TPTP Problem File: SLH0068^1.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 : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_00125_003610__16100414_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1442 ( 474 unt; 169 typ; 0 def)
% Number of atoms : 4075 ( 972 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11205 ( 412 ~; 80 |; 281 &;8356 @)
% ( 0 <=>;2076 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 609 ( 609 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 155 usr; 26 con; 0-3 aty)
% Number of variables : 3653 ( 343 ^;3152 !; 158 ?;3653 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:47:57.059
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $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__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (155)
thf(sy_c_Assumptions__and__Approximations_OL0,type,
assumptions_and_L0: nat ).
thf(sy_c_Assumptions__and__Approximations_OL0_H,type,
assumptions_and_L02: nat ).
thf(sy_c_Assumptions__and__Approximations_OM0,type,
assumptions_and_M0: nat ).
thf(sy_c_Assumptions__and__Approximations_OM0_H,type,
assumptions_and_M02: nat ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions,type,
assump2881078719466019805ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions__axioms,type,
assump8934899134041091456axioms: nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Othird__assumptions,type,
assump2119784843035796504ptions: nat > nat > nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
clique5786534781347292306Graphs: set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Complex__Ocomplex,type,
clique7858167266224639776omplex: set_complex > set_complex > set_set_complex ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
clique8906516429304539640et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
complete_Sup_Sup_int: set_int > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
comple6569609367425551173et_nat: set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_Discrete_Olog,type,
log: nat > nat ).
thf(sy_c_Discrete_Osqrt,type,
sqrt: nat > nat ).
thf(sy_c_Finite__Product__Measure_Ofinite__product__sigma__finite__axioms_001t__Nat__Onat,type,
finite8532872553530597074ms_nat: set_nat > $o ).
thf(sy_c_Finite__Product__Measure_Ofinite__product__sigma__finite__axioms_001t__Set__Oset_It__Nat__Onat_J,type,
finite4265250653382689416et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
finite_card_complex: set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
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_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Int__Oint,type,
measur1300132876559666865at_int: set_set_nat > ( set_nat > int ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Nat__Onat,type,
measur1302623347068717141at_nat: set_set_nat > ( set_nat > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Real__Oreal,type,
measur5905188192028735665t_real: set_set_nat > ( set_nat > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
measur5248428813077667851et_nat: set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
measur496615480034414785et_nat: set_set_nat > ( set_nat > set_set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
measur1680303437913248359at_int: set_set_set_nat > ( set_set_nat > int ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
measur1682793908422298635at_nat: set_set_set_nat > ( set_set_nat > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
measur7326116447087509863t_real: set_set_set_nat > ( set_set_nat > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
measur6219391137901972417et_nat: set_set_set_nat > ( set_set_nat > set_nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
measur8782999752332551287et_nat: set_set_set_nat > ( set_set_nat > set_set_nat ) > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set_nat > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Orderings_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_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo6227097192321305471_nat_o: set_set_nat > $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_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le466346588697744319_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
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__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Int__Oint_J,type,
ord_less_eq_o_int: ( $o > int ) > ( $o > int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J,type,
ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le6539261115178940645et_nat: ( $o > set_set_nat ) > ( $o > set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le3616423863276227763_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_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__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
order_1279421399067128355et_nat: ( set_set_nat > $o ) > set_set_nat ).
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_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Complex__Ocomplex,type,
set_or7194820819169546315omplex: complex > set_complex ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or6631954706645296601et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Transcendental_Olog,type,
log2: real > real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1269)
thf(fact_0_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_1_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_2_sameprod__finite,axiom,
! [X2: set_set_nat] :
( ( finite1152437895449049373et_nat @ X2 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X2 @ X2 ) ) ) ).
% sameprod_finite
thf(fact_3_sameprod__finite,axiom,
! [X2: set_nat] :
( ( finite_finite_nat @ X2 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X2 @ X2 ) ) ) ).
% sameprod_finite
thf(fact_4_finite__nat__bounded,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_5_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S2: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_6_numbers__def,axiom,
clique3652268606331196573umbers = set_ord_lessThan_nat ).
% numbers_def
thf(fact_7_finite__product__sigma__finite__axioms_Ointro,axiom,
! [I: set_nat] :
( ( finite_finite_nat @ I )
=> ( finite8532872553530597074ms_nat @ I ) ) ).
% finite_product_sigma_finite_axioms.intro
thf(fact_8_finite__product__sigma__finite__axioms_Ointro,axiom,
! [I: set_set_nat] :
( ( finite1152437895449049373et_nat @ I )
=> ( finite4265250653382689416et_nat @ I ) ) ).
% finite_product_sigma_finite_axioms.intro
thf(fact_9_finite__product__sigma__finite__axioms__def,axiom,
finite8532872553530597074ms_nat = finite_finite_nat ).
% finite_product_sigma_finite_axioms_def
thf(fact_10_finite__product__sigma__finite__axioms__def,axiom,
finite4265250653382689416et_nat = finite1152437895449049373et_nat ).
% finite_product_sigma_finite_axioms_def
thf(fact_11_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_12_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_13_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_14_sameprod__mono,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X2 @ X2 ) @ ( clique8906516429304539640et_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_15_sameprod__mono,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X2 @ X2 ) @ ( clique6722202388162463298od_nat @ Y2 @ Y2 ) ) ) ).
% sameprod_mono
thf(fact_16_finite__has__minimal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_eq_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_17_finite__has__minimal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( ord_le6893508408891458716et_nat @ X3 @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_18_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ord_less_eq_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_19_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_20_finite__has__minimal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ( ord_less_eq_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_21_finite__has__maximal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_eq_real @ A2 @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_22_finite__has__maximal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( ord_le6893508408891458716et_nat @ A2 @ X3 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_23_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ord_less_eq_set_nat @ A2 @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_24_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_25_finite__has__maximal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ( ord_less_eq_int @ A2 @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_26_rev__finite__subset,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_27_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_28_infinite__super,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T ) ) ) ).
% infinite_super
thf(fact_29_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_30_finite__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_subset
thf(fact_31_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_32_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( member_set_set_nat @ X3 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_33_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_34_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_35_subset__antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_36_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_37_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_38_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_39_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_40_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_41_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_42_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_43_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_44_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_45_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_46_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_47_finite__indexed__bound,axiom,
! [A: set_set_set_nat,P: set_set_nat > real > $o] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M: real] :
! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_48_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M: real] :
! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_49_finite__indexed__bound,axiom,
! [A: set_set_nat,P: set_nat > real > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M: real] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ? [K2: real] :
( ( ord_less_eq_real @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_50_finite__indexed__bound,axiom,
! [A: set_set_set_nat,P: set_set_nat > nat > $o] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M: nat] :
! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_51_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_52_finite__indexed__bound,axiom,
! [A: set_set_nat,P: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M: nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_53_finite__indexed__bound,axiom,
! [A: set_set_set_nat,P: set_set_nat > int > $o] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [X_1: int] : ( P @ X3 @ X_1 ) )
=> ? [M: int] :
! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ? [K2: int] :
( ( ord_less_eq_int @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_54_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > int > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [X_1: int] : ( P @ X3 @ X_1 ) )
=> ? [M: int] :
! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [K2: int] :
( ( ord_less_eq_int @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_55_finite__indexed__bound,axiom,
! [A: set_set_nat,P: set_nat > int > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [X_1: int] : ( P @ X3 @ X_1 ) )
=> ? [M: int] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ? [K2: int] :
( ( ord_less_eq_int @ K2 @ M )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_56_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X @ A )
=> ( member_set_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_57_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_58_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_59_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_60_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_61_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_62_equalityE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_63_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_64_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A3 )
=> ( member_set_set_nat @ X5 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_65_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ A3 )
=> ( member_set_nat @ X5 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_66_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( member_nat @ X5 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_67_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_68_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_69_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_70_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_71_numbers2__mono,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y ) @ ( clique3652268606331196573umbers @ Y ) ) ) ) ).
% numbers2_mono
thf(fact_72_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_73_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M2: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N2 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_74_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_75_order__antisym__conv,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_76_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_77_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_78_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_79_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_80_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_81_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_82_ord__le__eq__subst,axiom,
! [A2: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A2: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A2: real,B3: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_87_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_88_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_ord__le__eq__subst,axiom,
! [A2: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_92_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A2: set_nat,F: real > set_nat,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_103_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_104_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_105_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_106_order__eq__refl,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( X = Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_107_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_108_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_109_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_110_order__subst2,axiom,
! [A2: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_111_order__subst2,axiom,
! [A2: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_112_order__subst2,axiom,
! [A2: real,B3: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_113_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_116_order__subst2,axiom,
! [A2: int,B3: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_order__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_order__subst2,axiom,
! [A2: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst1,axiom,
! [A2: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst1,axiom,
! [A2: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst1,axiom,
! [A2: real,F: int > real,B3: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst1,axiom,
! [A2: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_order__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst1,axiom,
! [A2: int,F: real > int,B3: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_order__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_order__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_order__subst1,axiom,
! [A2: real,F: set_nat > real,B3: set_nat,C: set_nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_130_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_131_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_132_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_133_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_134_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_135_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
! [A2: complex,P: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X5: set_set_nat] : ( member_set_set_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X5: complex] : ( member_complex @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_143_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_144_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_145_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_146_antisym,axiom,
! [A2: real,B3: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_147_antisym,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_148_antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_149_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_150_antisym,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_151_dual__order_Otrans,axiom,
! [B3: real,A2: real,C: real] :
( ( ord_less_eq_real @ B3 @ A2 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_152_dual__order_Otrans,axiom,
! [B3: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B3 )
=> ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_153_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_154_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_155_dual__order_Otrans,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_156_dual__order_Oantisym,axiom,
! [B3: real,A2: real] :
( ( ord_less_eq_real @ B3 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_157_dual__order_Oantisym,axiom,
! [B3: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_158_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_159_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_160_dual__order_Oantisym,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_161_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_162_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_163_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_164_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B3: real] :
( ! [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_167_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_168_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B3: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_169_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_170_order__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_171_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_172_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_173_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_174_order_Otrans,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_175_order_Otrans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_176_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_177_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_178_order_Otrans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_179_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_180_order__antisym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_181_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_182_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_183_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_184_ord__le__eq__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_185_ord__le__eq__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_186_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_187_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_188_ord__le__eq__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_189_ord__eq__le__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( A2 = B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_190_ord__eq__le__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( A2 = B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_191_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_192_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_193_ord__eq__le__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( A2 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_194_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_195_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
& ( ord_le6893508408891458716et_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_196_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_197_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_198_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_199_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_200_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_201_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_202_nle__le,axiom,
! [A2: real,B3: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B3 ) )
= ( ( ord_less_eq_real @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_203_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_204_nle__le,axiom,
! [A2: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_205_Collect__mono__iff,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X5: set_set_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_206_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X5: complex] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_207_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X5: set_nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_208_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_209_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_210_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_211_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_212_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_213_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_214_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_215_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_216_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_217_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_218_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_219_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
! [T2: set_set_nat] :
( ( member_set_set_nat @ T2 @ A3 )
=> ( member_set_set_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_220_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A3 )
=> ( member_set_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_221_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_222_v__mono,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_223_Fpow__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ ( finite_Fpow_set_nat @ A ) @ ( finite_Fpow_set_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_224_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_225_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_226_Greatest__equality,axiom,
! [P: set_set_nat > $o,X: set_set_nat] :
( ( P @ X )
=> ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X ) )
=> ( ( order_1279421399067128355et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_227_Greatest__equality,axiom,
! [P: set_nat > $o,X: set_nat] :
( ( P @ X )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_228_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_229_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_230_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y6: real] :
( ( P @ Y6 )
=> ( ord_less_eq_real @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_231_GreatestI2__order,axiom,
! [P: set_set_nat > $o,X: set_set_nat,Q: set_set_nat > $o] :
( ( P @ X )
=> ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X ) )
=> ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( ! [Y6: set_set_nat] :
( ( P @ Y6 )
=> ( ord_le6893508408891458716et_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1279421399067128355et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_232_GreatestI2__order,axiom,
! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P @ X )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( ! [Y6: set_nat] :
( ( P @ Y6 )
=> ( ord_less_eq_set_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_233_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y6: int] :
( ( P @ Y6 )
=> ( ord_less_eq_int @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_234_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_235_increasing__def,axiom,
( measur7326116447087509863t_real
= ( ^ [M3: set_set_set_nat,Mu: set_set_nat > real] :
! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ M3 )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M3 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_236_increasing__def,axiom,
( measur8782999752332551287et_nat
= ( ^ [M3: set_set_set_nat,Mu: set_set_nat > set_set_nat] :
! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ M3 )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M3 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_237_increasing__def,axiom,
( measur6219391137901972417et_nat
= ( ^ [M3: set_set_set_nat,Mu: set_set_nat > set_nat] :
! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ M3 )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M3 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_238_increasing__def,axiom,
( measur1682793908422298635at_nat
= ( ^ [M3: set_set_set_nat,Mu: set_set_nat > nat] :
! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ M3 )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M3 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_239_increasing__def,axiom,
( measur1680303437913248359at_int
= ( ^ [M3: set_set_set_nat,Mu: set_set_nat > int] :
! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ M3 )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M3 )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_240_increasing__def,axiom,
( measur5905188192028735665t_real
= ( ^ [M3: set_set_nat,Mu: set_nat > real] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ M3 )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M3 )
=> ( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_241_increasing__def,axiom,
( measur496615480034414785et_nat
= ( ^ [M3: set_set_nat,Mu: set_nat > set_set_nat] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ M3 )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M3 )
=> ( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_242_increasing__def,axiom,
( measur5248428813077667851et_nat
= ( ^ [M3: set_set_nat,Mu: set_nat > set_nat] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ M3 )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M3 )
=> ( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_nat @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_243_increasing__def,axiom,
( measur1302623347068717141at_nat
= ( ^ [M3: set_set_nat,Mu: set_nat > nat] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ M3 )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M3 )
=> ( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_244_increasing__def,axiom,
( measur1300132876559666865at_int
= ( ^ [M3: set_set_nat,Mu: set_nat > int] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ M3 )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M3 )
=> ( ( ord_less_eq_set_nat @ X5 @ Y5 )
=> ( ord_less_eq_int @ ( Mu @ X5 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_245_increasingD,axiom,
! [M4: set_set_set_nat,F: set_set_nat > real,X: set_set_nat,Y: set_set_nat] :
( ( measur7326116447087509863t_real @ M4 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( member_set_set_nat @ X @ M4 )
=> ( ( member_set_set_nat @ Y @ M4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_246_increasingD,axiom,
! [M4: set_set_set_nat,F: set_set_nat > set_set_nat,X: set_set_nat,Y: set_set_nat] :
( ( measur8782999752332551287et_nat @ M4 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( member_set_set_nat @ X @ M4 )
=> ( ( member_set_set_nat @ Y @ M4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_247_increasingD,axiom,
! [M4: set_set_set_nat,F: set_set_nat > set_nat,X: set_set_nat,Y: set_set_nat] :
( ( measur6219391137901972417et_nat @ M4 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( member_set_set_nat @ X @ M4 )
=> ( ( member_set_set_nat @ Y @ M4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_248_increasingD,axiom,
! [M4: set_set_set_nat,F: set_set_nat > nat,X: set_set_nat,Y: set_set_nat] :
( ( measur1682793908422298635at_nat @ M4 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( member_set_set_nat @ X @ M4 )
=> ( ( member_set_set_nat @ Y @ M4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_249_increasingD,axiom,
! [M4: set_set_set_nat,F: set_set_nat > int,X: set_set_nat,Y: set_set_nat] :
( ( measur1680303437913248359at_int @ M4 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( member_set_set_nat @ X @ M4 )
=> ( ( member_set_set_nat @ Y @ M4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_250_increasingD,axiom,
! [M4: set_set_nat,F: set_nat > real,X: set_nat,Y: set_nat] :
( ( measur5905188192028735665t_real @ M4 @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( member_set_nat @ X @ M4 )
=> ( ( member_set_nat @ Y @ M4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_251_increasingD,axiom,
! [M4: set_set_nat,F: set_nat > set_set_nat,X: set_nat,Y: set_nat] :
( ( measur496615480034414785et_nat @ M4 @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( member_set_nat @ X @ M4 )
=> ( ( member_set_nat @ Y @ M4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_252_increasingD,axiom,
! [M4: set_set_nat,F: set_nat > set_nat,X: set_nat,Y: set_nat] :
( ( measur5248428813077667851et_nat @ M4 @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( member_set_nat @ X @ M4 )
=> ( ( member_set_nat @ Y @ M4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_253_increasingD,axiom,
! [M4: set_set_nat,F: set_nat > nat,X: set_nat,Y: set_nat] :
( ( measur1302623347068717141at_nat @ M4 @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( member_set_nat @ X @ M4 )
=> ( ( member_set_nat @ Y @ M4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_254_increasingD,axiom,
! [M4: set_set_nat,F: set_nat > int,X: set_nat,Y: set_nat] :
( ( measur1300132876559666865at_int @ M4 @ F )
=> ( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( member_set_nat @ X @ M4 )
=> ( ( member_set_nat @ Y @ M4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_255_le__left__mono,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ A2 )
=> ( ord_less_eq_real @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_256_le__left__mono,axiom,
! [X: set_set_nat,Y: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_257_le__left__mono,axiom,
! [X: set_nat,Y: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ A2 )
=> ( ord_less_eq_set_nat @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_258_le__left__mono,axiom,
! [X: nat,Y: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ A2 )
=> ( ord_less_eq_nat @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_259_le__left__mono,axiom,
! [X: int,Y: int,A2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ A2 )
=> ( ord_less_eq_int @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_260_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_real
= ( ^ [X6: $o > real,Y7: $o > real] :
( ( ord_less_eq_real @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_real @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_261_le__rel__bool__arg__iff,axiom,
( ord_le6539261115178940645et_nat
= ( ^ [X6: $o > set_set_nat,Y7: $o > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le6893508408891458716et_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_262_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X6: $o > set_nat,Y7: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_263_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X6: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_264_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_int
= ( ^ [X6: $o > int,Y7: $o > int] :
( ( ord_less_eq_int @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_int @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_265_verit__la__disequality,axiom,
! [A2: real,B3: real] :
( ( A2 = B3 )
| ~ ( ord_less_eq_real @ A2 @ B3 )
| ~ ( ord_less_eq_real @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_266_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_267_verit__la__disequality,axiom,
! [A2: int,B3: int] :
( ( A2 = B3 )
| ~ ( ord_less_eq_int @ A2 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_268_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M: nat] :
( ( P @ M )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_269_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_270_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_271_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_272_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_273_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_274_first__assumptions_Ov__mono,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_275_enumerate__mono__le__iff,axiom,
! [S: set_nat,M5: nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M5 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_276_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_277_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_278_eq__imp__le,axiom,
! [M5: nat,N3: nat] :
( ( M5 = N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% eq_imp_le
thf(fact_279_le__antisym,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M5 )
=> ( M5 = N3 ) ) ) ).
% le_antisym
thf(fact_280_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_281_nat__le__linear,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
| ( ord_less_eq_nat @ N3 @ M5 ) ) ).
% nat_le_linear
thf(fact_282_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_283_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_284_enumerate__in__set,axiom,
! [S: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ S ) ) ).
% enumerate_in_set
thf(fact_285_enumerate__Ex,axiom,
! [S: set_nat,S3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S3 @ S )
=> ? [N4: nat] :
( ( infini8530281810654367211te_nat @ S @ N4 )
= S3 ) ) ) ).
% enumerate_Ex
thf(fact_286_le__enumerate,axiom,
! [S: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N3 @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ).
% le_enumerate
thf(fact_287_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_288_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_289_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_complex,C2: nat] :
( ! [G2: set_complex] :
( ( ord_le211207098394363844omplex @ G2 @ F2 )
=> ( ( finite3207457112153483333omplex @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_complex @ G2 ) @ C2 ) ) )
=> ( ( finite3207457112153483333omplex @ F2 )
& ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_290_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_nat,C2: nat] :
( ! [G2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G2 @ F2 )
=> ( ( finite1152437895449049373et_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G2 ) @ C2 ) ) )
=> ( ( finite1152437895449049373et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_291_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_292_obtain__subset__with__card__n,axiom,
! [N3: nat,S: set_complex] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_complex @ S ) )
=> ~ ! [T3: set_complex] :
( ( ord_le211207098394363844omplex @ T3 @ S )
=> ( ( ( finite_card_complex @ T3 )
= N3 )
=> ~ ( finite3207457112153483333omplex @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_293_obtain__subset__with__card__n,axiom,
! [N3: nat,S: set_set_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_set_nat @ S ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S )
=> ( ( ( finite_card_set_nat @ T3 )
= N3 )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_294_obtain__subset__with__card__n,axiom,
! [N3: nat,S: set_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N3 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_295_exists__subset__between,axiom,
! [A: set_complex,N3: nat,C2: set_complex] :
( ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( finite_card_complex @ C2 ) )
=> ( ( ord_le211207098394363844omplex @ A @ C2 )
=> ( ( finite3207457112153483333omplex @ C2 )
=> ? [B6: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B6 )
& ( ord_le211207098394363844omplex @ B6 @ C2 )
& ( ( finite_card_complex @ B6 )
= N3 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_296_exists__subset__between,axiom,
! [A: set_set_nat,N3: nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( finite_card_set_nat @ C2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( finite1152437895449049373et_nat @ C2 )
=> ? [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ C2 )
& ( ( finite_card_set_nat @ B6 )
= N3 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_297_exists__subset__between,axiom,
! [A: set_nat,N3: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N3 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_298_card__seteq,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ B ) @ ( finite_card_complex @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_299_card__seteq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_set_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_300_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_301_card__mono,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B ) ) ) ) ).
% card_mono
thf(fact_302_card__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% card_mono
thf(fact_303_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_304_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_305_finite__subset__Union,axiom,
! [A: set_set_nat,B7: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ ( comple548664676211718543et_nat @ B7 ) )
=> ~ ! [F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( ord_le9131159989063066194et_nat @ F3 @ B7 )
=> ~ ( ord_le6893508408891458716et_nat @ A @ ( comple548664676211718543et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_306_finite__subset__Union,axiom,
! [A: set_nat,B7: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B7 ) )
=> ~ ! [F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ B7 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_307_finite__Union,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [M6: set_set_nat] :
( ( member_set_set_nat @ M6 @ A )
=> ( finite1152437895449049373et_nat @ M6 ) )
=> ( finite1152437895449049373et_nat @ ( comple548664676211718543et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_308_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M6: set_nat] :
( ( member_set_nat @ M6 @ A )
=> ( finite_finite_nat @ M6 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_309_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_310_card__numbers,axiom,
! [N3: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N3 ) )
= N3 ) ).
% card_numbers
thf(fact_311_card__subset__eq,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ( finite_card_complex @ A )
= ( finite_card_complex @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_312_card__subset__eq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( finite_card_set_nat @ A )
= ( finite_card_set_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_313_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_314_infinite__arbitrarily__large,axiom,
! [A: set_complex,N3: nat] :
( ~ ( finite3207457112153483333omplex @ A )
=> ? [B6: set_complex] :
( ( finite3207457112153483333omplex @ B6 )
& ( ( finite_card_complex @ B6 )
= N3 )
& ( ord_le211207098394363844omplex @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_315_infinite__arbitrarily__large,axiom,
! [A: set_set_nat,N3: nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ( finite_card_set_nat @ B6 )
= N3 )
& ( ord_le6893508408891458716et_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_316_infinite__arbitrarily__large,axiom,
! [A: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N3 )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_317_card__le__if__inj__on__rel,axiom,
! [B: set_complex,A: set_nat,R: nat > complex > $o] :
( ( finite3207457112153483333omplex @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: complex] :
( ( member_complex @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: complex] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_complex @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_318_card__le__if__inj__on__rel,axiom,
! [B: set_complex,A: set_complex,R: complex > complex > $o] :
( ( finite3207457112153483333omplex @ B )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ? [B8: complex] :
( ( member_complex @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: complex,A22: complex,B5: complex] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_complex @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_319_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_320_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_complex,R: complex > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: complex,A22: complex,B5: nat] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_321_card__le__if__inj__on__rel,axiom,
! [B: set_complex,A: set_set_nat,R: set_nat > complex > $o] :
( ( finite3207457112153483333omplex @ B )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [B8: complex] :
( ( member_complex @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B5: complex] :
( ( member_set_nat @ A1 @ A )
=> ( ( member_set_nat @ A22 @ A )
=> ( ( member_complex @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_322_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_set_nat,R: set_nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B5: nat] :
( ( member_set_nat @ A1 @ A )
=> ( ( member_set_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_323_card__le__if__inj__on__rel,axiom,
! [B: set_set_nat,A: set_nat,R: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: set_nat] :
( ( member_set_nat @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_324_card__le__if__inj__on__rel,axiom,
! [B: set_set_nat,A: set_complex,R: complex > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ? [B8: set_nat] :
( ( member_set_nat @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: complex,A22: complex,B5: set_nat] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_325_card__le__if__inj__on__rel,axiom,
! [B: set_complex,A: set_set_set_nat,R: set_set_nat > complex > $o] :
( ( finite3207457112153483333omplex @ B )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A )
=> ? [B8: complex] :
( ( member_complex @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B5: complex] :
( ( member_set_set_nat @ A1 @ A )
=> ( ( member_set_set_nat @ A22 @ A )
=> ( ( member_complex @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_326_card__le__if__inj__on__rel,axiom,
! [B: set_set_set_nat,A: set_nat,R: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B8: set_set_nat] :
( ( member_set_set_nat @ B8 @ B )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_set_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_327_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_328_finite__UnionD,axiom,
! [A: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ ( comple548664676211718543et_nat @ A ) )
=> ( finite6739761609112101331et_nat @ A ) ) ).
% finite_UnionD
thf(fact_329_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_330_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_331_Union__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_mono
thf(fact_332_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_333_ex__card,axiom,
! [N3: nat,A: set_complex] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_complex @ A ) )
=> ? [S4: set_complex] :
( ( ord_le211207098394363844omplex @ S4 @ A )
& ( ( finite_card_complex @ S4 )
= N3 ) ) ) ).
% ex_card
thf(fact_334_ex__card,axiom,
! [N3: nat,A: set_set_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_set_nat @ A ) )
=> ? [S4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S4 @ A )
& ( ( finite_card_set_nat @ S4 )
= N3 ) ) ) ).
% ex_card
thf(fact_335_ex__card,axiom,
! [N3: nat,A: set_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_nat @ A ) )
=> ? [S4: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ A )
& ( ( finite_card_nat @ S4 )
= N3 ) ) ) ).
% ex_card
thf(fact_336_le__cSup__finite,axiom,
! [X2: set_real,X: real] :
( ( finite_finite_real @ X2 )
=> ( ( member_real @ X @ X2 )
=> ( ord_less_eq_real @ X @ ( comple1385675409528146559p_real @ X2 ) ) ) ) ).
% le_cSup_finite
thf(fact_337_le__cSup__finite,axiom,
! [X2: set_set_set_nat,X: set_set_nat] :
( ( finite6739761609112101331et_nat @ X2 )
=> ( ( member_set_set_nat @ X @ X2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( comple548664676211718543et_nat @ X2 ) ) ) ) ).
% le_cSup_finite
thf(fact_338_le__cSup__finite,axiom,
! [X2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ X2 )
=> ( ( member_set_nat @ X @ X2 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ X2 ) ) ) ) ).
% le_cSup_finite
thf(fact_339_le__cSup__finite,axiom,
! [X2: set_int,X: int] :
( ( finite_finite_int @ X2 )
=> ( ( member_int @ X @ X2 )
=> ( ord_less_eq_int @ X @ ( complete_Sup_Sup_int @ X2 ) ) ) ) ).
% le_cSup_finite
thf(fact_340_le__cSup__finite,axiom,
! [X2: set_nat,X: nat] :
( ( finite_finite_nat @ X2 )
=> ( ( member_nat @ X @ X2 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X2 ) ) ) ) ).
% le_cSup_finite
thf(fact_341_Sup__subset__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_342_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_343_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ C2 )
& ( member_nat @ A @ X5 ) ) ) ) ).
% Union_iff
thf(fact_344_Union__iff,axiom,
! [A: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) )
= ( ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ C2 )
& ( member_set_set_nat @ A @ X5 ) ) ) ) ).
% Union_iff
thf(fact_345_Union__iff,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
= ( ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ C2 )
& ( member_set_nat @ A @ X5 ) ) ) ) ).
% Union_iff
thf(fact_346_UnionI,axiom,
! [X2: set_set_set_nat,C2: set_set_set_set_nat,A: set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ C2 )
=> ( ( member_set_set_nat @ A @ X2 )
=> ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_347_UnionI,axiom,
! [X2: set_set_nat,C2: set_set_set_nat,A: set_nat] :
( ( member_set_set_nat @ X2 @ C2 )
=> ( ( member_set_nat @ A @ X2 )
=> ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_348_UnionI,axiom,
! [X2: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X2 @ C2 )
=> ( ( member_nat @ A @ X2 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_349_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_350_UnionE,axiom,
! [A: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) )
=> ~ ! [X7: set_set_set_nat] :
( ( member_set_set_nat @ A @ X7 )
=> ~ ( member2946998982187404937et_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_351_UnionE,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
=> ~ ! [X7: set_set_nat] :
( ( member_set_nat @ A @ X7 )
=> ~ ( member_set_set_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_352_Sup__eqI,axiom,
! [A: set_set_set_nat,X: set_set_nat] :
( ! [Y3: set_set_nat] :
( ( member_set_set_nat @ Y3 @ A )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X ) )
=> ( ! [Y3: set_set_nat] :
( ! [Z3: set_set_nat] :
( ( member_set_set_nat @ Z3 @ A )
=> ( ord_le6893508408891458716et_nat @ Z3 @ Y3 ) )
=> ( ord_le6893508408891458716et_nat @ X @ Y3 ) )
=> ( ( comple548664676211718543et_nat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_353_Sup__eqI,axiom,
! [A: set_set_nat,X: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [Y3: set_nat] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ A )
=> ( ord_less_eq_set_nat @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_354_Sup__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
& ( ord_le6893508408891458716et_nat @ A5 @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_355_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_356_Sup__least,axiom,
! [A: set_set_set_nat,Z2: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( ord_le6893508408891458716et_nat @ X3 @ Z2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_357_Sup__least,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_358_Sup__upper,axiom,
! [X: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( ord_le6893508408891458716et_nat @ X @ ( comple548664676211718543et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_359_Sup__upper,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_360_Sup__le__iff,axiom,
! [A: set_set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ B3 )
= ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( ord_le6893508408891458716et_nat @ X5 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_361_Sup__le__iff,axiom,
! [A: set_set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B3 )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ X5 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_362_Sup__upper2,axiom,
! [U: set_set_nat,A: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A )
=> ( ( ord_le6893508408891458716et_nat @ V @ U )
=> ( ord_le6893508408891458716et_nat @ V @ ( comple548664676211718543et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_363_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_364_cSup__eq__maximum,axiom,
! [Z2: real,X2: set_real] :
( ( member_real @ Z2 @ X2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ( comple1385675409528146559p_real @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_365_cSup__eq__maximum,axiom,
! [Z2: set_set_nat,X2: set_set_set_nat] :
( ( member_set_set_nat @ Z2 @ X2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ Z2 ) )
=> ( ( comple548664676211718543et_nat @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_366_cSup__eq__maximum,axiom,
! [Z2: set_nat,X2: set_set_nat] :
( ( member_set_nat @ Z2 @ X2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_367_cSup__eq__maximum,axiom,
! [Z2: int,X2: set_int] :
( ( member_int @ Z2 @ X2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_int @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_368_cSup__eq__maximum,axiom,
! [Z2: nat,X2: set_nat] :
( ( member_nat @ Z2 @ X2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X2 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_369_cSup__eq,axiom,
! [X2: set_real,A2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ( ord_less_eq_real @ X3 @ A2 ) )
=> ( ! [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X2 )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
=> ( ord_less_eq_real @ A2 @ Y3 ) )
=> ( ( comple1385675409528146559p_real @ X2 )
= A2 ) ) ) ).
% cSup_eq
thf(fact_370_cSup__eq,axiom,
! [X2: set_int,A2: int] :
( ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ( ord_less_eq_int @ X3 @ A2 ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X2 )
=> ( ord_less_eq_int @ X4 @ Y3 ) )
=> ( ord_less_eq_int @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_int @ X2 )
= A2 ) ) ) ).
% cSup_eq
thf(fact_371_Union__subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ B )
& ( ord_le6893508408891458716et_nat @ X3 @ Y6 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_372_Union__subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B )
& ( ord_less_eq_set_nat @ X3 @ Y6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_373_Union__upper,axiom,
! [B: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ ( comple548664676211718543et_nat @ A ) ) ) ).
% Union_upper
thf(fact_374_Union__upper,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_375_Union__least,axiom,
! [A: set_set_set_nat,C2: set_set_nat] :
( ! [X7: set_set_nat] :
( ( member_set_set_nat @ X7 @ A )
=> ( ord_le6893508408891458716et_nat @ X7 @ C2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_376_Union__least,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ X7 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_377_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_378_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_379_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_380_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_381_finite__enum__subset,axiom,
! [X2: set_nat,Y2: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X2 ) )
=> ( ( infini8530281810654367211te_nat @ X2 @ I3 )
= ( infini8530281810654367211te_nat @ Y2 @ I3 ) ) )
=> ( ( finite_finite_nat @ X2 )
=> ( ( finite_finite_nat @ Y2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X2 ) @ ( finite_card_nat @ Y2 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_382_finite__le__enumerate,axiom,
! [S: set_nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N3 @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ) ).
% finite_le_enumerate
thf(fact_383__092_060G_062__def,axiom,
( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
= ( collect_set_set_nat
@ ^ [G3: set_set_nat] : ( ord_le6893508408891458716et_nat @ G3 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% \<G>_def
thf(fact_384__092_060K_062__def,axiom,
( ( clique3326749438856946062irst_K @ k )
= ( collect_set_set_nat
@ ^ [K4: set_set_nat] :
( ( member_set_set_nat @ K4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K4 ) )
= k )
& ( K4
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K4 ) @ ( clique5033774636164728513irst_v @ K4 ) ) ) ) ) ) ).
% \<K>_def
thf(fact_385_sameprod___092_060G_062,axiom,
! [X2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X2 ) )
=> ( member_set_set_nat @ ( clique6722202388162463298od_nat @ X2 @ X2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% sameprod_\<G>
thf(fact_386_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_387_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X5: set_set_nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_388_empty__Collect__eq,axiom,
! [P: complex > $o] :
( ( bot_bot_set_complex
= ( collect_complex @ P ) )
= ( ! [X5: complex] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_389_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X5: set_nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_390_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X5: nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_391_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X5: set_set_nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_392_Collect__empty__eq,axiom,
! [P: complex > $o] :
( ( ( collect_complex @ P )
= bot_bot_set_complex )
= ( ! [X5: complex] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_393_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X5: set_nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_394_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X5: nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_395_all__not__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ! [X5: set_set_nat] :
~ ( member_set_set_nat @ X5 @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_396_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X5: set_nat] :
~ ( member_set_nat @ X5 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_397_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X5: nat] :
~ ( member_nat @ X5 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_398_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_399_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_400_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_401_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_402_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_403_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_404_k2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ k ).
% k2
thf(fact_405_finite__Collect__disjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X5: set_set_nat] :
( ( P @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
& ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_406_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( P @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_407_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( P @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_408_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( P @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_409_finite__Collect__conjI,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
| ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ Q ) ) )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [X5: set_set_nat] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_410_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_411_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_412_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_413_l2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ l ).
% l2
thf(fact_414_p,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).
% p
thf(fact_415_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_416_v__mem__sub,axiom,
! [E: set_nat,G: set_set_nat] :
( ( ( finite_card_nat @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( member_set_nat @ E @ G )
=> ( ord_less_eq_set_nat @ E @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_mem_sub
thf(fact_417_m2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ k ) ).
% m2
thf(fact_418_subset__empty,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_419_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_420_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_421_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_422_Sup__bot__conv_I2_J,axiom,
! [A: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( comple548664676211718543et_nat @ A ) )
= ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_423_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_424_Sup__bot__conv_I1_J,axiom,
! [A: set_set_set_nat] :
( ( ( comple548664676211718543et_nat @ A )
= bot_bot_set_set_nat )
= ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_425_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_426_lessThan__iff,axiom,
! [I2: set_set_nat,K: set_set_nat] :
( ( member_set_set_nat @ I2 @ ( set_or6631954706645296601et_nat @ K ) )
= ( ord_less_set_set_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_427_lessThan__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_428_lessThan__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_429_lessThan__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_430_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_431_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_432_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_433_card__Collect__less__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N3 ) ) )
= N3 ) ).
% card_Collect_less_nat
thf(fact_434_Sup__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% Sup_empty
thf(fact_435_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_436_finite__Collect__subsets,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_437_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_438_enumerate__mono__iff,axiom,
! [S: set_nat,M5: nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M5 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ) ).
% enumerate_mono_iff
thf(fact_439_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M5: nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M5 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M5 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_440_v__sameprod,axiom,
! [X2: set_nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X2 ) )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ X2 @ X2 ) )
= X2 ) ) ).
% v_sameprod
thf(fact_441_v__numbers2,axiom,
! [X: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) )
= ( clique3652268606331196573umbers @ X ) ) ) ).
% v_numbers2
thf(fact_442_less__cSupD,axiom,
! [X2: set_int,Z2: int] :
( ( X2 != bot_bot_set_int )
=> ( ( ord_less_int @ Z2 @ ( complete_Sup_Sup_int @ X2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ X2 )
& ( ord_less_int @ Z2 @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_443_less__cSupD,axiom,
! [X2: set_real,Z2: real] :
( ( X2 != bot_bot_set_real )
=> ( ( ord_less_real @ Z2 @ ( comple1385675409528146559p_real @ X2 ) )
=> ? [X3: real] :
( ( member_real @ X3 @ X2 )
& ( ord_less_real @ Z2 @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_444_less__cSupD,axiom,
! [X2: set_nat,Z2: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z2 @ ( complete_Sup_Sup_nat @ X2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X2 )
& ( ord_less_nat @ Z2 @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_445_less__cSupE,axiom,
! [Y: int,X2: set_int] :
( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X2 ) )
=> ( ( X2 != bot_bot_set_int )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ~ ( ord_less_int @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_446_less__cSupE,axiom,
! [Y: real,X2: set_real] :
( ( ord_less_real @ Y @ ( comple1385675409528146559p_real @ X2 ) )
=> ( ( X2 != bot_bot_set_real )
=> ~ ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ~ ( ord_less_real @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_447_less__cSupE,axiom,
! [Y: nat,X2: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X2 ) )
=> ( ( X2 != bot_bot_set_nat )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_448_Union__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% Union_empty
thf(fact_449_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_450_card__2__iff_H,axiom,
! [S: set_nat] :
( ( ( finite_card_nat @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ? [Y5: nat] :
( ( member_nat @ Y5 @ S )
& ( X5 != Y5 )
& ! [Z4: nat] :
( ( member_nat @ Z4 @ S )
=> ( ( Z4 = X5 )
| ( Z4 = Y5 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_451_card__2__iff_H,axiom,
! [S: set_complex] :
( ( ( finite_card_complex @ S )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ S )
& ? [Y5: complex] :
( ( member_complex @ Y5 @ S )
& ( X5 != Y5 )
& ! [Z4: complex] :
( ( member_complex @ Z4 @ S )
=> ( ( Z4 = X5 )
| ( Z4 = Y5 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_452_sameprod__altdef,axiom,
! [X2: set_complex] :
( ( clique7858167266224639776omplex @ X2 @ X2 )
= ( collect_set_complex
@ ^ [Y7: set_complex] :
( ( ord_le211207098394363844omplex @ Y7 @ X2 )
& ( ( finite_card_complex @ Y7 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_453_sameprod__altdef,axiom,
! [X2: set_set_nat] :
( ( clique8906516429304539640et_nat @ X2 @ X2 )
= ( collect_set_set_nat
@ ^ [Y7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y7 @ X2 )
& ( ( finite_card_set_nat @ Y7 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_454_sameprod__altdef,axiom,
! [X2: set_nat] :
( ( clique6722202388162463298od_nat @ X2 @ X2 )
= ( collect_set_nat
@ ^ [Y7: set_nat] :
( ( ord_less_eq_set_nat @ Y7 @ X2 )
& ( ( finite_card_nat @ Y7 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sameprod_altdef
thf(fact_455_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_456_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N4 )
& ~ ( P @ M7 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_457_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M7: nat] :
( ( ord_less_nat @ M7 @ N4 )
=> ( P @ M7 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_458_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_459_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less_nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_460_less__not__refl2,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ N3 @ M5 )
=> ( M5 != N3 ) ) ).
% less_not_refl2
thf(fact_461_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_462_nat__neq__iff,axiom,
! [M5: nat,N3: nat] :
( ( M5 != N3 )
= ( ( ord_less_nat @ M5 @ N3 )
| ( ord_less_nat @ N3 @ M5 ) ) ) ).
% nat_neq_iff
thf(fact_463_Iio__eq__empty__iff,axiom,
! [N3: nat] :
( ( ( set_ord_lessThan_nat @ N3 )
= bot_bot_set_nat )
= ( N3 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_464_lessThan__strict__subset__iff,axiom,
! [M5: int,N3: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M5 ) @ ( set_ord_lessThan_int @ N3 ) )
= ( ord_less_int @ M5 @ N3 ) ) ).
% lessThan_strict_subset_iff
thf(fact_465_lessThan__strict__subset__iff,axiom,
! [M5: real,N3: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M5 ) @ ( set_or5984915006950818249n_real @ N3 ) )
= ( ord_less_real @ M5 @ N3 ) ) ).
% lessThan_strict_subset_iff
thf(fact_466_lessThan__strict__subset__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M5 ) @ ( set_ord_lessThan_nat @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% lessThan_strict_subset_iff
thf(fact_467_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_468_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_469_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_470_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_471_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_472_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_473_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_474_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_475_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_476_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_477_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_478_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_479_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_480_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_481_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_482_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_483_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_484_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_485_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_486_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_487_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_488_order__less__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_489_order__less__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_490_order__less__subst2,axiom,
! [A2: int,B3: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_491_order__less__subst2,axiom,
! [A2: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_492_order__less__subst2,axiom,
! [A2: real,B3: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_493_order__less__subst2,axiom,
! [A2: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_494_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_495_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_496_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_497_order__less__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_498_order__less__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_499_order__less__subst1,axiom,
! [A2: int,F: real > int,B3: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_500_order__less__subst1,axiom,
! [A2: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_501_order__less__subst1,axiom,
! [A2: real,F: int > real,B3: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_502_order__less__subst1,axiom,
! [A2: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_503_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_504_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_505_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_506_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_507_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_508_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_509_ord__less__eq__subst,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_510_ord__less__eq__subst,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_511_ord__less__eq__subst,axiom,
! [A2: int,B3: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_512_ord__less__eq__subst,axiom,
! [A2: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_513_ord__less__eq__subst,axiom,
! [A2: real,B3: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_514_ord__less__eq__subst,axiom,
! [A2: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_515_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_516_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_517_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_518_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_519_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_520_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_521_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_522_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_523_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B3: real,C: real] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_524_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_525_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_526_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_527_order__less__asym_H,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_528_order__less__asym_H,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ~ ( ord_less_int @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_529_order__less__asym_H,axiom,
! [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
=> ~ ( ord_less_real @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_530_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_531_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_532_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_533_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_534_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_535_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_536_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_537_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_538_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_539_ex__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ? [X5: set_set_nat] : ( member_set_set_nat @ X5 @ A ) )
= ( A != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_540_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X5: set_nat] : ( member_set_nat @ X5 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_541_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X5: nat] : ( member_nat @ X5 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_542_empty__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat
@ ^ [X5: set_set_nat] : $false ) ) ).
% empty_def
thf(fact_543_empty__def,axiom,
( bot_bot_set_complex
= ( collect_complex
@ ^ [X5: complex] : $false ) ) ).
% empty_def
thf(fact_544_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X5: set_nat] : $false ) ) ).
% empty_def
thf(fact_545_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X5: nat] : $false ) ) ).
% empty_def
thf(fact_546_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y3: set_set_nat] :
~ ( member_set_set_nat @ Y3 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_547_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_548_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_549_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_550_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_551_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_552_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_553_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_554_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_555_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_556_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_557_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A2: real] :
( ( ord_less_real @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_558_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_559_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_560_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_561_dual__order_Ostrict__trans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_562_dual__order_Ostrict__trans,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_563_dual__order_Ostrict__trans,axiom,
! [B3: real,A2: real,C: real] :
( ( ord_less_real @ B3 @ A2 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_564_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_565_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_566_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_567_bot_Onot__eq__extremum,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
= ( ord_less_set_set_nat @ bot_bot_set_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_568_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_569_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_570_bot_Oextremum__strict,axiom,
! [A2: set_set_nat] :
~ ( ord_less_set_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% bot.extremum_strict
thf(fact_571_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_572_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_573_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_574_order_Ostrict__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_575_order_Ostrict__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_576_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_577_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B3: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_578_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B3: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_579_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X8: nat] : ( P3 @ X8 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_580_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_581_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_582_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_583_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_584_dual__order_Oasym,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ~ ( ord_less_int @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_585_dual__order_Oasym,axiom,
! [B3: real,A2: real] :
( ( ord_less_real @ B3 @ A2 )
=> ~ ( ord_less_real @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_586_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_587_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_588_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_589_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_590_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_591_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_592_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_593_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_594_ord__less__eq__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_595_ord__less__eq__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_596_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_597_ord__eq__less__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( A2 = B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_598_ord__eq__less__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( A2 = B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_599_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_600_order_Oasym,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ~ ( ord_less_int @ B3 @ A2 ) ) ).
% order.asym
thf(fact_601_order_Oasym,axiom,
! [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
=> ~ ( ord_less_real @ B3 @ A2 ) ) ).
% order.asym
thf(fact_602_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_603_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_604_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_605_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z5: real] :
( ( ord_less_real @ X @ Z5 )
& ( ord_less_real @ Z5 @ Y ) ) ) ).
% dense
thf(fact_606_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_607_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_608_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_609_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_610_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_611_pigeonhole__infinite__rel,axiom,
! [A: set_set_set_nat,B: set_nat,R2: set_set_nat > nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_612_pigeonhole__infinite__rel,axiom,
! [A: set_complex,B: set_nat,R2: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_613_pigeonhole__infinite__rel,axiom,
! [A: set_set_set_nat,B: set_set_nat,R2: set_set_nat > set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ~ ( finite6739761609112101331et_nat
@ ( collect_set_set_nat
@ ^ [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_614_pigeonhole__infinite__rel,axiom,
! [A: set_complex,B: set_set_nat,R2: complex > set_nat > $o] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_615_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_616_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_set_nat,R2: nat > set_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_617_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_nat,R2: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_618_pigeonhole__infinite__rel,axiom,
! [A: set_set_nat,B: set_set_nat,R2: set_nat > set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_619_not__finite__existsD,axiom,
! [P: set_set_nat > $o] :
( ~ ( finite6739761609112101331et_nat @ ( collect_set_set_nat @ P ) )
=> ? [X_12: set_set_nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_620_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_12: complex] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_621_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_12: nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_622_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_12: set_nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_623_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_624_lessThan__def,axiom,
( set_or6631954706645296601et_nat
= ( ^ [U2: set_set_nat] :
( collect_set_set_nat
@ ^ [X5: set_set_nat] : ( ord_less_set_set_nat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_625_lessThan__def,axiom,
( set_or7194820819169546315omplex
= ( ^ [U2: complex] :
( collect_complex
@ ^ [X5: complex] : ( ord_less_complex @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_626_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X5: int] : ( ord_less_int @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_627_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X5: real] : ( ord_less_real @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_628_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_629_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_630_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_631_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_632_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_633_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X5: set_set_nat] : ( member_set_set_nat @ X5 @ A3 )
@ ^ [X5: set_set_nat] : ( member_set_set_nat @ X5 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_634_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A3 )
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_635_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A3 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_636_Collect__subset,axiom,
! [A: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_637_Collect__subset,axiom,
! [A: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_638_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_639_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A )
& ( P @ X5 ) ) )
@ A ) ).
% Collect_subset
thf(fact_640_infinite__growing,axiom,
! [X2: set_nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X2 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X2 ) ) ) ).
% infinite_growing
thf(fact_641_infinite__growing,axiom,
! [X2: set_int] :
( ( X2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X2 )
& ( ord_less_int @ X3 @ Xa ) ) )
=> ~ ( finite_finite_int @ X2 ) ) ) ).
% infinite_growing
thf(fact_642_infinite__growing,axiom,
! [X2: set_real] :
( ( X2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X2 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X2 ) ) ) ).
% infinite_growing
thf(fact_643_ex__min__if__finite,axiom,
! [S: set_set_nat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( S != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ S )
& ~ ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ S )
& ( ord_less_set_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_644_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_645_ex__min__if__finite,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S )
& ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_646_ex__min__if__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_647_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_648_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_649_finite__Sup__less__iff,axiom,
! [X2: set_int,A2: int] :
( ( finite_finite_int @ X2 )
=> ( ( X2 != bot_bot_set_int )
=> ( ( ord_less_int @ ( complete_Sup_Sup_int @ X2 ) @ A2 )
= ( ! [X5: int] :
( ( member_int @ X5 @ X2 )
=> ( ord_less_int @ X5 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_650_finite__Sup__less__iff,axiom,
! [X2: set_real,A2: real] :
( ( finite_finite_real @ X2 )
=> ( ( X2 != bot_bot_set_real )
=> ( ( ord_less_real @ ( comple1385675409528146559p_real @ X2 ) @ A2 )
= ( ! [X5: real] :
( ( member_real @ X5 @ X2 )
=> ( ord_less_real @ X5 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_651_finite__Sup__less__iff,axiom,
! [X2: set_nat,A2: nat] :
( ( finite_finite_nat @ X2 )
=> ( ( X2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X2 ) @ A2 )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ X2 )
=> ( ord_less_nat @ X5 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_652_bot_Oextremum,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% bot.extremum
thf(fact_653_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_654_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_655_bot_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_656_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_657_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_658_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_659_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_660_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_661_enumerate__mono,axiom,
! [M5: nat,N3: nat,S: set_nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M5 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ) ).
% enumerate_mono
thf(fact_662_verit__comp__simplify1_I3_J,axiom,
! [B9: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B9 @ A6 ) )
= ( ord_less_real @ A6 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_663_verit__comp__simplify1_I3_J,axiom,
! [B9: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B9 @ A6 ) )
= ( ord_less_nat @ A6 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_664_verit__comp__simplify1_I3_J,axiom,
! [B9: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B9 @ A6 ) )
= ( ord_less_int @ A6 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_665_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_666_leD,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ~ ( ord_less_set_set_nat @ X @ Y ) ) ).
% leD
thf(fact_667_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_668_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_669_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_670_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_671_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_672_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_673_nless__le,axiom,
! [A2: real,B3: real] :
( ( ~ ( ord_less_real @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_674_nless__le,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ~ ( ord_less_set_set_nat @ A2 @ B3 ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_675_nless__le,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_676_nless__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_677_nless__le,axiom,
! [A2: int,B3: int] :
( ( ~ ( ord_less_int @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_678_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_679_antisym__conv1,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ~ ( ord_less_set_set_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_680_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_681_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_682_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_683_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_684_antisym__conv2,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_685_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_686_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_687_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_688_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_689_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_690_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_691_less__le__not__le,axiom,
( ord_less_set_set_nat
= ( ^ [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
& ~ ( ord_le6893508408891458716et_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_692_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_693_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_694_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_695_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_696_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_697_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_698_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_699_order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_700_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_701_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_702_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_703_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_704_order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_705_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_706_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_707_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_708_order_Ostrict__trans1,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_709_order_Ostrict__trans1,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_less_set_set_nat @ B3 @ C )
=> ( ord_less_set_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_710_order_Ostrict__trans1,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_711_order_Ostrict__trans1,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_712_order_Ostrict__trans1,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_713_order_Ostrict__trans2,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_714_order_Ostrict__trans2,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C )
=> ( ord_less_set_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_715_order_Ostrict__trans2,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_716_order_Ostrict__trans2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_717_order_Ostrict__trans2,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_718_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_719_order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ~ ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_720_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_721_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_722_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_723_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_724_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_725_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_726_dual__order_Oorder__iff__strict,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( ord_less_set_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_727_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_728_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_729_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_730_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_731_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_732_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_733_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_734_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_735_dual__order_Ostrict__trans1,axiom,
! [B3: real,A2: real,C: real] :
( ( ord_less_eq_real @ B3 @ A2 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_736_dual__order_Ostrict__trans1,axiom,
! [B3: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( ord_less_set_set_nat @ C @ B3 )
=> ( ord_less_set_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_737_dual__order_Ostrict__trans1,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_738_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_739_dual__order_Ostrict__trans1,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_740_dual__order_Ostrict__trans2,axiom,
! [B3: real,A2: real,C: real] :
( ( ord_less_real @ B3 @ A2 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_741_dual__order_Ostrict__trans2,axiom,
! [B3: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ B3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B3 )
=> ( ord_less_set_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_742_dual__order_Ostrict__trans2,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_743_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_744_dual__order_Ostrict__trans2,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_745_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_746_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ~ ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_747_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_748_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_749_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_750_order_Ostrict__implies__order,axiom,
! [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ord_less_eq_real @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_751_order_Ostrict__implies__order,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_752_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_753_order_Ostrict__implies__order,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_754_order_Ostrict__implies__order,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ord_less_eq_int @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_755_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A2: real] :
( ( ord_less_real @ B3 @ A2 )
=> ( ord_less_eq_real @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_756_dual__order_Ostrict__implies__order,axiom,
! [B3: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B3 @ A2 )
=> ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_757_dual__order_Ostrict__implies__order,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B3 @ A2 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_758_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_759_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ord_less_eq_int @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_760_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_761_order__le__less,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_less_set_set_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_762_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X5: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_763_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_764_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_765_order__less__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_766_order__less__le,axiom,
( ord_less_set_set_nat
= ( ^ [X5: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_767_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X5: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_768_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_769_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_770_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_771_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_772_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_773_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_774_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_775_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_776_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_777_order__less__imp__le,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_778_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_779_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_780_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_781_order__le__neq__trans,axiom,
! [A2: real,B3: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_real @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_782_order__le__neq__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_set_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_783_order__le__neq__trans,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_784_order__le__neq__trans,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_785_order__le__neq__trans,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_int @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_786_order__neq__le__trans,axiom,
! [A2: real,B3: real] :
( ( A2 != B3 )
=> ( ( ord_less_eq_real @ A2 @ B3 )
=> ( ord_less_real @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_787_order__neq__le__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 != B3 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_less_set_set_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_788_order__neq__le__trans,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_789_order__neq__le__trans,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_790_order__neq__le__trans,axiom,
! [A2: int,B3: int] :
( ( A2 != B3 )
=> ( ( ord_less_eq_int @ A2 @ B3 )
=> ( ord_less_int @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_791_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_792_order__le__less__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_less_set_set_nat @ Y @ Z2 )
=> ( ord_less_set_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_793_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_794_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_795_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_796_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_797_order__less__le__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_less_set_set_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
=> ( ord_less_set_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_798_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_799_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_800_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_801_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_802_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B3: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_803_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_804_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_805_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_806_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_807_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_808_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_809_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B3: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_810_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_811_order__le__less__subst2,axiom,
! [A2: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_812_order__le__less__subst2,axiom,
! [A2: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_813_order__le__less__subst2,axiom,
! [A2: real,B3: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_814_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_815_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_816_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_817_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_818_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_819_order__le__less__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_820_order__le__less__subst2,axiom,
! [A2: real,B3: real,F: real > set_nat,C: set_nat] :
( ( ord_less_eq_real @ A2 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_821_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_822_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_823_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B3: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_824_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_825_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_826_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_827_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B3: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_828_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_829_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_830_order__less__le__subst1,axiom,
! [A2: set_nat,F: real > set_nat,B3: real,C: real] :
( ( ord_less_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_831_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_832_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_833_order__less__le__subst2,axiom,
! [A2: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_834_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_835_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_836_order__less__le__subst2,axiom,
! [A2: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_837_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_838_order__less__le__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_839_order__less__le__subst2,axiom,
! [A2: real,B3: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_840_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_841_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_842_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_843_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_844_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_845_order__le__imp__less__or__eq,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_less_set_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_846_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_847_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_848_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_849_complete__interval,axiom,
! [A2: real,B3: real,P: real > $o] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A2 @ C3 )
& ( ord_less_eq_real @ C3 @ B3 )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_real @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_850_complete__interval,axiom,
! [A2: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_851_complete__interval,axiom,
! [A2: int,B3: int,P: int > $o] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B3 )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_852_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_853_less__imp__le__nat,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_854_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_855_less__or__eq__imp__le,axiom,
! [M5: nat,N3: nat] :
( ( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_856_le__neq__implies__less,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( M5 != N3 )
=> ( ord_less_nat @ M5 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_857_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_858_finite__transitivity__chain,axiom,
! [A: set_set_set_nat,R2: set_set_nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [X3: set_set_nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: set_set_nat,Y3: set_set_nat,Z5: set_set_nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z5 )
=> ( R2 @ X3 @ Z5 ) ) )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ? [Y6: set_set_nat] :
( ( member_set_set_nat @ Y6 @ A )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A = bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_859_finite__transitivity__chain,axiom,
! [A: set_set_nat,R2: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X3: set_nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: set_nat,Y3: set_nat,Z5: set_nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z5 )
=> ( R2 @ X3 @ Z5 ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_860_finite__transitivity__chain,axiom,
! [A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z5: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z5 )
=> ( R2 @ X3 @ Z5 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_861_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_862_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_863_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_864_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_865_Union__empty__conv,axiom,
! [A: set_set_set_nat] :
( ( ( comple548664676211718543et_nat @ A )
= bot_bot_set_set_nat )
= ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_866_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_867_empty__Union__conv,axiom,
! [A: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( comple548664676211718543et_nat @ A ) )
= ( ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_868_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( X5 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_869_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M2: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N2 )
=> ( ord_less_nat @ X5 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_870_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N: nat] :
( ( ord_less_nat @ M2 @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_871_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N3: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N3 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_872_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M: nat] :
( ( ord_less_nat @ K @ M )
=> ? [N6: nat] :
( ( ord_less_nat @ M @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_873_Fpow__def,axiom,
( finite_Fpow_set_nat
= ( ^ [A3: set_set_nat] :
( collect_set_set_nat
@ ^ [X6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X6 @ A3 )
& ( finite1152437895449049373et_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_874_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [X6: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ A3 )
& ( finite_finite_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_875_finite__enumerate__mono,axiom,
! [M5: nat,N3: nat,S: set_nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M5 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_876_first__assumptions_Ov__mem__sub,axiom,
! [L: nat,P2: nat,K: nat,E: set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ( finite_card_nat @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( member_set_nat @ E @ G )
=> ( ord_less_eq_set_nat @ E @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_mem_sub
thf(fact_877_empty__in__Fpow,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ bot_bot_set_set_nat @ ( finite_Fpow_set_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_878_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_879_Fpow__not__empty,axiom,
! [A: set_nat] :
( ( finite_Fpow_nat @ A )
!= bot_bot_set_set_nat ) ).
% Fpow_not_empty
thf(fact_880_Graphs__def,axiom,
( clique5786534781347292306Graphs
= ( ^ [V2: set_nat] :
( collect_set_set_nat
@ ^ [G3: set_set_nat] : ( ord_le6893508408891458716et_nat @ G3 @ ( clique6722202388162463298od_nat @ V2 @ V2 ) ) ) ) ) ).
% Graphs_def
thf(fact_881_first__assumptions_Ov__sameprod,axiom,
! [L: nat,P2: nat,K: nat,X2: set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X2 ) )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ X2 @ X2 ) )
= X2 ) ) ) ).
% first_assumptions.v_sameprod
thf(fact_882_first__assumptions_Ov__numbers2,axiom,
! [L: nat,P2: nat,K: nat,X: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) )
= ( clique3652268606331196573umbers @ X ) ) ) ) ).
% first_assumptions.v_numbers2
thf(fact_883_finite__imp__Sup__less,axiom,
! [X2: set_int,X: int,A2: int] :
( ( finite_finite_int @ X2 )
=> ( ( member_int @ X @ X2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ( ord_less_int @ X3 @ A2 ) )
=> ( ord_less_int @ ( complete_Sup_Sup_int @ X2 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_884_finite__imp__Sup__less,axiom,
! [X2: set_real,X: real,A2: real] :
( ( finite_finite_real @ X2 )
=> ( ( member_real @ X @ X2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ( ord_less_real @ X3 @ A2 ) )
=> ( ord_less_real @ ( comple1385675409528146559p_real @ X2 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_885_finite__imp__Sup__less,axiom,
! [X2: set_nat,X: nat,A2: nat] :
( ( finite_finite_nat @ X2 )
=> ( ( member_nat @ X @ X2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ( ord_less_nat @ X3 @ A2 ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X2 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_886_finite__has__maximal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_887_finite__has__maximal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_888_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_889_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_890_finite__has__maximal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_891_finite__has__minimal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_892_finite__has__minimal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_893_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_894_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_895_finite__has__minimal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_896_cSup__eq__non__empty,axiom,
! [X2: set_real,A2: real] :
( ( X2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ( ord_less_eq_real @ X3 @ A2 ) )
=> ( ! [Y3: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X2 )
=> ( ord_less_eq_real @ X4 @ Y3 ) )
=> ( ord_less_eq_real @ A2 @ Y3 ) )
=> ( ( comple1385675409528146559p_real @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_897_cSup__eq__non__empty,axiom,
! [X2: set_set_set_nat,A2: set_set_nat] :
( ( X2 != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ A2 ) )
=> ( ! [Y3: set_set_nat] :
( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ X2 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Y3 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ Y3 ) )
=> ( ( comple548664676211718543et_nat @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_898_cSup__eq__non__empty,axiom,
! [X2: set_set_nat,A2: set_nat] :
( ( X2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X2 )
=> ( ord_less_eq_set_nat @ X3 @ A2 ) )
=> ( ! [Y3: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X2 )
=> ( ord_less_eq_set_nat @ X4 @ Y3 ) )
=> ( ord_less_eq_set_nat @ A2 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_899_cSup__eq__non__empty,axiom,
! [X2: set_int,A2: int] :
( ( X2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ( ord_less_eq_int @ X3 @ A2 ) )
=> ( ! [Y3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X2 )
=> ( ord_less_eq_int @ X4 @ Y3 ) )
=> ( ord_less_eq_int @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_int @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_900_cSup__eq__non__empty,axiom,
! [X2: set_nat,A2: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ( ord_less_eq_nat @ X3 @ A2 ) )
=> ( ! [Y3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X2 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) )
=> ( ord_less_eq_nat @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_901_cSup__least,axiom,
! [X2: set_real,Z2: real] :
( ( X2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X2 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_902_cSup__least,axiom,
! [X2: set_set_set_nat,Z2: set_set_nat] :
( ( X2 != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X2 )
=> ( ord_le6893508408891458716et_nat @ X3 @ Z2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ X2 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_903_cSup__least,axiom,
! [X2: set_set_nat,Z2: set_nat] :
( ( X2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X2 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_904_cSup__least,axiom,
! [X2: set_int,Z2: int] :
( ( X2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X2 )
=> ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X2 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_905_cSup__least,axiom,
! [X2: set_nat,Z2: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X2 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_906_less__eq__Sup,axiom,
! [A: set_set_set_nat,U: set_set_nat] :
( ! [V3: set_set_nat] :
( ( member_set_set_nat @ V3 @ A )
=> ( ord_le6893508408891458716et_nat @ U @ V3 ) )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple548664676211718543et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_907_less__eq__Sup,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V3: set_nat] :
( ( member_set_nat @ V3 @ A )
=> ( ord_less_eq_set_nat @ U @ V3 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_908_first__assumptions_O_092_060K_062__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3326749438856946062irst_K @ K )
= ( collect_set_set_nat
@ ^ [K4: set_set_nat] :
( ( member_set_set_nat @ K4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
& ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ K4 ) )
= K )
& ( K4
= ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ K4 ) @ ( clique5033774636164728513irst_v @ K4 ) ) ) ) ) ) ) ).
% first_assumptions.\<K>_def
thf(fact_909_first__assumptions_Osameprod___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,X2: set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_less_eq_set_nat @ X2 @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X2 ) )
=> ( member_set_set_nat @ ( clique6722202388162463298od_nat @ X2 @ X2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.sameprod_\<G>
thf(fact_910_first__assumptions_O_092_060G_062__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) )
= ( collect_set_set_nat
@ ^ [G3: set_set_nat] : ( ord_le6893508408891458716et_nat @ G3 @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.\<G>_def
thf(fact_911_finite__enumerate__in__set,axiom,
! [S: set_nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_912_finite__enumerate__Ex,axiom,
! [S: set_nat,S3: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat @ S3 @ S )
=> ? [N4: nat] :
( ( ord_less_nat @ N4 @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N4 )
= S3 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_913_finite__enum__ext,axiom,
! [X2: set_nat,Y2: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X2 ) )
=> ( ( infini8530281810654367211te_nat @ X2 @ I3 )
= ( infini8530281810654367211te_nat @ Y2 @ I3 ) ) )
=> ( ( finite_finite_nat @ X2 )
=> ( ( finite_finite_nat @ Y2 )
=> ( ( ( finite_card_nat @ X2 )
= ( finite_card_nat @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_914_ccSup__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% ccSup_empty
thf(fact_915_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_916_first__assumptions_Om2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.m2
thf(fact_917_first__assumptions__def,axiom,
( assump5453534214990993103ptions
= ( ^ [L2: nat,P5: nat,K3: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L2 )
& ( ord_less_nat @ L2 @ P5 )
& ( ord_less_nat @ P5 @ K3 ) ) ) ) ).
% first_assumptions_def
thf(fact_918_first__assumptions_Ointro,axiom,
! [L: nat,P2: nat,K: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L )
=> ( ( ord_less_nat @ L @ P2 )
=> ( ( ord_less_nat @ P2 @ K )
=> ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ) ) ).
% first_assumptions.intro
thf(fact_919_first__assumptions_Ol2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) ) ).
% first_assumptions.l2
thf(fact_920_first__assumptions_Ok2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) ) ).
% first_assumptions.k2
thf(fact_921_first__assumptions_Op,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P2 ) ) ).
% first_assumptions.p
thf(fact_922_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_923_numeral__le__iff,axiom,
! [M5: num,N3: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M5 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( ord_less_eq_num @ M5 @ N3 ) ) ).
% numeral_le_iff
thf(fact_924_numeral__le__iff,axiom,
! [M5: num,N3: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M5 ) @ ( numeral_numeral_real @ N3 ) )
= ( ord_less_eq_num @ M5 @ N3 ) ) ).
% numeral_le_iff
thf(fact_925_numeral__le__iff,axiom,
! [M5: num,N3: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) )
= ( ord_less_eq_num @ M5 @ N3 ) ) ).
% numeral_le_iff
thf(fact_926_numeral__le__iff,axiom,
! [M5: num,N3: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) )
= ( ord_less_eq_num @ M5 @ N3 ) ) ).
% numeral_le_iff
thf(fact_927_diff__diff__cancel,axiom,
! [I2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_928_psubsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_929_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_930_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A3 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_931_less__set__def,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( ord_le466346588697744319_nat_o
@ ^ [X5: set_set_nat] : ( member_set_set_nat @ X5 @ A3 )
@ ^ [X5: set_set_nat] : ( member_set_set_nat @ X5 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_932_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A3 )
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_933_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_934_psubsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_935_psubsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_936_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_937_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_938_bot__set__def,axiom,
( bot_bot_set_complex
= ( collect_complex @ bot_bot_complex_o ) ) ).
% bot_set_def
thf(fact_939_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_940_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_941_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_942_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_943_subset__psubset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C2 )
=> ( ord_less_set_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_944_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_945_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
& ~ ( ord_le6893508408891458716et_nat @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_946_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_947_psubset__subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_less_set_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_948_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_949_psubset__imp__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_950_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_951_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_952_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_953_psubsetE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_954_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_955_not__psubset__empty,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_956_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_957_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B10: set_nat] :
( ( ord_less_set_nat @ B10 @ A7 )
=> ( P @ B10 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_958_finite__psubset__induct,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [A7: set_set_nat] :
( ( finite1152437895449049373et_nat @ A7 )
=> ( ! [B10: set_set_nat] :
( ( ord_less_set_set_nat @ B10 @ A7 )
=> ( P @ B10 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_959_diff__le__mono2,axiom,
! [M5: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_960_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_961_diff__le__self,axiom,
! [M5: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N3 ) @ M5 ) ).
% diff_le_self
thf(fact_962_diff__le__mono,axiom,
! [M5: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_963_Nat_Odiff__diff__eq,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_964_le__diff__iff,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_965_eq__diff__iff,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M5 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M5 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_966_diff__less__mono2,axiom,
! [M5: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ( ord_less_nat @ M5 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M5 ) ) ) ) ).
% diff_less_mono2
thf(fact_967_less__imp__diff__less,axiom,
! [J: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_968_diff__less__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_969_less__diff__iff,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_nat @ M5 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_970_first__assumptions_Okml,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_971_psubset__card__mono,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_less_set_complex @ A @ B )
=> ( ord_less_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_972_psubset__card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_973_psubset__card__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_less_set_set_nat @ A @ B )
=> ( ord_less_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_974_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_975_card__psubset,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B ) )
=> ( ord_less_set_complex @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_976_card__psubset,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_977_card__psubset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A @ B ) ) ) ) ).
% card_psubset
thf(fact_978_first__assumptions_Opl,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ P2 ) ) ).
% first_assumptions.pl
thf(fact_979_first__assumptions_Okp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ K ) ) ).
% first_assumptions.kp
thf(fact_980_first__assumptions_Ok,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_981_first__assumptions_Okm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_982_first__assumptions_Omp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_983_enat__ord__number_I1_J,axiom,
! [M5: num,N3: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M5 ) @ ( numera1916890842035813515d_enat @ N3 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).
% enat_ord_number(1)
thf(fact_984_m__def,axiom,
( ( assump1710595444109740334irst_m @ k )
= ( power_power_nat @ k @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% m_def
thf(fact_985_diff__shunt__var,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X @ Y )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_986_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_987_pred__subset__eq,axiom,
! [R2: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ R2 )
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ S ) )
= ( ord_le6893508408891458716et_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_988_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ R2 )
@ ^ [X5: nat] : ( member_nat @ X5 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_989_first__assumptions_Om__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( assump1710595444109740334irst_m @ K )
= ( power_power_nat @ K @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% first_assumptions.m_def
thf(fact_990_sqrt__aux_I1_J,axiom,
! [N3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M2: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 ) ) ) ).
% sqrt_aux(1)
thf(fact_991_diff__le__diff__pow,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N3 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M5 ) @ ( power_power_nat @ K @ N3 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_992_sqrt__aux_I2_J,axiom,
! [N3: nat] :
( ( collect_nat
@ ^ [M2: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 ) )
!= bot_bot_set_nat ) ).
% sqrt_aux(2)
thf(fact_993_power2__nat__le__imp__le,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% power2_nat_le_imp_le
thf(fact_994_power2__nat__le__eq__le,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% power2_nat_le_eq_le
thf(fact_995_self__le__ge2__pow,axiom,
! [K: nat,M5: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M5 @ ( power_power_nat @ K @ M5 ) ) ) ).
% self_le_ge2_pow
thf(fact_996_second__assumptions__axioms_Ointro,axiom,
! [K: nat,L: nat] :
( ( K
= ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L )
=> ( assump8934899134041091456axioms @ L @ K ) ) ) ).
% second_assumptions_axioms.intro
thf(fact_997_second__assumptions__axioms__def,axiom,
( assump8934899134041091456axioms
= ( ^ [L2: nat,K3: nat] :
( ( K3
= ( power_power_nat @ L2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
& ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L2 ) ) ) ) ).
% second_assumptions_axioms_def
thf(fact_998_le__sqrtI,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_eq_nat @ X @ ( sqrt @ Y ) ) ) ).
% le_sqrtI
thf(fact_999_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1000_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1001_int__ops_I3_J,axiom,
! [N3: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N3 ) )
= ( numeral_numeral_int @ N3 ) ) ).
% int_ops(3)
thf(fact_1002_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1003_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1004_sqrt__le,axiom,
! [N3: nat] : ( ord_less_eq_nat @ ( sqrt @ N3 ) @ N3 ) ).
% sqrt_le
thf(fact_1005_mono__sqrt_H,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( sqrt @ M5 ) @ ( sqrt @ N3 ) ) ) ).
% mono_sqrt'
thf(fact_1006_sqrt__power2__le,axiom,
! [N3: nat] : ( ord_less_eq_nat @ ( power_power_nat @ ( sqrt @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 ) ).
% sqrt_power2_le
thf(fact_1007_sqrt__le__iff,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ ( sqrt @ Y ) @ X )
= ( ! [Z4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ Z4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_eq_nat @ Z4 @ X ) ) ) ) ).
% sqrt_le_iff
thf(fact_1008_le__sqrt__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ ( sqrt @ Y ) )
= ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% le_sqrt_iff
thf(fact_1009_sqrt__leI,axiom,
! [Y: nat,X: nat] :
( ! [Z5: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ Z5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_eq_nat @ Z5 @ X ) )
=> ( ord_less_eq_nat @ ( sqrt @ Y ) @ X ) ) ).
% sqrt_leI
thf(fact_1010_numeral__le__real__of__nat__iff,axiom,
! [N3: num,M5: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ ( semiri5074537144036343181t_real @ M5 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ M5 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1011_sqrt__unique,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 )
=> ( ( ord_less_nat @ N3 @ ( power_power_nat @ ( suc @ M5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( sqrt @ N3 )
= M5 ) ) ) ).
% sqrt_unique
thf(fact_1012_nat__descend__induct,axiom,
! [N3: nat,P: nat > $o,M5: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N3 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M5 ) ) ) ).
% nat_descend_induct
thf(fact_1013_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1014_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1015_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1016_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1017_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1018_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_1019_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1020_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1021_diff__self__eq__0,axiom,
! [M5: nat] :
( ( minus_minus_nat @ M5 @ M5 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1022_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1023_Suc__le__mono,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M5 ) )
= ( ord_less_eq_nat @ N3 @ M5 ) ) ).
% Suc_le_mono
thf(fact_1024_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_1025_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1026_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_1027_Suc__mono,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( suc @ M5 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_1028_Suc__less__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_less_eq
thf(fact_1029_diff__Suc__Suc,axiom,
! [M5: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_1030_Suc__diff__diff,axiom,
! [M5: nat,N3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M5 ) @ N3 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M5 @ N3 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1031_diff__is__0__eq_H,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( minus_minus_nat @ M5 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1032_diff__is__0__eq,axiom,
! [M5: nat,N3: nat] :
( ( ( minus_minus_nat @ M5 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1033_zero__less__diff,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M5 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% zero_less_diff
thf(fact_1034_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1035_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_1036_card__Collect__le__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N3 ) ) )
= ( suc @ N3 ) ) ).
% card_Collect_le_nat
thf(fact_1037_nat__one__le__power,axiom,
! [I2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N3 ) ) ) ).
% nat_one_le_power
thf(fact_1038_diff__Suc__less,axiom,
! [N3: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I2 ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1039_int__if,axiom,
! [P: $o,A2: nat,B3: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B3 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B3 ) )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% int_if
thf(fact_1040_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1041_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1042_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1043_less__Suc__eq__0__disj,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
= ( ( M5 = zero_zero_nat )
| ? [J3: nat] :
( ( M5
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1044_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M: nat] :
( N3
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_1045_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1046_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M2: nat] :
( N3
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1047_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1048_ex__least__nat__less,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N3 )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1049_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1050_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1051_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1052_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1053_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1054_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1055_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_1056_diff__induct,axiom,
! [P: nat > nat > $o,M5: nat,N3: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M5 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_1057_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_1058_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1059_Suc__neq__Zero,axiom,
! [M5: nat] :
( ( suc @ M5 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1060_Zero__neq__Suc,axiom,
! [M5: nat] :
( zero_zero_nat
!= ( suc @ M5 ) ) ).
% Zero_neq_Suc
thf(fact_1061_Zero__not__Suc,axiom,
! [M5: nat] :
( zero_zero_nat
!= ( suc @ M5 ) ) ).
% Zero_not_Suc
thf(fact_1062_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M: nat] :
( N3
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_1063_not__less__less__Suc__eq,axiom,
! [N3: nat,M5: nat] :
( ~ ( ord_less_nat @ N3 @ M5 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M5 ) )
= ( N3 = M5 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1064_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1065_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1066_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1067_Suc__less__SucD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_1068_less__antisym,axiom,
! [N3: nat,M5: nat] :
( ~ ( ord_less_nat @ N3 @ M5 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M5 ) )
=> ( M5 = N3 ) ) ) ).
% less_antisym
thf(fact_1069_Suc__less__eq2,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M5 )
= ( ? [M8: nat] :
( ( M5
= ( suc @ M8 ) )
& ( ord_less_nat @ N3 @ M8 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1070_All__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1071_not__less__eq,axiom,
! [M5: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M5 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M5 ) ) ) ).
% not_less_eq
thf(fact_1072_less__Suc__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_1073_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
& ( P @ I4 ) ) )
= ( ( P @ N3 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1074_less__SucI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ M5 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_1075_less__SucE,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M5 @ N3 )
=> ( M5 = N3 ) ) ) ).
% less_SucE
thf(fact_1076_Suc__lessI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ( ( suc @ M5 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M5 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_1077_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1078_Suc__lessD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M5 ) @ N3 )
=> ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_lessD
thf(fact_1079_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1080_Suc__leD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% Suc_leD
thf(fact_1081_le__SucE,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M5 @ N3 )
=> ( M5
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_1082_le__SucI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ M5 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_1083_Suc__le__D,axiom,
! [N3: nat,M9: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M9 )
=> ? [M: nat] :
( M9
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_1084_le__Suc__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M5 @ N3 )
| ( M5
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_1085_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_1086_not__less__eq__eq,axiom,
! [M5: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M5 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M5 ) ) ).
% not_less_eq_eq
thf(fact_1087_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M7: nat] :
( ( ord_less_eq_nat @ ( suc @ M7 ) @ N4 )
=> ( P @ M7 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_1088_nat__induct__at__least,axiom,
! [M5: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( P @ M5 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1089_transitive__stepwise__le,axiom,
! [M5: nat,N3: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z5: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z5 )
=> ( R2 @ X3 @ Z5 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M5 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1090_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1091_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1092_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1093_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1094_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1095_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_1096_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1097_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_1098_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1099_gr__implies__not0,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1100_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N4 )
& ~ ( P @ M7 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_1101_int__ops_I6_J,axiom,
! [A2: nat,B3: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1102_minus__nat_Odiff__0,axiom,
! [M5: nat] :
( ( minus_minus_nat @ M5 @ zero_zero_nat )
= M5 ) ).
% minus_nat.diff_0
thf(fact_1103_diffs0__imp__equal,axiom,
! [M5: nat,N3: nat] :
( ( ( minus_minus_nat @ M5 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M5 )
= zero_zero_nat )
=> ( M5 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1104_card__less,axiom,
! [M4: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M4 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M4 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1105_card__less__Suc,axiom,
! [M4: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M4 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M4 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M4 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1106_card__less__Suc2,axiom,
! [M4: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M4 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M4 )
& ( ord_less_nat @ K3 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M4 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1107_le__imp__less__Suc,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_nat @ M5 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_1108_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1109_less__Suc__eq__le,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_1110_le__less__Suc__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M5 ) )
= ( N3 = M5 ) ) ) ).
% le_less_Suc_eq
thf(fact_1111_Suc__le__lessD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_1112_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1113_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1114_Suc__le__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_le_eq
thf(fact_1115_Suc__leI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 ) ) ).
% Suc_leI
thf(fact_1116_Suc__diff__le,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_eq_nat @ N3 @ M5 )
=> ( ( minus_minus_nat @ ( suc @ M5 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M5 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_1117_diff__less__Suc,axiom,
! [M5: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M5 @ N3 ) @ ( suc @ M5 ) ) ).
% diff_less_Suc
thf(fact_1118_Suc__diff__Suc,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ N3 @ M5 )
=> ( ( suc @ ( minus_minus_nat @ M5 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_1119_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N3 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1120_diff__less,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_nat @ ( minus_minus_nat @ M5 @ N3 ) @ M5 ) ) ) ).
% diff_less
thf(fact_1121_lessThan__empty__iff,axiom,
! [N3: nat] :
( ( ( set_ord_lessThan_nat @ N3 )
= bot_bot_set_nat )
= ( N3 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1122_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1123_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1124_zle__int,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% zle_int
thf(fact_1125_log__exp2__le,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( log @ N3 ) ) @ N3 ) ) ).
% log_exp2_le
thf(fact_1126_card__nth__roots,axiom,
! [C: complex,N3: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N3 )
= C ) ) )
= N3 ) ) ) ).
% card_nth_roots
thf(fact_1127_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_1128_Discrete_Olog__le__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( log @ M5 ) @ ( log @ N3 ) ) ) ).
% Discrete.log_le_iff
thf(fact_1129_Least__Suc2,axiom,
! [P: nat > $o,N3: nat,Q: nat > $o,M5: nat] :
( ( P @ N3 )
=> ( ( Q @ M5 )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_1130_Least__Suc,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1131_log__eqI,axiom,
! [N3: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N3 )
=> ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) ) )
=> ( ( log @ N3 )
= K ) ) ) ) ).
% log_eqI
thf(fact_1132_log2__of__power__le,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).
% log2_of_power_le
thf(fact_1133_mult__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ K )
= ( times_times_nat @ N3 @ K ) )
= ( ( M5 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1134_mult__cancel1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ( times_times_nat @ K @ M5 )
= ( times_times_nat @ K @ N3 ) )
= ( ( M5 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1135_mult__0__right,axiom,
! [M5: nat] :
( ( times_times_nat @ M5 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1136_mult__is__0,axiom,
! [M5: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ N3 )
= zero_zero_nat )
= ( ( M5 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1137_mult__eq__1__iff,axiom,
! [M5: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( M5
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1138_one__eq__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M5 @ N3 ) )
= ( ( M5
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1139_nat__0__less__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M5 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M5 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1140_mult__less__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M5 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_1141_one__le__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M5 @ N3 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M5 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).
% one_le_mult_iff
thf(fact_1142_mult__le__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1143_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1144_diff__mult__distrib,axiom,
! [M5: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M5 @ N3 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1145_diff__mult__distrib2,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M5 @ N3 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_1146_Suc__mult__cancel1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M5 )
= ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( M5 = N3 ) ) ).
% Suc_mult_cancel1
thf(fact_1147_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1148_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1149_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1150_le__square,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ).
% le_square
thf(fact_1151_le__cube,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ) ).
% le_cube
thf(fact_1152_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1153_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1154_Suc__mult__le__cancel1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M5 ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1155_Suc__mult__less__cancel1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M5 ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1156_n__less__n__mult__m,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M5 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ N3 @ M5 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1157_n__less__m__mult__n,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M5 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ M5 @ N3 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1158_one__less__mult,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M5 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M5 @ N3 ) ) ) ) ).
% one_less_mult
thf(fact_1159_le__log2__of__power,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M5 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% le_log2_of_power
thf(fact_1160_log__exp2__ge,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( log @ N3 ) ) ) ) ).
% log_exp2_ge
thf(fact_1161_nat__mult__le__cancel__disj,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1162_nat__mult__le__cancel1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1163_int__ops_I7_J,axiom,
! [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B3 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(7)
thf(fact_1164_third__assumptions_Opllog_I1_J,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P2 @ K )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ L ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( assump1710595444109740334irst_m @ K ) ) ) ) @ ( semiri5074537144036343181t_real @ P2 ) ) ) ).
% third_assumptions.pllog(1)
thf(fact_1165_seq__mono__lemma,axiom,
! [M5: nat,D2: nat > real,E: nat > real] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_real @ ( D2 @ N4 ) @ ( E @ N4 ) ) )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_real @ ( E @ N4 ) @ ( E @ M5 ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M5 @ N6 )
=> ( ord_less_real @ ( D2 @ N6 ) @ ( E @ M5 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1166_third__assumptions_OM0,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% third_assumptions.M0
thf(fact_1167_third__assumptions_OM0_H,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% third_assumptions.M0'
thf(fact_1168_third__assumptions_OL0,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ assumptions_and_L0 @ L ) ) ).
% third_assumptions.L0
thf(fact_1169_third__assumptions_OL0_H,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ assumptions_and_L02 @ L ) ) ).
% third_assumptions.L0'
thf(fact_1170_second__assumptions_Ol8,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L ) ) ).
% second_assumptions.l8
thf(fact_1171_dvd__1__iff__1,axiom,
! [M5: nat] :
( ( dvd_dvd_nat @ M5 @ ( suc @ zero_zero_nat ) )
= ( M5
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1172_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1173_dvd__antisym,axiom,
! [M5: nat,N3: nat] :
( ( dvd_dvd_nat @ M5 @ N3 )
=> ( ( dvd_dvd_nat @ N3 @ M5 )
=> ( M5 = N3 ) ) ) ).
% dvd_antisym
thf(fact_1174_dvd__diff__nat,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( dvd_dvd_nat @ K @ M5 )
=> ( ( dvd_dvd_nat @ K @ N3 )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M5 @ N3 ) ) ) ) ).
% dvd_diff_nat
thf(fact_1175_nat__dvd__not__less,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ( ord_less_nat @ M5 @ N3 )
=> ~ ( dvd_dvd_nat @ N3 @ M5 ) ) ) ).
% nat_dvd_not_less
thf(fact_1176_dvd__diffD,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M5 @ N3 ) )
=> ( ( dvd_dvd_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M5 )
=> ( dvd_dvd_nat @ K @ M5 ) ) ) ) ).
% dvd_diffD
thf(fact_1177_dvd__diffD1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M5 @ N3 ) )
=> ( ( dvd_dvd_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ N3 @ M5 )
=> ( dvd_dvd_nat @ K @ N3 ) ) ) ) ).
% dvd_diffD1
thf(fact_1178_less__eq__dvd__minus,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( dvd_dvd_nat @ M5 @ N3 )
= ( dvd_dvd_nat @ M5 @ ( minus_minus_nat @ N3 @ M5 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1179_dvd__minus__self,axiom,
! [M5: nat,N3: nat] :
( ( dvd_dvd_nat @ M5 @ ( minus_minus_nat @ N3 @ M5 ) )
= ( ( ord_less_nat @ N3 @ M5 )
| ( dvd_dvd_nat @ M5 @ N3 ) ) ) ).
% dvd_minus_self
thf(fact_1180_second__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ).
% second_assumptions.axioms(1)
thf(fact_1181_dvd__imp__le,axiom,
! [K: nat,N3: nat] :
( ( dvd_dvd_nat @ K @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_eq_nat @ K @ N3 ) ) ) ).
% dvd_imp_le
thf(fact_1182_dvd__mult__cancel,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M5 @ N3 ) ) ) ).
% dvd_mult_cancel
thf(fact_1183_second__assumptions_Ointro,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( assump8934899134041091456axioms @ L @ K )
=> ( assump2881078719466019805ptions @ L @ P2 @ K ) ) ) ).
% second_assumptions.intro
thf(fact_1184_second__assumptions__def,axiom,
( assump2881078719466019805ptions
= ( ^ [L2: nat,P5: nat,K3: nat] :
( ( assump5453534214990993103ptions @ L2 @ P5 @ K3 )
& ( assump8934899134041091456axioms @ L2 @ K3 ) ) ) ) ).
% second_assumptions_def
thf(fact_1185_dvd__power__iff__le,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M5 ) @ ( power_power_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% dvd_power_iff_le
thf(fact_1186_finite__divisors__nat,axiom,
! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M5 ) ) ) ) ).
% finite_divisors_nat
thf(fact_1187_even__set__encode__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A ) )
= ( ~ ( member_nat @ zero_zero_nat @ A ) ) ) ) ).
% even_set_encode_iff
thf(fact_1188_set__encode__eq,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ( nat_set_encode @ A )
= ( nat_set_encode @ B ) )
= ( A = B ) ) ) ) ).
% set_encode_eq
thf(fact_1189_set__encode__inf,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( nat_set_encode @ A )
= zero_zero_nat ) ) ).
% set_encode_inf
thf(fact_1190_nat__mult__eq__1__iff,axiom,
! [M5: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ N3 )
= one_one_nat )
= ( ( M5 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1191_nat__1__eq__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M5 @ N3 ) )
= ( ( M5 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1192_nat__dvd__1__iff__1,axiom,
! [M5: nat] :
( ( dvd_dvd_nat @ M5 @ one_one_nat )
= ( M5 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1193_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_1194_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_1195_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_1196_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1197_int__ops_I8_J,axiom,
! [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B3 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(8)
thf(fact_1198_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times_nat @ N3 @ one_one_nat )
= N3 ) ).
% nat_mult_1_right
thf(fact_1199_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times_nat @ one_one_nat @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_1200_diff__Suc__eq__diff__pred,axiom,
! [M5: nat,N3: nat] :
( ( minus_minus_nat @ M5 @ ( suc @ N3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1201_mult__eq__self__implies__10,axiom,
! [M5: nat,N3: nat] :
( ( M5
= ( times_times_nat @ M5 @ N3 ) )
=> ( ( N3 = one_one_nat )
| ( M5 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1202_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1203_log__induct,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
=> ( ( P @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( P @ N4 ) ) )
=> ( P @ N3 ) ) ) ) ).
% log_induct
thf(fact_1204_nat__induct__non__zero,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1205_Suc__pred_H,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( N3
= ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1206_Suc__diff__eq__diff__pred,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( minus_minus_nat @ ( suc @ M5 ) @ N3 )
= ( minus_minus_nat @ M5 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1207_dvd__mult__cancel2,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N3 @ M5 ) @ M5 )
= ( N3 = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1208_dvd__mult__cancel1,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M5 @ N3 ) @ M5 )
= ( N3 = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1209_power__dvd__imp__le,axiom,
! [I2: nat,M5: nat,N3: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M5 ) @ ( power_power_nat @ I2 @ N3 ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% power_dvd_imp_le
thf(fact_1210_card__roots__unity__eq,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N3 )
= one_one_complex ) ) )
= N3 ) ) ).
% card_roots_unity_eq
thf(fact_1211_log__rec,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
=> ( ( log @ N3 )
= ( suc @ ( log @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% log_rec
thf(fact_1212_card__complex__roots__unity,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N3 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N3 )
= one_one_complex ) ) )
= N3 ) ) ).
% card_complex_roots_unity
thf(fact_1213_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L3 @ X4 @ I5 ) @ one_one_nat )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( X4 @ I5 )
= zero_zero_real ) )
=> ( ( L3 @ X4 @ I5 )
= zero_zero_nat ) )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( X4 @ I5 )
= one_one_real ) )
=> ( ( L3 @ X4 @ I5 )
= one_one_nat ) )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( L3 @ X4 @ I5 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I5 ) @ ( F @ X4 @ I5 ) ) )
& ! [X4: nat > real,I5: nat] :
( ( ( P @ X4 )
& ( Q @ I5 )
& ( ( L3 @ X4 @ I5 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I5 ) @ ( X4 @ I5 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1214_split__div_H,axiom,
! [P: nat > $o,M5: nat,N3: nat] :
( ( P @ ( divide_divide_nat @ M5 @ N3 ) )
= ( ( ( N3 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q2 ) @ M5 )
& ( ord_less_nat @ M5 @ ( times_times_nat @ N3 @ ( suc @ Q2 ) ) )
& ( P @ Q2 ) ) ) ) ).
% split_div'
thf(fact_1215_le__div__geq,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M5 )
=> ( ( divide_divide_nat @ M5 @ N3 )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).
% le_div_geq
thf(fact_1216_div__le__mono,axiom,
! [M5: nat,N3: nat,K: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M5 @ K ) @ ( divide_divide_nat @ N3 @ K ) ) ) ).
% div_le_mono
thf(fact_1217_div__le__dividend,axiom,
! [M5: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M5 @ N3 ) @ M5 ) ).
% div_le_dividend
thf(fact_1218_Suc__div__le__mono,axiom,
! [M5: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M5 @ N3 ) @ ( divide_divide_nat @ ( suc @ M5 ) @ N3 ) ) ).
% Suc_div_le_mono
thf(fact_1219_div__times__less__eq__dividend,axiom,
! [M5: nat,N3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N3 ) @ N3 ) @ M5 ) ).
% div_times_less_eq_dividend
thf(fact_1220_times__div__less__eq__dividend,axiom,
! [N3: nat,M5: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M5 @ N3 ) ) @ M5 ) ).
% times_div_less_eq_dividend
thf(fact_1221_div__le__mono2,axiom,
! [M5: nat,N3: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N3 ) @ ( divide_divide_nat @ K @ M5 ) ) ) ) ).
% div_le_mono2
thf(fact_1222_div__greater__zero__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M5 @ N3 ) )
= ( ( ord_less_eq_nat @ N3 @ M5 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% div_greater_zero_iff
thf(fact_1223_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M5 @ ( divide_divide_nat @ N3 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M5 @ Q3 ) @ N3 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1224_div__nat__eqI,axiom,
! [N3: nat,Q3: nat,M5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q3 ) @ M5 )
=> ( ( ord_less_nat @ M5 @ ( times_times_nat @ N3 @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M5 @ N3 )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1225_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1226_Nat_Oadd__0__right,axiom,
! [M5: nat] :
( ( plus_plus_nat @ M5 @ zero_zero_nat )
= M5 ) ).
% Nat.add_0_right
thf(fact_1227_add__is__0,axiom,
! [M5: nat,N3: nat] :
( ( ( plus_plus_nat @ M5 @ N3 )
= zero_zero_nat )
= ( ( M5 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1228_add__Suc__right,axiom,
! [M5: nat,N3: nat] :
( ( plus_plus_nat @ M5 @ ( suc @ N3 ) )
= ( suc @ ( plus_plus_nat @ M5 @ N3 ) ) ) ).
% add_Suc_right
thf(fact_1229_nat__add__left__cancel__le,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_1230_nat__add__left__cancel__less,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_1231_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1232_add__gr__0,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M5 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M5 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_1233_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1234_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1235_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1236_mult__Suc__right,axiom,
! [M5: nat,N3: nat] :
( ( times_times_nat @ M5 @ ( suc @ N3 ) )
= ( plus_plus_nat @ M5 @ ( times_times_nat @ M5 @ N3 ) ) ) ).
% mult_Suc_right
thf(fact_1237_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1238_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1239_add__eq__self__zero,axiom,
! [M5: nat,N3: nat] :
( ( ( plus_plus_nat @ M5 @ N3 )
= M5 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1240_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_1241_diff__add__0,axiom,
! [N3: nat,M5: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M5 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1242_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1243_add__is__1,axiom,
! [M5: nat,N3: nat] :
( ( ( plus_plus_nat @ M5 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M5
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M5 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1244_one__is__add,axiom,
! [M5: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M5 @ N3 ) )
= ( ( ( M5
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M5 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1245_add__mult__distrib,axiom,
! [M5: nat,N3: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M5 @ N3 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1246_add__mult__distrib2,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M5 @ N3 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N3 ) ) ) ).
% add_mult_distrib2
thf(fact_1247_less__add__eq__less,axiom,
! [K: nat,L: nat,M5: nat,N3: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M5 @ L )
= ( plus_plus_nat @ K @ N3 ) )
=> ( ord_less_nat @ M5 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_1248_trans__less__add2,axiom,
! [I2: nat,J: nat,M5: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% trans_less_add2
thf(fact_1249_trans__less__add1,axiom,
! [I2: nat,J: nat,M5: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% trans_less_add1
thf(fact_1250_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1251_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1252_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1253_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1254_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1255_Nat_Odiff__cancel,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_1256_diff__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M5 @ K ) @ ( plus_plus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ).
% diff_cancel2
thf(fact_1257_diff__add__inverse,axiom,
! [N3: nat,M5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M5 ) @ N3 )
= M5 ) ).
% diff_add_inverse
thf(fact_1258_diff__add__inverse2,axiom,
! [M5: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M5 @ N3 ) @ N3 )
= M5 ) ).
% diff_add_inverse2
thf(fact_1259_add__Suc__shift,axiom,
! [M5: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M5 ) @ N3 )
= ( plus_plus_nat @ M5 @ ( suc @ N3 ) ) ) ).
% add_Suc_shift
thf(fact_1260_add__Suc,axiom,
! [M5: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M5 ) @ N3 )
= ( suc @ ( plus_plus_nat @ M5 @ N3 ) ) ) ).
% add_Suc
thf(fact_1261_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1262_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1263_trans__le__add2,axiom,
! [I2: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% trans_le_add2
thf(fact_1264_trans__le__add1,axiom,
! [I2: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% trans_le_add1
thf(fact_1265_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1266_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1267_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1268_add__leD2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N3 )
=> ( ord_less_eq_nat @ K @ N3 ) ) ).
% add_leD2
% 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,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_nat @ ( set_ord_lessThan_nat @ n ) ).
%------------------------------------------------------------------------------