ( member5128974058612258834omplex @ X3 @ T2 )
& ( ord_le6271439605799870481omplex @ T2 @ S3 ) ) ) ) ) ).
% open_subopen
thf(fact_285_open__subopen,axiom,
( topolo4110288021797289639omplex
= ( ^ [S3: set_complex] :
! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ? [T2: set_complex] :
( ( topolo4110288021797289639omplex @ T2 )
& ( member_complex @ X3 @ T2 )
& ( ord_le211207098394363844omplex @ T2 @ S3 ) ) ) ) ) ).
% open_subopen
thf(fact_286_topological__space__class_OopenI,axiom,
! [S4: set_complex_complex] :
( ! [X2: complex > complex] :
( ( member5128974058612258834omplex @ X2 @ S4 )
=> ? [T3: set_complex_complex] :
( ( topolo7782564052695852148omplex @ T3 )
& ( member5128974058612258834omplex @ X2 @ T3 )
& ( ord_le6271439605799870481omplex @ T3 @ S4 ) ) )
=> ( topolo7782564052695852148omplex @ S4 ) ) ).
% topological_space_class.openI
thf(fact_287_topological__space__class_OopenI,axiom,
! [S4: set_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ S4 )
=> ? [T3: set_complex] :
( ( topolo4110288021797289639omplex @ T3 )
& ( member_complex @ X2 @ T3 )
& ( ord_le211207098394363844omplex @ T3 @ S4 ) ) )
=> ( topolo4110288021797289639omplex @ S4 ) ) ).
% topological_space_class.openI
thf(fact_288_is__singleton__the__elem,axiom,
( is_sin8098092515724343303omplex
= ( ^ [A3: set_complex_complex] :
( A3
= ( insert2420511349202589739omplex @ ( the_el7055940373909865542omplex @ A3 ) @ bot_bo8693375350852365381omplex ) ) ) ) ).
% is_singleton_the_elem
thf(fact_289_is__singleton__the__elem,axiom,
( is_singleton_complex
= ( ^ [A3: set_complex] :
( A3
= ( insert_complex @ ( the_elem_complex @ A3 ) @ bot_bot_set_complex ) ) ) ) ).
% is_singleton_the_elem
thf(fact_290_dual__order_Orefl,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ A ) ).
% dual_order.refl
thf(fact_291_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_292_dual__order_Orefl,axiom,
! [A: set_complex_complex] : ( ord_le6271439605799870481omplex @ A @ A ) ).
% dual_order.refl
thf(fact_293_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_294_order__refl,axiom,
! [X: set_complex] : ( ord_le211207098394363844omplex @ X @ X ) ).
% order_refl
thf(fact_295_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_296_order__refl,axiom,
! [X: set_complex_complex] : ( ord_le6271439605799870481omplex @ X @ X ) ).
% order_refl
thf(fact_297_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_298_is__singletonI,axiom,
! [X: complex > complex] : ( is_sin8098092515724343303omplex @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) ) ).
% is_singletonI
thf(fact_299_is__singletonI,axiom,
! [X: complex] : ( is_singleton_complex @ ( insert_complex @ X @ bot_bot_set_complex ) ) ).
% is_singletonI
thf(fact_300_holomorphic__on__subset,axiom,
! [F: complex > complex,S: set_complex,T4: set_complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( ( ord_le211207098394363844omplex @ T4 @ S )
=> ( comple7700996537433184370hic_on @ F @ T4 ) ) ) ).
% holomorphic_on_subset
thf(fact_301_holomorphic__on__empty,axiom,
! [F: complex > complex] : ( comple7700996537433184370hic_on @ F @ bot_bot_set_complex ) ).
% holomorphic_on_empty
thf(fact_302_Ints__minus,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_303_Ints__minus,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% Ints_minus
thf(fact_304_minus__in__Ints__iff,axiom,
! [X: real] :
( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_305_minus__in__Ints__iff,axiom,
! [X: complex] :
( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
= ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% minus_in_Ints_iff
thf(fact_306_is__singletonI_H,axiom,
! [A2: set_complex_complex] :
( ( A2 != bot_bo8693375350852365381omplex )
=> ( ! [X2: complex > complex,Y2: complex > complex] :
( ( member5128974058612258834omplex @ X2 @ A2 )
=> ( ( member5128974058612258834omplex @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_sin8098092515724343303omplex @ A2 ) ) ) ).
% is_singletonI'
thf(fact_307_is__singletonI_H,axiom,
! [A2: set_complex] :
( ( A2 != bot_bot_set_complex )
=> ( ! [X2: complex,Y2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_complex @ A2 ) ) ) ).
% is_singletonI'
thf(fact_308_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_309_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_310_le__cases3,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_311_le__cases3,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_312_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_complex,Z: set_complex] : ( Y = Z ) )
= ( ^ [X3: set_complex,Y4: set_complex] :
( ( ord_le211207098394363844omplex @ X3 @ Y4 )
& ( ord_le211207098394363844omplex @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_313_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_314_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_complex_complex,Z: set_complex_complex] : ( Y = Z ) )
= ( ^ [X3: set_complex_complex,Y4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X3 @ Y4 )
& ( ord_le6271439605799870481omplex @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_315_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_316_ord__eq__le__trans,axiom,
! [A: set_complex,B: set_complex,C: set_complex] :
( ( A = B )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ord_le211207098394363844omplex @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_317_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_318_ord__eq__le__trans,axiom,
! [A: set_complex_complex,B: set_complex_complex,C: set_complex_complex] :
( ( A = B )
=> ( ( ord_le6271439605799870481omplex @ B @ C )
=> ( ord_le6271439605799870481omplex @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_319_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_320_ord__le__eq__trans,axiom,
! [A: set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( B = C )
=> ( ord_le211207098394363844omplex @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_321_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_322_ord__le__eq__trans,axiom,
! [A: set_complex_complex,B: set_complex_complex,C: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ B )
=> ( ( B = C )
=> ( ord_le6271439605799870481omplex @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_323_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_324_order__antisym,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y3 )
=> ( ( ord_le211207098394363844omplex @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_325_order__antisym,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_326_order__antisym,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X @ Y3 )
=> ( ( ord_le6271439605799870481omplex @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_327_order__antisym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_328_order_Otrans,axiom,
! [A: set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ord_le211207098394363844omplex @ A @ C ) ) ) ).
% order.trans
thf(fact_329_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_330_order_Otrans,axiom,
! [A: set_complex_complex,B: set_complex_complex,C: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ B )
=> ( ( ord_le6271439605799870481omplex @ B @ C )
=> ( ord_le6271439605799870481omplex @ A @ C ) ) ) ).
% order.trans
thf(fact_331_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_332_order__trans,axiom,
! [X: set_complex,Y3: set_complex,Z2: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y3 )
=> ( ( ord_le211207098394363844omplex @ Y3 @ Z2 )
=> ( ord_le211207098394363844omplex @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_333_order__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_334_order__trans,axiom,
! [X: set_complex_complex,Y3: set_complex_complex,Z2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X @ Y3 )
=> ( ( ord_le6271439605799870481omplex @ Y3 @ Z2 )
=> ( ord_le6271439605799870481omplex @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_335_order__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_336_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A6: real,B6: real] :
( ( ord_less_eq_real @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: real,B6: real] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_337_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_338_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_complex,Z: set_complex] : ( Y = Z ) )
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ B4 @ A4 )
& ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_339_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_340_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_complex_complex,Z: set_complex_complex] : ( Y = Z ) )
= ( ^ [A4: set_complex_complex,B4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ B4 @ A4 )
& ( ord_le6271439605799870481omplex @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_341_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_342_dual__order_Oantisym,axiom,
! [B: set_complex,A: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_343_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_344_dual__order_Oantisym,axiom,
! [B: set_complex_complex,A: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ B @ A )
=> ( ( ord_le6271439605799870481omplex @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_345_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_346_dual__order_Otrans,axiom,
! [B: set_complex,A: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A )
=> ( ( ord_le211207098394363844omplex @ C @ B )
=> ( ord_le211207098394363844omplex @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_347_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_348_dual__order_Otrans,axiom,
! [B: set_complex_complex,A: set_complex_complex,C: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ B @ A )
=> ( ( ord_le6271439605799870481omplex @ C @ B )
=> ( ord_le6271439605799870481omplex @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_349_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_350_antisym,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_le211207098394363844omplex @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_351_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_352_antisym,axiom,
! [A: set_complex_complex,B: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ B )
=> ( ( ord_le6271439605799870481omplex @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_353_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_354_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_complex,Z: set_complex] : ( Y = Z ) )
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B4 )
& ( ord_le211207098394363844omplex @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_355_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_356_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_complex_complex,Z: set_complex_complex] : ( Y = Z ) )
= ( ^ [A4: set_complex_complex,B4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A4 @ B4 )
& ( ord_le6271439605799870481omplex @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_357_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_358_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_359_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_360_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_361_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_362_order__subst1,axiom,
! [A: set_complex,F: real > set_complex,B: real,C: real] :
( ( ord_le211207098394363844omplex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_363_order__subst1,axiom,
! [A: set_complex,F: nat > set_complex,B: nat,C: nat] :
( ( ord_le211207098394363844omplex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_364_order__subst1,axiom,
! [A: real,F: set_complex > real,B: set_complex,C: set_complex] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_365_order__subst1,axiom,
! [A: nat,F: set_complex > nat,B: set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_366_order__subst1,axiom,
! [A: set_complex,F: set_complex > set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_367_order__subst1,axiom,
! [A: real,F: set_complex_complex > real,B: set_complex_complex,C: set_complex_complex] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le6271439605799870481omplex @ B @ C )
=> ( ! [X2: set_complex_complex,Y2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_368_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_369_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_370_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_371_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_372_order__subst2,axiom,
! [A: set_complex,B: set_complex,F: set_complex > real,C: real] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_373_order__subst2,axiom,
! [A: set_complex,B: set_complex,F: set_complex > nat,C: nat] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_374_order__subst2,axiom,
! [A: real,B: real,F: real > set_complex,C: set_complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le211207098394363844omplex @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_375_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le211207098394363844omplex @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_376_order__subst2,axiom,
! [A: set_complex,B: set_complex,F: set_complex > set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_le211207098394363844omplex @ ( F @ B ) @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_377_order__subst2,axiom,
! [A: real,B: real,F: real > set_complex_complex,C: set_complex_complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le6271439605799870481omplex @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le6271439605799870481omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6271439605799870481omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_378_order__eq__refl,axiom,
! [X: set_complex,Y3: set_complex] :
( ( X = Y3 )
=> ( ord_le211207098394363844omplex @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_379_order__eq__refl,axiom,
! [X: real,Y3: real] :
( ( X = Y3 )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_380_order__eq__refl,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( X = Y3 )
=> ( ord_le6271439605799870481omplex @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_381_order__eq__refl,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_382_linorder__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
| ( ord_less_eq_real @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_383_linorder__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_384_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_385_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_386_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_387_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_388_ord__eq__le__subst,axiom,
! [A: real,F: set_complex > real,B: set_complex,C: set_complex] :
( ( A
= ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_389_ord__eq__le__subst,axiom,
! [A: nat,F: set_complex > nat,B: set_complex,C: set_complex] :
( ( A
= ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_390_ord__eq__le__subst,axiom,
! [A: set_complex,F: real > set_complex,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_391_ord__eq__le__subst,axiom,
! [A: set_complex,F: nat > set_complex,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_392_ord__eq__le__subst,axiom,
! [A: set_complex,F: set_complex > set_complex,B: set_complex,C: set_complex] :
( ( A
= ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_393_ord__eq__le__subst,axiom,
! [A: set_complex_complex,F: real > set_complex_complex,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le6271439605799870481omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6271439605799870481omplex @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_394_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_395_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_396_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_397_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_398_ord__le__eq__subst,axiom,
! [A: set_complex,B: set_complex,F: set_complex > real,C: real] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_399_ord__le__eq__subst,axiom,
! [A: set_complex,B: set_complex,F: set_complex > nat,C: nat] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_400_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_complex,C: set_complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_401_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_402_ord__le__eq__subst,axiom,
! [A: set_complex,B: set_complex,F: set_complex > set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_403_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_complex_complex,C: set_complex_complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le6271439605799870481omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6271439605799870481omplex @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_404_linorder__le__cases,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_405_linorder__le__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_406_order__antisym__conv,axiom,
! [Y3: set_complex,X: set_complex] :
( ( ord_le211207098394363844omplex @ Y3 @ X )
=> ( ( ord_le211207098394363844omplex @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_407_order__antisym__conv,axiom,
! [Y3: real,X: real] :
( ( ord_less_eq_real @ Y3 @ X )
=> ( ( ord_less_eq_real @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_408_order__antisym__conv,axiom,
! [Y3: set_complex_complex,X: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ Y3 @ X )
=> ( ( ord_le6271439605799870481omplex @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_409_order__antisym__conv,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_410_holomorphic__transform,axiom,
! [F: complex > complex,S: set_complex,G: complex > complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( comple7700996537433184370hic_on @ G @ S ) ) ) ).
% holomorphic_transform
thf(fact_411_holomorphic__cong,axiom,
! [S: set_complex,T4: set_complex,F: complex > complex,G: complex > complex] :
( ( S = T4 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( comple7700996537433184370hic_on @ F @ S )
= ( comple7700996537433184370hic_on @ G @ T4 ) ) ) ) ).
% holomorphic_cong
thf(fact_412_is__singleton__def,axiom,
( is_sin8098092515724343303omplex
= ( ^ [A3: set_complex_complex] :
? [X3: complex > complex] :
( A3
= ( insert2420511349202589739omplex @ X3 @ bot_bo8693375350852365381omplex ) ) ) ) ).
% is_singleton_def
thf(fact_413_is__singleton__def,axiom,
( is_singleton_complex
= ( ^ [A3: set_complex] :
? [X3: complex] :
( A3
= ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ).
% is_singleton_def
thf(fact_414_is__singletonE,axiom,
! [A2: set_complex_complex] :
( ( is_sin8098092515724343303omplex @ A2 )
=> ~ ! [X2: complex > complex] :
( A2
!= ( insert2420511349202589739omplex @ X2 @ bot_bo8693375350852365381omplex ) ) ) ).
% is_singletonE
thf(fact_415_is__singletonE,axiom,
! [A2: set_complex] :
( ( is_singleton_complex @ A2 )
=> ~ ! [X2: complex] :
( A2
!= ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ).
% is_singletonE
thf(fact_416_bot_Oextremum__uniqueI,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ bot_bot_set_complex )
=> ( A = bot_bot_set_complex ) ) ).
% bot.extremum_uniqueI
thf(fact_417_bot_Oextremum__uniqueI,axiom,
! [A: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ bot_bo8693375350852365381omplex )
=> ( A = bot_bo8693375350852365381omplex ) ) ).
% bot.extremum_uniqueI
thf(fact_418_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_419_bot_Oextremum__unique,axiom,
! [A: set_complex] :
( ( ord_le211207098394363844omplex @ A @ bot_bot_set_complex )
= ( A = bot_bot_set_complex ) ) ).
% bot.extremum_unique
thf(fact_420_bot_Oextremum__unique,axiom,
! [A: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ bot_bo8693375350852365381omplex )
= ( A = bot_bo8693375350852365381omplex ) ) ).
% bot.extremum_unique
thf(fact_421_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_422_bot_Oextremum,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ bot_bot_set_complex @ A ) ).
% bot.extremum
thf(fact_423_bot_Oextremum,axiom,
! [A: set_complex_complex] : ( ord_le6271439605799870481omplex @ bot_bo8693375350852365381omplex @ A ) ).
% bot.extremum
thf(fact_424_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_425_Ints__0,axiom,
member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% Ints_0
thf(fact_426_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_427_Ints__diff,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_428_Ints__diff,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_diff
thf(fact_429_Collect__empty__eq__bot,axiom,
! [P: complex > $o] :
( ( ( collect_complex @ P )
= bot_bot_set_complex )
= ( P = bot_bot_complex_o ) ) ).
% Collect_empty_eq_bot
thf(fact_430_bot__empty__eq,axiom,
( bot_bo466812550819818624plex_o
= ( ^ [X3: complex > complex] : ( member5128974058612258834omplex @ X3 @ bot_bo8693375350852365381omplex ) ) ) ).
% bot_empty_eq
thf(fact_431_bot__empty__eq,axiom,
( bot_bot_complex_o
= ( ^ [X3: complex] : ( member_complex @ X3 @ bot_bot_set_complex ) ) ) ).
% bot_empty_eq
thf(fact_432_insert__subsetI,axiom,
! [X: complex,A2: set_complex,X4: set_complex] :
( ( member_complex @ X @ A2 )
=> ( ( ord_le211207098394363844omplex @ X4 @ A2 )
=> ( ord_le211207098394363844omplex @ ( insert_complex @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_433_insert__subsetI,axiom,
! [X: complex > complex,A2: set_complex_complex,X4: set_complex_complex] :
( ( member5128974058612258834omplex @ X @ A2 )
=> ( ( ord_le6271439605799870481omplex @ X4 @ A2 )
=> ( ord_le6271439605799870481omplex @ ( insert2420511349202589739omplex @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_434_subset__emptyI,axiom,
! [A2: set_complex] :
( ! [X2: complex] :
~ ( member_complex @ X2 @ A2 )
=> ( ord_le211207098394363844omplex @ A2 @ bot_bot_set_complex ) ) ).
% subset_emptyI
thf(fact_435_subset__emptyI,axiom,
! [A2: set_complex_complex] :
( ! [X2: complex > complex] :
~ ( member5128974058612258834omplex @ X2 @ A2 )
=> ( ord_le6271439605799870481omplex @ A2 @ bot_bo8693375350852365381omplex ) ) ).
% subset_emptyI
thf(fact_436_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_437_minus__diff__minus,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_438_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_439_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_440_remove__def,axiom,
( remove_complex
= ( ^ [X3: complex,A3: set_complex] : ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ).
% remove_def
thf(fact_441_remove__def,axiom,
( remove423721587600764758omplex
= ( ^ [X3: complex > complex,A3: set_complex_complex] : ( minus_3522879524658371850omplex @ A3 @ ( insert2420511349202589739omplex @ X3 @ bot_bo8693375350852365381omplex ) ) ) ) ).
% remove_def
thf(fact_442_Set_Ois__empty__def,axiom,
( is_empty_complex
= ( ^ [A3: set_complex] : ( A3 = bot_bot_set_complex ) ) ) ).
% Set.is_empty_def
thf(fact_443_member__remove,axiom,
! [X: complex,Y3: complex,A2: set_complex] :
( ( member_complex @ X @ ( remove_complex @ Y3 @ A2 ) )
= ( ( member_complex @ X @ A2 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_444_member__remove,axiom,
! [X: complex > complex,Y3: complex > complex,A2: set_complex_complex] :
( ( member5128974058612258834omplex @ X @ ( remove423721587600764758omplex @ Y3 @ A2 ) )
= ( ( member5128974058612258834omplex @ X @ A2 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_445_Gcd__factorial__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( factor8539158941071730396al_nat @ A2 )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_factorial_eq_0_iff
thf(fact_446_psubset__insert__iff,axiom,
! [A2: set_complex,X: complex,B2: set_complex] :
( ( ord_less_set_complex @ A2 @ ( insert_complex @ X @ B2 ) )
= ( ( ( member_complex @ X @ B2 )
=> ( ord_less_set_complex @ A2 @ B2 ) )
& ( ~ ( member_complex @ X @ B2 )
=> ( ( ( member_complex @ X @ A2 )
=> ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
& ( ~ ( member_complex @ X @ A2 )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_447_psubset__insert__iff,axiom,
! [A2: set_complex_complex,X: complex > complex,B2: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A2 @ ( insert2420511349202589739omplex @ X @ B2 ) )
= ( ( ( member5128974058612258834omplex @ X @ B2 )
=> ( ord_le3207539288156484613omplex @ A2 @ B2 ) )
& ( ~ ( member5128974058612258834omplex @ X @ B2 )
=> ( ( ( member5128974058612258834omplex @ X @ A2 )
=> ( ord_le3207539288156484613omplex @ ( minus_3522879524658371850omplex @ A2 @ ( insert2420511349202589739omplex @ X @ bot_bo8693375350852365381omplex ) ) @ B2 ) )
& ( ~ ( member5128974058612258834omplex @ X @ A2 )
=> ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_448_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_449_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_450_isolated__singularity__at__holomorphic,axiom,
! [F: complex > complex,S: set_complex,Z2: complex] :
( ( comple7700996537433184370hic_on @ F @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ Z2 @ bot_bot_set_complex ) ) )
=> ( ( topolo4110288021797289639omplex @ S )
=> ( ( member_complex @ Z2 @ S )
=> ( comple1891072044276206784ity_at @ F @ Z2 ) ) ) ) ).
% isolated_singularity_at_holomorphic
thf(fact_451_analytic__at,axiom,
! [F: complex > complex,Z2: complex] :
( ( comple673786817313641009tic_on @ F @ ( insert_complex @ Z2 @ bot_bot_set_complex ) )
= ( ? [S5: set_complex] :
( ( topolo4110288021797289639omplex @ S5 )
& ( member_complex @ Z2 @ S5 )
& ( comple7700996537433184370hic_on @ F @ S5 ) ) ) ) ).
% analytic_at
thf(fact_452_analytic__at__two,axiom,
! [F: complex > complex,Z2: complex,G: complex > complex] :
( ( ( comple673786817313641009tic_on @ F @ ( insert_complex @ Z2 @ bot_bot_set_complex ) )
& ( comple673786817313641009tic_on @ G @ ( insert_complex @ Z2 @ bot_bot_set_complex ) ) )
= ( ? [S5: set_complex] :
( ( topolo4110288021797289639omplex @ S5 )
& ( member_complex @ Z2 @ S5 )
& ( comple7700996537433184370hic_on @ F @ S5 )
& ( comple7700996537433184370hic_on @ G @ S5 ) ) ) ) ).
% analytic_at_two
thf(fact_453_totally__bounded__empty,axiom,
topolo7663995495656592641omplex @ bot_bot_set_complex ).
% totally_bounded_empty
thf(fact_454_set__add__0,axiom,
! [A2: set_real] :
( ( plus_plus_set_real @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_455_set__add__0,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_456_set__add__0,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_457_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_458_add__right__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_459_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_460_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_461_add__left__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_462_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_463_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_464_add__le__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_465_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_466_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_467_add__le__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_468_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_469_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_470_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_471_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_472_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_473_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_474_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_475_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_476_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_477_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_478_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_479_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_480_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_481_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_482_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_483_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_484_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_485_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_486_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_487_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_488_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_489_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_490_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_491_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_492_add__less__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( ord_less_complex @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_493_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_494_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_495_add__less__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( ord_less_complex @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_496_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_497_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_498_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_499_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_500_add__diff__cancel__right_H,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_501_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_502_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_503_add__diff__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( minus_minus_complex @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_504_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_505_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_506_add__diff__cancel__left_H,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_507_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_508_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_509_add__diff__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( minus_minus_complex @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_510_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_511_diff__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_512_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_513_add__diff__cancel,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_514_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_515_neg__less__iff__less,axiom,
! [B: complex,A: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_complex @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_516_compl__less__compl__iff,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_less_set_complex @ ( uminus8566677241136511917omplex @ X ) @ ( uminus8566677241136511917omplex @ Y3 ) )
= ( ord_less_set_complex @ Y3 @ X ) ) ).
% compl_less_compl_iff
thf(fact_517_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_518_minus__add__distrib,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_add_distrib
thf(fact_519_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_520_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_521_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_522_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_523_psubsetI,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_complex @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_524_psubsetI,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le3207539288156484613omplex @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_525_Ints__add__iff2,axiom,
! [Y3: real,X: real] :
( ( member_real @ Y3 @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y3 ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff2
thf(fact_526_Ints__add__iff2,axiom,
! [Y3: complex,X: complex] :
( ( member_complex @ Y3 @ ring_1_Ints_complex )
=> ( ( member_complex @ ( plus_plus_complex @ X @ Y3 ) @ ring_1_Ints_complex )
= ( member_complex @ X @ ring_1_Ints_complex ) ) ) ).
% Ints_add_iff2
thf(fact_527_Ints__add__iff1,axiom,
! [X: real,Y3: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y3 ) @ ring_1_Ints_real )
= ( member_real @ Y3 @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff1
thf(fact_528_Ints__add__iff1,axiom,
! [X: complex,Y3: complex] :
( ( member_complex @ X @ ring_1_Ints_complex )
=> ( ( member_complex @ ( plus_plus_complex @ X @ Y3 ) @ ring_1_Ints_complex )
= ( member_complex @ Y3 @ ring_1_Ints_complex ) ) ) ).
% Ints_add_iff1
thf(fact_529_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_530_set__plus__intro,axiom,
! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
( ( member_complex @ A @ C2 )
=> ( ( member_complex @ B @ D2 )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_p7052360327008956141omplex @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_531_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_532_set__plus__mono2,axiom,
! [C2: set_complex,D2: set_complex,E: set_complex,F2: set_complex] :
( ( ord_le211207098394363844omplex @ C2 @ D2 )
=> ( ( ord_le211207098394363844omplex @ E @ F2 )
=> ( ord_le211207098394363844omplex @ ( plus_p7052360327008956141omplex @ C2 @ E ) @ ( plus_p7052360327008956141omplex @ D2 @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_533_sumset__empty_I1_J,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ A2 @ bot_bot_set_complex )
= bot_bot_set_complex ) ).
% sumset_empty(1)
thf(fact_534_sumset__empty_I2_J,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ bot_bot_set_complex @ A2 )
= bot_bot_set_complex ) ).
% sumset_empty(2)
thf(fact_535_add__le__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_536_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_537_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_538_add__le__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_539_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_540_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_541_le__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel1
thf(fact_542_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_543_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_544_le__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel2
thf(fact_545_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_546_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_547_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_548_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_549_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_550_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_551_less__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel2
thf(fact_552_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_553_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_554_less__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel1
thf(fact_555_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_556_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_557_add__less__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel2
thf(fact_558_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_559_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_560_add__less__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel1
thf(fact_561_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_562_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_563_diff__gt__0__iff__gt,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( minus_minus_complex @ A @ B ) )
= ( ord_less_complex @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_564_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_565_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_566_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_567_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_568_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_569_neg__less__0__iff__less,axiom,
! [A: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A ) @ zero_zero_complex )
= ( ord_less_complex @ zero_zero_complex @ A ) ) ).
% neg_less_0_iff_less
thf(fact_570_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_571_neg__0__less__iff__less,axiom,
! [A: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% neg_0_less_iff_less
thf(fact_572_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_573_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_574_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_575_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_576_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_577_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_578_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_579_uminus__add__conv__diff,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( minus_minus_complex @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_580_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_581_diff__minus__eq__add,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( plus_plus_complex @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_582_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_583_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_584_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_585_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_586_set__add__0__right,axiom,
! [A2: set_real] :
( ( plus_plus_set_real @ A2 @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
= A2 ) ).
% set_add_0_right
thf(fact_587_set__add__0__right,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= A2 ) ).
% set_add_0_right
thf(fact_588_set__add__0__right,axiom,
! [A2: set_complex] :
( ( plus_p7052360327008956141omplex @ A2 @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) )
= A2 ) ).
% set_add_0_right
thf(fact_589_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_590_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_591_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_592_is__num__normalize_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_593_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_594_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_595_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_596_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_597_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_598_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_599_add__le__less__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ( ord_less_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_600_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_601_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_602_add__less__le__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_complex @ A @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_603_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_604_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_605_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_606_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_607_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_608_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_609_set__plus__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A6: real,B6: real] :
( ( X
= ( plus_plus_real @ A6 @ B6 ) )
=> ( ( member_real @ A6 @ A2 )
=> ~ ( member_real @ B6 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_610_set__plus__elim,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ X @ ( plus_p7052360327008956141omplex @ A2 @ B2 ) )
=> ~ ! [A6: complex,B6: complex] :
( ( X
= ( plus_plus_complex @ A6 @ B6 ) )
=> ( ( member_complex @ A6 @ A2 )
=> ~ ( member_complex @ B6 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_611_set__plus__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A6: nat,B6: nat] :
( ( X
= ( plus_plus_nat @ A6 @ B6 ) )
=> ( ( member_nat @ A6 @ A2 )
=> ~ ( member_nat @ B6 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_612_pos__add__strict,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_613_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_614_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_615_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_616_add__pos__pos,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_617_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_618_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_619_add__neg__neg,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ zero_zero_complex )
=> ( ( ord_less_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_neg_neg
thf(fact_620_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_621_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_622_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_623_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_624_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_625_psubsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_626_psubsetD,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,C: complex > complex] :
( ( ord_le3207539288156484613omplex @ A2 @ B2 )
=> ( ( member5128974058612258834omplex @ C @ A2 )
=> ( member5128974058612258834omplex @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_627_add__less__imp__less__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
=> ( ord_less_complex @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_628_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_629_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_630_add__less__imp__less__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
=> ( ord_less_complex @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_631_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_632_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_633_add__strict__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_634_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_635_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_636_add__strict__left__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_637_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_638_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_639_add__strict__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_complex @ A @ B )
=> ( ( ord_less_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_640_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_641_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_642_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_643_add__right__imp__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_644_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_645_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_646_add__left__imp__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_647_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_648_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_649_add_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( plus_plus_complex @ B @ ( plus_plus_complex @ A @ C ) )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% add.left_commute
thf(fact_650_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_651_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_652_add_Ocommute,axiom,
( plus_plus_complex
= ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_653_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_654_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_655_add_Oright__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_656_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_657_add_Oleft__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_658_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_659_add_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% add.assoc
thf(fact_660_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_661_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_662_group__cancel_Oadd2,axiom,
! [B2: complex,K: complex,B: complex,A: complex] :
( ( B2
= ( plus_plus_complex @ K @ B ) )
=> ( ( plus_plus_complex @ A @ B2 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_663_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_664_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_665_group__cancel_Oadd1,axiom,
! [A2: complex,K: complex,A: complex,B: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( plus_plus_complex @ A2 @ B )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_666_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_667_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I3 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_668_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_complex @ I3 @ K )
= ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_669_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_670_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( K = L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_671_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_672_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_673_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_674_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( I3 = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_675_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_676_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I3 @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_677_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_678_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_679_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_680_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_681_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_682_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_683_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_684_order__less__imp__not__less,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_685_order__less__imp__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_686_order__less__imp__not__eq2,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_687_order__less__imp__not__eq2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_688_order__less__imp__not__eq,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_689_order__less__imp__not__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_690_linorder__less__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_real @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_691_linorder__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_692_order__less__imp__triv,axiom,
! [X: real,Y3: real,P: $o] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_693_order__less__imp__triv,axiom,
! [X: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_694_order__less__not__sym,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_695_order__less__not__sym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_696_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_697_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_698_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_699_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_700_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_701_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_702_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_703_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_704_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_705_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_706_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_707_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_708_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_709_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_710_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_711_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_712_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_713_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_714_order__less__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_715_order__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_716_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_717_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_718_linorder__neq__iff,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
= ( ( ord_less_real @ X @ Y3 )
| ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_719_linorder__neq__iff,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
= ( ( ord_less_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_720_order__less__asym,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_721_order__less__asym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_722_linorder__neqE,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_723_linorder__neqE,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_724_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_725_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_726_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_727_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_728_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_729_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_730_not__less__iff__gr__or__eq,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( ( ord_less_real @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_731_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_732_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_733_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_734_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A6: real,B6: real] :
( ( ord_less_real @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: real] : ( P @ A6 @ A6 )
=> ( ! [A6: real,B6: real] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_735_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_736_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_737_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_738_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_739_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_740_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_741_linorder__cases,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_742_linorder__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_743_antisym__conv3,axiom,
! [Y3: real,X: real] :
( ~ ( ord_less_real @ Y3 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_744_antisym__conv3,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_nat @ Y3 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_745_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_746_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_747_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_748_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_749_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_750_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_751_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_752_less__imp__neq,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_753_less__imp__neq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_754_dense,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y3 ) ) ) ).
% dense
thf(fact_755_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_756_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_757_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_758_diff__less__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ ( minus_minus_complex @ A @ B ) @ C )
= ( ord_less_complex @ A @ ( plus_plus_complex @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_759_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_760_less__diff__eq,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ A @ ( minus_minus_complex @ C @ B ) )
= ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_761_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_762_Ints__odd__less__0,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% Ints_odd_less_0
thf(fact_763_add__neg__nonpos,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_neg_nonpos
thf(fact_764_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_765_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_766_add__nonneg__pos,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_767_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_768_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_769_add__nonpos__neg,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_nonpos_neg
thf(fact_770_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_771_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_772_add__pos__nonneg,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_773_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_774_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_775_add__strict__increasing,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_776_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_777_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_778_add__strict__increasing2,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_779_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_780_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_781_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_782_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_783_set__one,axiom,
( one_one_set_real
= ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).
% set_one
thf(fact_784_set__one,axiom,
( one_one_set_nat
= ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).
% set_one
thf(fact_785_set__one,axiom,
( one_one_set_complex
= ( insert_complex @ one_one_complex @ bot_bot_set_complex ) ) ).
% set_one
thf(fact_786_analytic__imp__holomorphic,axiom,
! [F: complex > complex,S4: set_complex] :
( ( comple673786817313641009tic_on @ F @ S4 )
=> ( comple7700996537433184370hic_on @ F @ S4 ) ) ).
% analytic_imp_holomorphic
thf(fact_787_Ints__odd__nonzero,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
!= zero_zero_complex ) ) ).
% Ints_odd_nonzero
thf(fact_788_Ints__odd__nonzero,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_789_analytic__on__subset,axiom,
! [F: complex > complex,S4: set_complex,T5: set_complex] :
( ( comple673786817313641009tic_on @ F @ S4 )
=> ( ( ord_le211207098394363844omplex @ T5 @ S4 )
=> ( comple673786817313641009tic_on @ F @ T5 ) ) ) ).
% analytic_on_subset
thf(fact_790_order__le__imp__less__or__eq,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y3 )
=> ( ( ord_less_set_complex @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_791_order__le__imp__less__or__eq,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_real @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_792_order__le__imp__less__or__eq,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X @ Y3 )
=> ( ( ord_le3207539288156484613omplex @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_793_order__le__imp__less__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_794_linorder__le__less__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
| ( ord_less_real @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_795_linorder__le__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_796_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_complex,C: set_complex] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le211207098394363844omplex @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_797_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_complex,C: set_complex] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le211207098394363844omplex @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_798_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_799_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_800_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_complex_complex,C: set_complex_complex] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le6271439605799870481omplex @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le3207539288156484613omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3207539288156484613omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_801_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_complex_complex,C: set_complex_complex] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6271439605799870481omplex @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le3207539288156484613omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3207539288156484613omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_802_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_803_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_804_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_805_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_806_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_807_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_808_order__less__le__subst1,axiom,
! [A: real,F: set_complex > real,B: set_complex,C: set_complex] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_809_order__less__le__subst1,axiom,
! [A: nat,F: set_complex > nat,B: set_complex,C: set_complex] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_810_order__less__le__subst1,axiom,
! [A: set_complex,F: real > set_complex,B: real,C: real] :
( ( ord_less_set_complex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_811_order__less__le__subst1,axiom,
! [A: set_complex,F: nat > set_complex,B: nat,C: nat] :
( ( ord_less_set_complex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_812_order__less__le__subst1,axiom,
! [A: set_complex,F: set_complex > set_complex,B: set_complex,C: set_complex] :
( ( ord_less_set_complex @ A @ ( F @ B ) )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_813_order__less__le__subst1,axiom,
! [A: set_complex_complex,F: real > set_complex_complex,B: real,C: real] :
( ( ord_le3207539288156484613omplex @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le6271439605799870481omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3207539288156484613omplex @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_814_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_815_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_816_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_817_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_818_order__le__less__subst2,axiom,
! [A: set_complex,B: set_complex,F: set_complex > real,C: real] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_819_order__le__less__subst2,axiom,
! [A: set_complex,B: set_complex,F: set_complex > nat,C: nat] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_820_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_complex,C: set_complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_complex @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_821_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_complex @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_822_order__le__less__subst2,axiom,
! [A: set_complex,B: set_complex,F: set_complex > set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_set_complex @ ( F @ B ) @ C )
=> ( ! [X2: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_823_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_complex_complex,C: set_complex_complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le3207539288156484613omplex @ ( F @ B ) @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_le6271439605799870481omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3207539288156484613omplex @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_824_order__le__less__subst1,axiom,
! [A: set_complex,F: real > set_complex,B: real,C: real] :
( ( ord_le211207098394363844omplex @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_set_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_825_order__le__less__subst1,axiom,
! [A: set_complex,F: nat > set_complex,B: nat,C: nat] :
( ( ord_le211207098394363844omplex @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_complex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_complex @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_826_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_827_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_828_order__le__less__subst1,axiom,
! [A: set_complex_complex,F: real > set_complex_complex,B: real,C: real] :
( ( ord_le6271439605799870481omplex @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le3207539288156484613omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3207539288156484613omplex @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_829_order__le__less__subst1,axiom,
! [A: set_complex_complex,F: nat > set_complex_complex,B: nat,C: nat] :
( ( ord_le6271439605799870481omplex @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le3207539288156484613omplex @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3207539288156484613omplex @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_830_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_831_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_832_order__less__le__trans,axiom,
! [X: set_complex,Y3: set_complex,Z2: set_complex] :
( ( ord_less_set_complex @ X @ Y3 )
=> ( ( ord_le211207098394363844omplex @ Y3 @ Z2 )
=> ( ord_less_set_complex @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_833_order__less__le__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_834_order__less__le__trans,axiom,
! [X: set_complex_complex,Y3: set_complex_complex,Z2: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ X @ Y3 )
=> ( ( ord_le6271439605799870481omplex @ Y3 @ Z2 )
=> ( ord_le3207539288156484613omplex @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_835_order__less__le__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_836_order__le__less__trans,axiom,
! [X: set_complex,Y3: set_complex,Z2: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y3 )
=> ( ( ord_less_set_complex @ Y3 @ Z2 )
=> ( ord_less_set_complex @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_837_order__le__less__trans,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_838_order__le__less__trans,axiom,
! [X: set_complex_complex,Y3: set_complex_complex,Z2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X @ Y3 )
=> ( ( ord_le3207539288156484613omplex @ Y3 @ Z2 )
=> ( ord_le3207539288156484613omplex @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_839_order__le__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_840_order__neq__le__trans,axiom,
! [A: set_complex,B: set_complex] :
( ( A != B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ord_less_set_complex @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_841_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_842_order__neq__le__trans,axiom,
! [A: set_complex_complex,B: set_complex_complex] :
( ( A != B )
=> ( ( ord_le6271439605799870481omplex @ A @ B )
=> ( ord_le3207539288156484613omplex @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_843_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_844_order__le__neq__trans,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_complex @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_845_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_846_order__le__neq__trans,axiom,
! [A: set_complex_complex,B: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ B )
=> ( ( A != B )
=> ( ord_le3207539288156484613omplex @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_847_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_848_order__less__imp__le,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_less_set_complex @ X @ Y3 )
=> ( ord_le211207098394363844omplex @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_849_order__less__imp__le,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_eq_real @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_850_order__less__imp__le,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ X @ Y3 )
=> ( ord_le6271439605799870481omplex @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_851_order__less__imp__le,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_852_linorder__not__less,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_853_linorder__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_854_linorder__not__le,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_eq_real @ X @ Y3 ) )
= ( ord_less_real @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_855_linorder__not__le,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
= ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_856_order__less__le,axiom,
( ord_less_set_complex
= ( ^ [X3: set_complex,Y4: set_complex] :
( ( ord_le211207098394363844omplex @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_857_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_858_order__less__le,axiom,
( ord_le3207539288156484613omplex
= ( ^ [X3: set_complex_complex,Y4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_859_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_860_order__le__less,axiom,
( ord_le211207098394363844omplex
= ( ^ [X3: set_complex,Y4: set_complex] :
( ( ord_less_set_complex @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_861_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_862_order__le__less,axiom,
( ord_le6271439605799870481omplex
= ( ^ [X3: set_complex_complex,Y4: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_863_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_864_dual__order_Ostrict__implies__order,axiom,
! [B: set_complex,A: set_complex] :
( ( ord_less_set_complex @ B @ A )
=> ( ord_le211207098394363844omplex @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_865_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_866_dual__order_Ostrict__implies__order,axiom,
! [B: set_complex_complex,A: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ B @ A )
=> ( ord_le6271439605799870481omplex @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_867_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_868_order_Ostrict__implies__order,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_less_set_complex @ A @ B )
=> ( ord_le211207098394363844omplex @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_869_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_870_order_Ostrict__implies__order,axiom,
! [A: set_complex_complex,B: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A @ B )
=> ( ord_le6271439605799870481omplex @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_871_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_872_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_complex
= ( ^ [B4: set_complex,A4: set_complex] :
( ( ord_le211207098394363844omplex @ B4 @ A4 )
& ~ ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_873_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_874_dual__order_Ostrict__iff__not,axiom,
( ord_le3207539288156484613omplex
= ( ^ [B4: set_complex_complex,A4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ B4 @ A4 )
& ~ ( ord_le6271439605799870481omplex @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_875_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_876_dual__order_Ostrict__trans2,axiom,
! [B: set_complex,A: set_complex,C: set_complex] :
( ( ord_less_set_complex @ B @ A )
=> ( ( ord_le211207098394363844omplex @ C @ B )
=> ( ord_less_set_complex @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_877_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_878_dual__order_Ostrict__trans2,axiom,
! [B: set_complex_complex,A: set_complex_complex,C: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ B @ A )
=> ( ( ord_le6271439605799870481omplex @ C @ B )
=> ( ord_le3207539288156484613omplex @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_879_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_880_dual__order_Ostrict__trans1,axiom,
! [B: set_complex,A: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ B @ A )
=> ( ( ord_less_set_complex @ C @ B )
=> ( ord_less_set_complex @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_881_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_882_dual__order_Ostrict__trans1,axiom,
! [B: set_complex_complex,A: set_complex_complex,C: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ B @ A )
=> ( ( ord_le3207539288156484613omplex @ C @ B )
=> ( ord_le3207539288156484613omplex @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_883_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_884_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_complex
= ( ^ [B4: set_complex,A4: set_complex] :
( ( ord_le211207098394363844omplex @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_885_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_886_dual__order_Ostrict__iff__order,axiom,
( ord_le3207539288156484613omplex
= ( ^ [B4: set_complex_complex,A4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_887_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_888_dual__order_Oorder__iff__strict,axiom,
( ord_le211207098394363844omplex
= ( ^ [B4: set_complex,A4: set_complex] :
( ( ord_less_set_complex @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_889_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_890_dual__order_Oorder__iff__strict,axiom,
( ord_le6271439605799870481omplex
= ( ^ [B4: set_complex_complex,A4: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_891_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_892_dense__le__bounded,axiom,
! [X: real,Y3: real,Z2: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y3 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_893_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y3: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y3 @ W ) ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_894_order_Ostrict__iff__not,axiom,
( ord_less_set_complex
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B4 )
& ~ ( ord_le211207098394363844omplex @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_895_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_896_order_Ostrict__iff__not,axiom,
( ord_le3207539288156484613omplex
= ( ^ [A4: set_complex_complex,B4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A4 @ B4 )
& ~ ( ord_le6271439605799870481omplex @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_897_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_898_order_Ostrict__trans2,axiom,
! [A: set_complex,B: set_complex,C: set_complex] :
( ( ord_less_set_complex @ A @ B )
=> ( ( ord_le211207098394363844omplex @ B @ C )
=> ( ord_less_set_complex @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_899_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_900_order_Ostrict__trans2,axiom,
! [A: set_complex_complex,B: set_complex_complex,C: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A @ B )
=> ( ( ord_le6271439605799870481omplex @ B @ C )
=> ( ord_le3207539288156484613omplex @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_901_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_902_order_Ostrict__trans1,axiom,
! [A: set_complex,B: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_set_complex @ B @ C )
=> ( ord_less_set_complex @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_903_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_904_order_Ostrict__trans1,axiom,
! [A: set_complex_complex,B: set_complex_complex,C: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A @ B )
=> ( ( ord_le3207539288156484613omplex @ B @ C )
=> ( ord_le3207539288156484613omplex @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_905_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_906_order_Ostrict__iff__order,axiom,
( ord_less_set_complex
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_907_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_908_order_Ostrict__iff__order,axiom,
( ord_le3207539288156484613omplex
= ( ^ [A4: set_complex_complex,B4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_909_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_910_order_Oorder__iff__strict,axiom,
( ord_le211207098394363844omplex
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_less_set_complex @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_911_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_912_order_Oorder__iff__strict,axiom,
( ord_le6271439605799870481omplex
= ( ^ [A4: set_complex_complex,B4: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_913_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_914_not__le__imp__less,axiom,
! [Y3: real,X: real] :
( ~ ( ord_less_eq_real @ Y3 @ X )
=> ( ord_less_real @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_915_not__le__imp__less,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X )
=> ( ord_less_nat @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_916_less__le__not__le,axiom,
( ord_less_set_complex
= ( ^ [X3: set_complex,Y4: set_complex] :
( ( ord_le211207098394363844omplex @ X3 @ Y4 )
& ~ ( ord_le211207098394363844omplex @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_917_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_918_less__le__not__le,axiom,
( ord_le3207539288156484613omplex
= ( ^ [X3: set_complex_complex,Y4: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X3 @ Y4 )
& ~ ( ord_le6271439605799870481omplex @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_919_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_920_dense__le,axiom,
! [Y3: real,Z2: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% dense_le
thf(fact_921_dense__ge,axiom,
! [Z2: real,Y3: real] :
( ! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ord_less_eq_real @ Y3 @ X2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% dense_ge
thf(fact_922_antisym__conv2,axiom,
! [X: set_complex,Y3: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y3 )
=> ( ( ~ ( ord_less_set_complex @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_923_antisym__conv2,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_924_antisym__conv2,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ X @ Y3 )
=> ( ( ~ ( ord_le3207539288156484613omplex @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_925_antisym__conv2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_926_antisym__conv1,axiom,
! [X: set_complex,Y3: set_complex] :
( ~ ( ord_less_set_complex @ X @ Y3 )
=> ( ( ord_le211207098394363844omplex @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_927_antisym__conv1,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_eq_real @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_928_antisym__conv1,axiom,
! [X: set_complex_complex,Y3: set_complex_complex] :
( ~ ( ord_le3207539288156484613omplex @ X @ Y3 )
=> ( ( ord_le6271439605799870481omplex @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_929_antisym__conv1,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_930_nless__le,axiom,
! [A: set_complex,B: set_complex] :
( ( ~ ( ord_less_set_complex @ A @ B ) )
= ( ~ ( ord_le211207098394363844omplex @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_931_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_932_nless__le,axiom,
! [A: set_complex_complex,B: set_complex_complex] :
( ( ~ ( ord_le3207539288156484613omplex @ A @ B ) )
= ( ~ ( ord_le6271439605799870481omplex @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_933_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_934_leI,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) ) ).
% leI
thf(fact_935_leI,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% leI
thf(fact_936_leD,axiom,
! [Y3: set_complex,X: set_complex] :
( ( ord_le211207098394363844omplex @ Y3 @ X )
=> ~ ( ord_less_set_complex @ X @ Y3 ) ) ).
% leD
thf(fact_937_leD,axiom,
! [Y3: real,X: real] :
( ( ord_less_eq_real @ Y3 @ X )
=> ~ ( ord_less_real @ X @ Y3 ) ) ).
% leD
thf(fact_938_leD,axiom,
! [Y3: set_complex_complex,X: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ Y3 @ X )
=> ~ ( ord_le3207539288156484613omplex @ X @ Y3 ) ) ).
% leD
thf(fact_939_leD,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_nat @ X @ Y3 ) ) ).
% leD
thf(fact_940_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
= ( ord_less_real @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_941_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_942_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_943_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_944_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_945_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_946_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_947_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_948_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_949_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_950_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_951_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_952_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( I3 = J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_953_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( I3 = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_954_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_955_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I3 @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I3 @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_956_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_957_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_958_add__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_mono
thf(fact_959_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_960_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_961_add__left__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) ) ) ).
% add_left_mono
thf(fact_962_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_963_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_964_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_965_add__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) ) ) ).
% add_right_mono
thf(fact_966_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_967_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_968_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_969_add__le__imp__le__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
=> ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_970_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_971_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_972_add__le__imp__le__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
=> ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_973_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_974_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_975_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_976_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_977_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_978_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_979_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_980_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_981_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_982_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_983_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_984_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_985_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_986_pth__d,axiom,
! [X: complex] :
( ( plus_plus_complex @ X @ zero_zero_complex )
= X ) ).
% pth_d
thf(fact_987_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_988_pth__7_I1_J,axiom,
! [X: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X )
= X ) ).
% pth_7(1)
thf(fact_989_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_990_diff__strict__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( minus_minus_complex @ A @ C ) @ ( minus_minus_complex @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_991_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_992_diff__strict__left__mono,axiom,
! [B: complex,A: complex,C: complex] :
( ( ord_less_complex @ B @ A )
=> ( ord_less_complex @ ( minus_minus_complex @ C @ A ) @ ( minus_minus_complex @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_993_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_994_diff__eq__diff__less,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( ord_less_complex @ A @ B )
= ( ord_less_complex @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_995_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_996_diff__strict__mono,axiom,
! [A: complex,B: complex,D: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ( ord_less_complex @ D @ C )
=> ( ord_less_complex @ ( minus_minus_complex @ A @ C ) @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_997_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_998_bot_Oextremum__strict,axiom,
! [A: set_complex] :
~ ( ord_less_set_complex @ A @ bot_bot_set_complex ) ).
% bot.extremum_strict
thf(fact_999_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1000_bot_Onot__eq__extremum,axiom,
! [A: set_complex] :
( ( A != bot_bot_set_complex )
= ( ord_less_set_complex @ bot_bot_set_complex @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1001_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1002_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1003_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1004_diff__diff__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1005_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1006_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1007_add__implies__diff,axiom,
! [C: complex,B: complex,A: complex] :
( ( ( plus_plus_complex @ C @ B )
= A )
=> ( C
= ( minus_minus_complex @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1008_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1009_diff__add__eq__diff__diff__swap,axiom,
! [A: complex,B: complex,C: complex] :
( ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) )
= ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1010_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1011_diff__add__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1012_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1013_diff__diff__eq2,axiom,
! [A: complex,B: complex,C: complex] :
( ( minus_minus_complex @ A @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1014_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1015_add__diff__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ A @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1016_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1017_eq__diff__eq,axiom,
! [A: complex,C: complex,B: complex] :
( ( A
= ( minus_minus_complex @ C @ B ) )
= ( ( plus_plus_complex @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1018_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1019_diff__eq__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( minus_minus_complex @ A @ B )
= C )
= ( A
= ( plus_plus_complex @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1020_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1021_group__cancel_Osub1,axiom,
! [A2: complex,K: complex,A: complex,B: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( minus_minus_complex @ A2 @ B )
= ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1022_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_1023_add__diff__add,axiom,
! [A: complex,C: complex,B: complex,D: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) )
= ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ ( minus_minus_complex @ C @ D ) ) ) ).
% add_diff_add
thf(fact_1024_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_1025_minus__less__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_1026_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_1027_less__minus__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( ord_less_complex @ B @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% less_minus_iff
thf(fact_1028_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1029_verit__negate__coefficient_I2_J,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1030_compl__less__swap2,axiom,
! [Y3: set_complex,X: set_complex] :
( ( ord_less_set_complex @ ( uminus8566677241136511917omplex @ Y3 ) @ X )
=> ( ord_less_set_complex @ ( uminus8566677241136511917omplex @ X ) @ Y3 ) ) ).
% compl_less_swap2
thf(fact_1031_compl__less__swap1,axiom,
! [Y3: set_complex,X: set_complex] :
( ( ord_less_set_complex @ Y3 @ ( uminus8566677241136511917omplex @ X ) )
=> ( ord_less_set_complex @ X @ ( uminus8566677241136511917omplex @ Y3 ) ) ) ).
% compl_less_swap1
thf(fact_1032_psubsetE,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ~ ( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ord_le211207098394363844omplex @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_1033_psubsetE,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A2 @ B2 )
=> ~ ( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ord_le6271439605799870481omplex @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_1034_psubset__eq,axiom,
( ord_less_set_complex
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1035_psubset__eq,axiom,
( ord_le3207539288156484613omplex
= ( ^ [A3: set_complex_complex,B3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1036_psubset__imp__subset,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1037_psubset__imp__subset,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A2 @ B2 )
=> ( ord_le6271439605799870481omplex @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1038_psubset__subset__trans,axiom,
! [A2: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ( ( ord_le211207098394363844omplex @ B2 @ C2 )
=> ( ord_less_set_complex @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1039_psubset__subset__trans,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,C2: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A2 @ B2 )
=> ( ( ord_le6271439605799870481omplex @ B2 @ C2 )
=> ( ord_le3207539288156484613omplex @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1040_subset__not__subset__eq,axiom,
( ord_less_set_complex
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B3 )
& ~ ( ord_le211207098394363844omplex @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1041_subset__not__subset__eq,axiom,
( ord_le3207539288156484613omplex
= ( ^ [A3: set_complex_complex,B3: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A3 @ B3 )
& ~ ( ord_le6271439605799870481omplex @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1042_subset__psubset__trans,axiom,
! [A2: set_complex,B2: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( ord_less_set_complex @ B2 @ C2 )
=> ( ord_less_set_complex @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1043_subset__psubset__trans,axiom,
! [A2: set_complex_complex,B2: set_complex_complex,C2: set_complex_complex] :
( ( ord_le6271439605799870481omplex @ A2 @ B2 )
=> ( ( ord_le3207539288156484613omplex @ B2 @ C2 )
=> ( ord_le3207539288156484613omplex @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1044_subset__iff__psubset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
( ( ord_less_set_complex @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1045_subset__iff__psubset__eq,axiom,
( ord_le6271439605799870481omplex
= ( ^ [A3: set_complex_complex,B3: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1046_not__psubset__empty,axiom,
! [A2: set_complex] :
~ ( ord_less_set_complex @ A2 @ bot_bot_set_complex ) ).
% not_psubset_empty
thf(fact_1047_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1048_add_Oinverse__distrib__swap,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1049_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_1050_group__cancel_Oneg1,axiom,
! [A2: complex,K: complex,A: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( uminus1482373934393186551omplex @ A2 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_1051_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_1052_is__num__normalize_I8_J,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_1053_psubset__imp__ex__mem,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ? [B6: complex] : ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1054_psubset__imp__ex__mem,axiom,
! [A2: set_complex_complex,B2: set_complex_complex] :
( ( ord_le3207539288156484613omplex @ A2 @ B2 )
=> ? [B6: complex > complex] : ( member5128974058612258834omplex @ B6 @ ( minus_3522879524658371850omplex @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1055_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_1056_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1057_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_1058_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_1059_Ints__add,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_add
thf(fact_1060_Ints__add,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_add
thf(fact_1061_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_1062_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
% Ints_1
thf(fact_1063_analytic__on__open,axiom,
! [S4: set_complex,F: complex > complex] :
( ( topolo4110288021797289639omplex @ S4 )
=> ( ( comple673786817313641009tic_on @ F @ S4 )
= ( comple7700996537433184370hic_on @ F @ S4 ) ) ) ).
% analytic_on_open
thf(fact_1064_analytic__on__analytic__at,axiom,
( comple673786817313641009tic_on
= ( ^ [F3: complex > complex,S5: set_complex] :
! [X3: complex] :
( ( member_complex @ X3 @ S5 )
=> ( comple673786817313641009tic_on @ F3 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ).
% analytic_on_analytic_at
thf(fact_1065_add__decreasing,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ C @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1066_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1067_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1068_add__increasing,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ B @ C )
=> ( ord_less_eq_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1069_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1070_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1071_add__decreasing2,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1072_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1073_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1074_add__increasing2,axiom,
! [C: complex,B: complex,A: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ( ord_less_eq_complex @ B @ A )
=> ( ord_less_eq_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1075_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1076_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1077_add__nonneg__nonneg,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1078_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1079_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1080_add__nonpos__nonpos,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_nonpos_nonpos
thf(fact_1081_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1082_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1083_add__nonneg__eq__0__iff,axiom,
! [X: complex,Y3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ Y3 )
=> ( ( ( plus_plus_complex @ X @ Y3 )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y3 = zero_zero_complex ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1084_add__nonneg__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1085_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1086_add__nonpos__eq__0__iff,axiom,
! [X: complex,Y3: complex] :
( ( ord_less_eq_complex @ X @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ Y3 @ zero_zero_complex )
=> ( ( ( plus_plus_complex @ X @ Y3 )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y3 = zero_zero_complex ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1087_add__nonpos__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1088_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1089_less__iff__diff__less__0,axiom,
( ord_less_complex
= ( ^ [A4: complex,B4: complex] : ( ord_less_complex @ ( minus_minus_complex @ A4 @ B4 ) @ zero_zero_complex ) ) ) ).
% less_iff_diff_less_0
thf(fact_1090_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_1091_diff__le__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ ( minus_minus_complex @ A @ B ) @ C )
= ( ord_less_eq_complex @ A @ ( plus_plus_complex @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1092_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1093_le__diff__eq,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( minus_minus_complex @ C @ B ) )
= ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1094_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1095_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1096_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1097_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1098_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1099_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1100_analytic__on__holomorphic,axiom,
( comple673786817313641009tic_on
= ( ^ [F3: complex > complex,S3: set_complex] :
? [T2: set_complex] :
( ( topolo4110288021797289639omplex @ T2 )
& ( ord_le211207098394363844omplex @ S3 @ T2 )
& ( comple7700996537433184370hic_on @ F3 @ T2 ) ) ) ) ).
% analytic_on_holomorphic
thf(fact_1101_isolated__singularity__at__def,axiom,
( comple1891072044276206784ity_at
= ( ^ [F3: complex > complex,Z4: complex] :
? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( comple673786817313641009tic_on @ F3 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ Z4 @ R ) @ ( insert_complex @ Z4 @ bot_bot_set_complex ) ) ) ) ) ) ).
% isolated_singularity_at_def
thf(fact_1102_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1103_real__add__le__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1104_real__0__le__add__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y3 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y3 ) ) ).
% real_0_le_add_iff
thf(fact_1105_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I3_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(3)
thf(fact_1106_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(1)
thf(fact_1107_holomorphic__on__balls__imp__entire_H,axiom,
! [F: complex > complex,C: complex,B2: set_complex] :
( ! [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ C @ R2 ) ) )
=> ( comple7700996537433184370hic_on @ F @ B2 ) ) ).
% holomorphic_on_balls_imp_entire'
thf(fact_1108_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1109_real__0__less__add__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y3 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y3 ) ) ).
% real_0_less_add_iff
thf(fact_1110_real__add__less__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
= ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1111_analytic__on__def,axiom,
( comple673786817313641009tic_on
= ( ^ [F3: complex > complex,S3: set_complex] :
! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ? [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
& ( comple7700996537433184370hic_on @ F3 @ ( elemen509638587668912547omplex @ X3 @ E3 ) ) ) ) ) ) ).
% analytic_on_def
thf(fact_1112_analytic__at__ball,axiom,
! [F: complex > complex,Z2: complex] :
( ( comple673786817313641009tic_on @ F @ ( insert_complex @ Z2 @ bot_bot_set_complex ) )
= ( ? [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
& ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ Z2 @ E3 ) ) ) ) ) ).
% analytic_at_ball
thf(fact_1113_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X3: real,Y4: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y4 ) ) ) ) ).
% minus_real_def
thf(fact_1114_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A6: real,B6: real,C4: real] :
( ( P @ A6 @ B6 )
=> ( ( P @ B6 @ C4 )
=> ( ( ord_less_eq_real @ A6 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ C4 )
=> ( P @ A6 @ C4 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A6: real,B6: real] :
( ( ( ord_less_eq_real @ A6 @ X2 )
& ( ord_less_eq_real @ X2 @ B6 )
& ( ord_less_real @ ( minus_minus_real @ B6 @ A6 ) @ D3 ) )
=> ( P @ A6 @ B6 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1115_isolated__zeros,axiom,
! [F: complex > complex,S4: set_complex,Xi: complex,Beta: complex] :
( ( comple7700996537433184370hic_on @ F @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( ( topolo3972588530358341399omplex @ S4 )
=> ( ( member_complex @ Xi @ S4 )
=> ( ( ( F @ Xi )
= zero_zero_complex )
=> ( ( member_complex @ Beta @ S4 )
=> ( ( ( F @ Beta )
!= zero_zero_complex )
=> ~ ! [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( ord_le211207098394363844omplex @ ( elemen509638587668912547omplex @ Xi @ R2 ) @ S4 )
=> ~ ! [Z5: complex] :
( ( member_complex @ Z5 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ Xi @ R2 ) @ ( insert_complex @ Xi @ bot_bot_set_complex ) ) )
=> ( ( F @ Z5 )
!= zero_zero_complex ) ) ) ) ) ) ) ) ) ) ) ).
% isolated_zeros
thf(fact_1116_GPicard5,axiom,
! [F: complex > complex] :
( ( comple7700996537433184370hic_on @ F @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
=> ( ( ( F @ Z3 )
!= zero_zero_complex )
& ( ( F @ Z3 )
!= one_one_complex ) ) )
=> ~ ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ( ord_less_real @ E2 @ one_one_real )
=> ! [B5: real] :
( ( ord_less_real @ zero_zero_real @ B5 )
=> ~ ( ! [X6: complex] :
( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ zero_zero_complex @ E2 ) @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) @ B5 ) )
| ! [X6: complex] :
( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ zero_zero_complex @ E2 ) @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
=> ( ord_less_eq_real @ B5 @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) ) ) ) ) ) ) ) ) ).
% GPicard5
thf(fact_1117_eq__diff__eq_H,axiom,
! [X: real,Y3: real,Z2: real] :
( ( X
= ( minus_minus_real @ Y3 @ Z2 ) )
= ( Y3
= ( plus_plus_real @ X @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_1118_GPicard1,axiom,
! [S4: set_complex,W2: complex,R3: real,Y6: set_complex_complex,X4: set_complex_complex] :
( ( topolo4110288021797289639omplex @ S4 )
=> ( ( topolo3972588530358341399omplex @ S4 )
=> ( ( member_complex @ W2 @ S4 )
=> ( ( ord_less_real @ zero_zero_real @ R3 )
=> ( ( ord_le6271439605799870481omplex @ Y6 @ X4 )
=> ( ! [H: complex > complex] :
( ( member5128974058612258834omplex @ H @ X4 )
=> ( comple7700996537433184370hic_on @ H @ S4 ) )
=> ( ! [H: complex > complex] :
( ( member5128974058612258834omplex @ H @ X4 )
=> ! [Z3: complex] :
( ( member_complex @ Z3 @ S4 )
=> ( ( ( H @ Z3 )
!= zero_zero_complex )
& ( ( H @ Z3 )
!= one_one_complex ) ) ) )
=> ( ! [H: complex > complex] :
( ( member5128974058612258834omplex @ H @ Y6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( H @ W2 ) ) @ R3 ) )
=> ~ ! [B5: real] :
( ( ord_less_real @ zero_zero_real @ B5 )
=> ! [Z6: set_complex] :
( ( topolo4110288021797289639omplex @ Z6 )
=> ( ( member_complex @ W2 @ Z6 )
=> ( ( ord_le211207098394363844omplex @ Z6 @ S4 )
=> ~ ! [H2: complex > complex] :
( ( member5128974058612258834omplex @ H2 @ Y6 )
=> ! [Z5: complex] :
( ( member_complex @ Z5 @ Z6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( H2 @ Z5 ) ) @ B5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% GPicard1
thf(fact_1119_analytic__continuation__open,axiom,
! [S: set_complex,S6: set_complex,F: complex > complex,G: complex > complex,Z2: complex] :
( ( topolo4110288021797289639omplex @ S )
=> ( ( topolo4110288021797289639omplex @ S6 )
=> ( ( S != bot_bot_set_complex )
=> ( ( topolo3972588530358341399omplex @ S6 )
=> ( ( ord_le211207098394363844omplex @ S @ S6 )
=> ( ( comple7700996537433184370hic_on @ F @ S6 )
=> ( ( comple7700996537433184370hic_on @ G @ S6 )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ S )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( member_complex @ Z2 @ S6 )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) ) ) ) ) ) ) ) ) ) ).
% analytic_continuation_open
thf(fact_1120_Schwarz__Lemma_I1_J,axiom,
! [F: complex > complex,Xi: complex] :
( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) )
=> ( ( ( F @ zero_zero_complex )
= zero_zero_complex )
=> ( ! [Z3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) @ one_one_real ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Xi ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ Xi ) ) @ ( real_V1022390504157884413omplex @ Xi ) ) ) ) ) ) ).
% Schwarz_Lemma(1)
thf(fact_1121_GPicard3,axiom,
! [S4: set_complex,W2: complex,Y6: set_complex_complex,X4: set_complex_complex,K2: set_complex] :
( ( topolo4110288021797289639omplex @ S4 )
=> ( ( topolo3972588530358341399omplex @ S4 )
=> ( ( member_complex @ W2 @ S4 )
=> ( ( ord_le6271439605799870481omplex @ Y6 @ X4 )
=> ( ! [H: complex > complex] :
( ( member5128974058612258834omplex @ H @ X4 )
=> ( comple7700996537433184370hic_on @ H @ S4 ) )
=> ( ! [H: complex > complex] :
( ( member5128974058612258834omplex @ H @ X4 )
=> ! [Z3: complex] :
( ( member_complex @ Z3 @ S4 )
=> ( ( ( H @ Z3 )
!= zero_zero_complex )
& ( ( H @ Z3 )
!= one_one_complex ) ) ) )
=> ( ! [H: complex > complex] :
( ( member5128974058612258834omplex @ H @ Y6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( H @ W2 ) ) @ one_one_real ) )
=> ( ( topolo702044070747558609omplex @ K2 )
=> ( ( ord_le211207098394363844omplex @ K2 @ S4 )
=> ~ ! [B5: real] :
~ ! [H2: complex > complex] :
( ( member5128974058612258834omplex @ H2 @ Y6 )
=> ! [Z5: complex] :
( ( member_complex @ Z5 @ K2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( H2 @ Z5 ) ) @ B5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% GPicard3
thf(fact_1122_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% complex_mod_triangle_ineq2
thf(fact_1123_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) )
& ( ord_less_eq_real @ ( X2 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I: nat] : ( ord_less_eq_nat @ ( L2 @ X6 @ I ) @ one_one_nat )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( X6 @ I )
= zero_zero_real ) )
=> ( ( L2 @ X6 @ I )
= zero_zero_nat ) )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( X6 @ I )
= one_one_real ) )
=> ( ( L2 @ X6 @ I )
= one_one_nat ) )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( L2 @ X6 @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I ) @ ( F @ X6 @ I ) ) )
& ! [X6: nat > real,I: nat] :
( ( ( P @ X6 )
& ( Q @ I )
& ( ( L2 @ X6 @ I )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I ) @ ( X6 @ I ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1124_kuhn__lemma,axiom,
! [P4: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P4 )
=> ( ! [X2: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_nat @ ( X2 @ I ) @ P4 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( Label @ X2 @ I2 )
= zero_zero_nat )
| ( ( Label @ X2 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_nat @ ( X2 @ I ) @ P4 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( X2 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X2 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X2: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_nat @ ( X2 @ I ) @ P4 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( X2 @ I2 )
= P4 )
=> ( ( Label @ X2 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_nat @ ( Q2 @ I ) @ P4 ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ? [R2: nat > nat] :
( ! [J2: nat] :
( ( ord_less_nat @ J2 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J2 ) @ ( R2 @ J2 ) )
& ( ord_less_eq_nat @ ( R2 @ J2 ) @ ( plus_plus_nat @ ( Q2 @ J2 ) @ one_one_nat ) ) ) )
& ? [S7: nat > nat] :
( ! [J2: nat] :
( ( ord_less_nat @ J2 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J2 ) @ ( S7 @ J2 ) )
& ( ord_less_eq_nat @ ( S7 @ J2 ) @ ( plus_plus_nat @ ( Q2 @ J2 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I )
!= ( Label @ S7 @ I ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1125_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1126_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E4: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E4 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E4 @ N3 ) @ ( E4 @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E4 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1127_diff__diff__left,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1128_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1129_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1130_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1131_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1132_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1133_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1134_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1135_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1136_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1137_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1138_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1139_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1140_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1141_less__diff__conv,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1142_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1143_trans__less__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1144_trans__less__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1145_add__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1146_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1147_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_1148_add__less__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1149_add__lessD1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_1150_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1151_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1152_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1153_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1154_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1155_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1156_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1157_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1158_less__not__refl3,axiom,
! [S: nat,T4: nat] :
( ( ord_less_nat @ S @ T4 )
=> ( S != T4 ) ) ).
% less_not_refl3
thf(fact_1159_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1160_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1161_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1162_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1163_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1164_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1165_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1166_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1167_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1168_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1169_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1170_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1171_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1172_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1173_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1174_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1175_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1176_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1177_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1178_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1179_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1180_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1181_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1182_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1183_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1184_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1185_less__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1186_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1187_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1188_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1189_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1190_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1191_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1192_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1193_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1194_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1195_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1196_trans__le__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1197_trans__le__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1198_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1199_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1200_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1201_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1202_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1203_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1204_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K3 )
=> ~ ( P @ I ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1205_GPicard4,axiom,
! [K: real,F: complex > complex,B2: real] :
( ( ord_less_real @ zero_zero_real @ K )
=> ( ( comple7700996537433184370hic_on @ F @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ zero_zero_complex @ K ) @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
=> ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ( ord_less_real @ E2 @ K )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ord_less_real @ D3 @ E2 )
& ! [X2: complex] :
( ( member_complex @ X2 @ ( elemen5844589719862857425omplex @ zero_zero_complex @ D3 ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ B2 ) ) ) ) )
=> ~ ! [Epsilon: real] :
( ( ord_less_real @ zero_zero_real @ Epsilon )
=> ( ( ord_less_real @ Epsilon @ K )
=> ~ ! [Z5: complex] :
( ( member_complex @ Z5 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ zero_zero_complex @ Epsilon ) @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ Z5 ) ) @ B2 ) ) ) ) ) ) ) ).
% GPicard4
thf(fact_1206_Schwarz2,axiom,
! [F: complex > complex,R3: real,S: real,W2: complex] :
( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ R3 ) )
=> ( ( ord_less_real @ zero_zero_real @ S )
=> ( ( ord_le211207098394363844omplex @ ( elemen509638587668912547omplex @ W2 @ S ) @ ( elemen509638587668912547omplex @ zero_zero_complex @ R3 ) )
=> ( ! [Z3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ W2 @ Z3 ) ) @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) @ ( real_V1022390504157884413omplex @ ( F @ W2 ) ) ) )
=> ( abstra1392856459910508790omplex @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ R3 ) ) ) ) ) ) ).
% Schwarz2
thf(fact_1207_Moebius__function__holomorphic,axiom,
! [W2: complex,T4: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( comple7700996537433184370hic_on @ ( rieman142549540920964168nction @ T4 @ W2 ) @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) ) ) ).
% Moebius_function_holomorphic
thf(fact_1208_maximum__modulus__principle,axiom,
! [F: complex > complex,S4: set_complex,U3: set_complex,Xi: complex] :
( ( comple7700996537433184370hic_on @ F @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( ( topolo3972588530358341399omplex @ S4 )
=> ( ( topolo4110288021797289639omplex @ U3 )
=> ( ( ord_le211207098394363844omplex @ U3 @ S4 )
=> ( ( member_complex @ Xi @ U3 )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ U3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) @ ( real_V1022390504157884413omplex @ ( F @ Xi ) ) ) )
=> ( abstra1392856459910508790omplex @ F @ S4 ) ) ) ) ) ) ) ) ).
% maximum_modulus_principle
thf(fact_1209_Moebius__function__eq__zero,axiom,
! [T4: real,W2: complex] :
( ( rieman142549540920964168nction @ T4 @ W2 @ W2 )
= zero_zero_complex ) ).
% Moebius_function_eq_zero
thf(fact_1210_constant__on__imp__holomorphic__on,axiom,
! [F: complex > complex,A2: set_complex] :
( ( abstra1392856459910508790omplex @ F @ A2 )
=> ( comple7700996537433184370hic_on @ F @ A2 ) ) ).
% constant_on_imp_holomorphic_on
thf(fact_1211_Moebius__function__norm__lt__1,axiom,
! [W2: complex,Z2: complex,T4: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( rieman142549540920964168nction @ T4 @ W2 @ Z2 ) ) @ one_one_real ) ) ) ).
% Moebius_function_norm_lt_1
thf(fact_1212_Moebius__function__compose,axiom,
! [W1: complex,W22: complex,Z2: complex] :
( ( ( uminus1482373934393186551omplex @ W1 )
= W22 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ W1 ) @ one_one_real )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( ( rieman142549540920964168nction @ zero_zero_real @ W1 @ ( rieman142549540920964168nction @ zero_zero_real @ W22 @ Z2 ) )
= Z2 ) ) ) ) ).
% Moebius_function_compose
thf(fact_1213_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B6 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1214_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K3 @ I )
=> ( P @ I ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1215_holomorphic__on__imp__continuous__on,axiom,
! [F: complex > complex,S: set_complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( topolo9015423870875150044omplex @ S @ F ) ) ).
% holomorphic_on_imp_continuous_on
thf(fact_1216_holomorphic__contract__to__zero,axiom,
! [Xi: complex,R3: real,F: complex > complex] :
( ( topolo9015423870875150044omplex @ ( elemen7827680097914048924omplex @ Xi @ R3 ) @ F )
=> ( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ Xi @ R3 ) )
=> ( ( ord_less_real @ zero_zero_real @ R3 )
=> ( ! [Z3: complex] :
( ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Xi @ Z3 ) )
= R3 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( F @ Xi ) ) @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) ) )
=> ~ ! [Z3: complex] :
( ( member_complex @ Z3 @ ( elemen509638587668912547omplex @ Xi @ R3 ) )
=> ( ( F @ Z3 )
!= zero_zero_complex ) ) ) ) ) ) ).
% holomorphic_contract_to_zero
thf(fact_1217_Landau__Picard,axiom,
~ ! [R4: complex > real] :
( ! [Z5: complex] : ( ord_less_real @ zero_zero_real @ ( R4 @ Z5 ) )
=> ~ ! [F4: complex > complex] :
( ( comple7700996537433184370hic_on @ F4 @ ( elemen7827680097914048924omplex @ zero_zero_complex @ ( R4 @ ( F4 @ zero_zero_complex ) ) ) )
=> ( ! [Z3: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( R4 @ ( F4 @ zero_zero_complex ) ) )
=> ( ( ( F4 @ Z3 )
!= zero_zero_complex )
& ( ( F4 @ Z3 )
!= one_one_complex ) ) )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( deriv_complex @ F4 @ zero_zero_complex ) ) @ one_one_real ) ) ) ) ).
% Landau_Picard
thf(fact_1218_holomorphic__factor__zero__nonconstant,axiom,
! [F: complex > complex,S4: set_complex,Xi: complex] :
( ( comple7700996537433184370hic_on @ F @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( ( topolo3972588530358341399omplex @ S4 )
=> ( ( member_complex @ Xi @ S4 )
=> ( ( ( F @ Xi )
= zero_zero_complex )
=> ( ~ ( abstra1392856459910508790omplex @ F @ S4 )
=> ~ ! [G2: complex > complex,R2: real,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( ord_le211207098394363844omplex @ ( elemen509638587668912547omplex @ Xi @ R2 ) @ S4 )
=> ( ( comple7700996537433184370hic_on @ G2 @ ( elemen509638587668912547omplex @ Xi @ R2 ) )
=> ( ! [W3: complex] :
( ( member_complex @ W3 @ ( elemen509638587668912547omplex @ Xi @ R2 ) )
=> ( ( F @ W3 )
= ( times_times_complex @ ( power_power_complex @ ( minus_minus_complex @ W3 @ Xi ) @ N3 ) @ ( G2 @ W3 ) ) ) )
=> ~ ! [W3: complex] :
( ( member_complex @ W3 @ ( elemen509638587668912547omplex @ Xi @ R2 ) )
=> ( ( G2 @ W3 )
!= zero_zero_complex ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% holomorphic_factor_zero_nonconstant
thf(fact_1219_Schwarz3,axiom,
! [F: complex > complex,R3: real] :
( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ R3 ) )
=> ( ( ( F @ zero_zero_complex )
= zero_zero_complex )
=> ~ ! [H: complex > complex] :
( ( comple7700996537433184370hic_on @ H @ ( elemen509638587668912547omplex @ zero_zero_complex @ R3 ) )
=> ( ! [Z5: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z5 ) @ R3 )
=> ( ( F @ Z5 )
= ( times_times_complex @ Z5 @ ( H @ Z5 ) ) ) )
=> ( ( deriv_complex @ F @ zero_zero_complex )
!= ( H @ zero_zero_complex ) ) ) ) ) ) ).
% Schwarz3
thf(fact_1220_holomorphic__on__linear,axiom,
! [C: complex,S: set_complex] : ( comple7700996537433184370hic_on @ ( times_times_complex @ C ) @ S ) ).
% holomorphic_on_linear
thf(fact_1221_analytic__on__linear,axiom,
! [C: complex,S4: set_complex] : ( comple673786817313641009tic_on @ ( times_times_complex @ C ) @ S4 ) ).
% analytic_on_linear
thf(fact_1222_complex__derivative__transform__within__open,axiom,
! [F: complex > complex,S: set_complex,G: complex > complex,Z2: complex] :
( ( comple7700996537433184370hic_on @ F @ S )
=> ( ( comple7700996537433184370hic_on @ G @ S )
=> ( ( topolo4110288021797289639omplex @ S )
=> ( ( member_complex @ Z2 @ S )
=> ( ! [W: complex] :
( ( member_complex @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( deriv_complex @ F @ Z2 )
= ( deriv_complex @ G @ Z2 ) ) ) ) ) ) ) ).
% complex_derivative_transform_within_open
thf(fact_1223_Schwarz__Lemma_I3_J,axiom,
! [F: complex > complex,Xi: complex] :
( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) )
=> ( ( ( F @ zero_zero_complex )
= zero_zero_complex )
=> ( ! [Z3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) @ one_one_real ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Xi ) @ one_one_real )
=> ( ( ? [Z5: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z5 ) @ one_one_real )
& ( Z5 != zero_zero_complex )
& ( ( real_V1022390504157884413omplex @ ( F @ Z5 ) )
= ( real_V1022390504157884413omplex @ Z5 ) ) )
| ( ( real_V1022390504157884413omplex @ ( deriv_complex @ F @ zero_zero_complex ) )
= one_one_real ) )
=> ? [Alpha: complex] :
( ! [Z5: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z5 ) @ one_one_real )
=> ( ( F @ Z5 )
= ( times_times_complex @ Alpha @ Z5 ) ) )
& ( ( real_V1022390504157884413omplex @ Alpha )
= one_one_real ) ) ) ) ) ) ) ).
% Schwarz_Lemma(3)
thf(fact_1224_Schwarz__Lemma_H,axiom,
! [F: complex > complex] :
( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) )
=> ( ( ( F @ zero_zero_complex )
= zero_zero_complex )
=> ( ! [Z3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) @ one_one_real ) )
=> ( ! [Xi2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Xi2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ Xi2 ) ) @ ( real_V1022390504157884413omplex @ Xi2 ) ) )
& ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( deriv_complex @ F @ zero_zero_complex ) ) @ one_one_real )
& ( ( ? [Z5: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z5 ) @ one_one_real )
& ( Z5 != zero_zero_complex )
& ( ( real_V1022390504157884413omplex @ ( F @ Z5 ) )
= ( real_V1022390504157884413omplex @ Z5 ) ) )
| ( ( real_V1022390504157884413omplex @ ( deriv_complex @ F @ zero_zero_complex ) )
= one_one_real ) )
=> ? [Alpha: complex] :
( ! [Z5: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z5 ) @ one_one_real )
=> ( ( F @ Z5 )
= ( times_times_complex @ Alpha @ Z5 ) ) )
& ( ( real_V1022390504157884413omplex @ Alpha )
= one_one_real ) ) ) ) ) ) ) ).
% Schwarz_Lemma'
thf(fact_1225_holomorphic__nonconstant,axiom,
! [F: complex > complex,S4: set_complex,Xi: complex] :
( ( comple7700996537433184370hic_on @ F @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( ( member_complex @ Xi @ S4 )
=> ( ( ( deriv_complex @ F @ Xi )
!= zero_zero_complex )
=> ~ ( abstra1392856459910508790omplex @ F @ S4 ) ) ) ) ) ).
% holomorphic_nonconstant
thf(fact_1226_Schwarz__Lemma_I2_J,axiom,
! [F: complex > complex,Xi: complex] :
( ( comple7700996537433184370hic_on @ F @ ( elemen509638587668912547omplex @ zero_zero_complex @ one_one_real ) )
=> ( ( ( F @ zero_zero_complex )
= zero_zero_complex )
=> ( ! [Z3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( F @ Z3 ) ) @ one_one_real ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Xi ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( deriv_complex @ F @ zero_zero_complex ) ) @ one_one_real ) ) ) ) ) ).
% Schwarz_Lemma(2)
thf(fact_1227_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1228_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1229_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1230_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1231_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1232_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1233_mult__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1234_mult__less__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1235_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1236_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1237_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1238_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1239_nat__power__less__imp__less,axiom,
! [I3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M2 ) @ ( power_power_nat @ I3 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1240_real__arch__pow,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1241_real__minus__mult__self__le,axiom,
! [U4: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U4 @ U4 ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1242_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1243_real__arch__pow__inv,axiom,
! [Y3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_1244_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N2: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1245_holomorphic__lower__bound__difference,axiom,
! [F: complex > complex,S4: set_complex,Xi: complex,Phi: complex] :
( ( comple7700996537433184370hic_on @ F @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( ( topolo3972588530358341399omplex @ S4 )
=> ( ( member_complex @ Xi @ S4 )
=> ( ( member_complex @ Phi @ S4 )
=> ( ( ( F @ Phi )
!= ( F @ Xi ) )
=> ~ ! [K3: real] :
( ( ord_less_real @ zero_zero_real @ K3 )
=> ! [N3: nat,R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( ord_le211207098394363844omplex @ ( elemen509638587668912547omplex @ Xi @ R2 ) @ S4 )
=> ~ ! [W3: complex] :
( ( member_complex @ W3 @ ( elemen509638587668912547omplex @ Xi @ R2 ) )
=> ( ord_less_eq_real @ ( times_times_real @ K3 @ ( power_power_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ W3 @ Xi ) ) @ N3 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( F @ W3 ) @ ( F @ Xi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% holomorphic_lower_bound_difference
thf(fact_1246_pole__theorem__analytic__0,axiom,
! [G: complex > complex,S4: set_complex,A: complex,F: complex > complex] :
( ( comple673786817313641009tic_on @ G @ S4 )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ S4 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [X2: complex] :
( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ ( elemen509638587668912547omplex @ Z3 @ D3 ) @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
=> ( ( G @ X2 )
= ( times_times_complex @ ( minus_minus_complex @ X2 @ A ) @ ( F @ X2 ) ) ) ) ) )
=> ( ( ( F @ A )
= ( deriv_complex @ G @ A ) )
=> ( ( ( G @ A )
= zero_zero_complex )
=> ( comple673786817313641009tic_on @ F @ S4 ) ) ) ) ) ).
% pole_theorem_analytic_0
thf(fact_1247_pole__theorem__analytic__open__superset__0,axiom,
! [G: complex > complex,S4: set_complex,T5: set_complex,A: complex,F: complex > complex] :
( ( comple673786817313641009tic_on @ G @ S4 )
=> ( ( ord_le211207098394363844omplex @ S4 @ T5 )
=> ( ( topolo4110288021797289639omplex @ T5 )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ ( minus_811609699411566653omplex @ T5 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
=> ( ( G @ Z3 )
= ( times_times_complex @ ( minus_minus_complex @ Z3 @ A ) @ ( F @ Z3 ) ) ) )
=> ( ( ( F @ A )
= ( deriv_complex @ G @ A ) )
=> ( ( ( G @ A )
= zero_zero_complex )
=> ( comple673786817313641009tic_on @ F @ S4 ) ) ) ) ) ) ) ).
% pole_theorem_analytic_open_superset_0
thf(fact_1248_analytic__deriv,axiom,
! [F: complex > complex,S4: set_complex] :
( ( comple673786817313641009tic_on @ F @ S4 )
=> ( comple673786817313641009tic_on @ ( deriv_complex @ F ) @ S4 ) ) ).
% analytic_deriv
thf(fact_1249_holomorphic__deriv,axiom,
! [F: complex > complex,S4: set_complex] :
( ( comple7700996537433184370hic_on @ F @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( comple7700996537433184370hic_on @ ( deriv_complex @ F ) @ S4 ) ) ) ).
% holomorphic_deriv
thf(fact_1250_pole__theorem__open__0,axiom,
! [G: complex > complex,S4: set_complex,A: complex,F: complex > complex] :
( ( comple7700996537433184370hic_on @ G @ S4 )
=> ( ( topolo4110288021797289639omplex @ S4 )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
=> ( ( G @ Z3 )
= ( times_times_complex @ ( minus_minus_complex @ Z3 @ A ) @ ( F @ Z3 ) ) ) )
=> ( ( ( F @ A )
= ( deriv_complex @ G @ A ) )
=> ( ( ( G @ A )
= zero_zero_complex )
=> ( comple7700996537433184370hic_on @ F @ S4 ) ) ) ) ) ) ).
% pole_theorem_open_0
thf(fact_1251_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1252_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1253_left__add__mult__distrib,axiom,
! [I3: nat,U4: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J ) @ U4 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1254_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1255_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1256_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1257_nat__eq__add__iff1,axiom,
! [J: nat,I3: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1258_nat__eq__add__iff2,axiom,
! [I3: nat,J: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1259_nat__le__add__iff1,axiom,
! [J: nat,I3: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1260_nat__le__add__iff2,axiom,
! [I3: nat,J: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1261_nat__diff__add__eq1,axiom,
! [J: nat,I3: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1262_nat__diff__add__eq2,axiom,
! [I3: nat,J: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1263_nat__less__add__iff1,axiom,
! [J: nat,I3: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1264_nat__less__add__iff2,axiom,
! [I3: nat,J: nat,U4: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1265_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1266_complex__not__root__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ? [U5: complex] :
( ( ( real_V1022390504157884413omplex @ U5 )
= one_one_real )
& ( ( power_power_complex @ U5 @ N )
!= one_one_complex ) ) ) ).
% complex_not_root_unity
% 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,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
comple7700996537433184370hic_on @ cotang8298477626502807258omplex @ ( uminus8566677241136511917omplex @ ( minus_811609699411566653omplex @ ring_1_Ints_complex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ) ).
%------------------------------------------------------------------------------