TPTP Problem File: SLH0309^1.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 : Hales_Jewett/0002_Hales_Jewett/prob_01037_043391__5795326_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1428 ( 502 unt; 151 typ; 0 def)
% Number of atoms : 3830 (1146 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11571 ( 290 ~; 84 |; 268 &;9247 @)
% ( 0 <=>;1682 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 1599 (1599 >; 0 *; 0 +; 0 <<)
% Number of symbols : 136 ( 133 usr; 10 con; 0-6 aty)
% Number of variables : 3855 ( 341 ^;3380 !; 134 ?;3855 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:06.633
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na3764207730537033026at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_na6626867396258451522at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8778986904112484418at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8843485148432118594at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_nat_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat3: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat4: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat5: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J_J,type,
set_nat_nat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (133)
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
the_in5300466440149791684at_nat: set_nat_nat > ( ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6840239867990089257at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat ) > set_na8843485148432118594at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat_nat5 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na7122919648973241129at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat_nat ) > set_na8778986904112484418at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat2: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat ) > set_nat_nat_nat_nat4 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_na7569501297962130601at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat2 ) > set_na6626867396258451522at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na8678869062391380393at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat ) > set_na3764207730537033026at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat3: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
piE_nat_nat_int: set_nat_nat > ( ( nat > nat ) > set_int ) > set_nat_nat_int ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat4: set_nat > ( nat > set_nat_nat_nat2 ) > set_nat_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_nat_nat_nat_nat5: set_nat > ( nat > set_nat_nat_nat ) > set_nat_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restri4446420529079022766at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restrict_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restrict_nat_nat_nat2: ( nat > nat > nat ) > set_nat > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: ( nat > nat ) > set_nat > nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
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_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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
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_Hales__Jewett_Oclasses,type,
hales_classes: nat > nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ohj,type,
hales_hj: nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001t__Int__Oint,type,
hales_4259056829518216709ce_int: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > int > ( ( nat > nat ) > int ) > $o ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001t__Nat__Onat,type,
hales_4261547300027266985ce_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > nat > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olhj,type,
hales_lhj: nat > nat > nat > $o ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le8812218136411540557_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le3977685358511927117_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_less_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_less_nat_nat_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_nat_nat_nat2: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
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__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $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__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le371403230139555384at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6871433888996735800at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $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__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le996020443555834177_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5384859702510996545_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le2017632242545079438at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3127000006974329230at_nat: ( nat > nat > nat ) > ( nat > nat > 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__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $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__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3190276326201062306at_nat: set_na8843485148432118594at_nat > set_na8843485148432118594at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3125778081881428130at_nat: set_na8778986904112484418at_nat > set_na8778986904112484418at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5260717879541182899at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le5934964663421696068at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3211623285424100676at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
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_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1262493855416953332at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_7809927846809980933at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1545173636400105204at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_913610194320715013at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_1991755285388994676at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3101123049818244468at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_nat_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_5809701139083627781at_nat: ( nat > ( nat > nat ) > nat ) > set_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6919068903512877573at_nat: ( nat > nat > nat > nat ) > set_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or5033131092550408871at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or6142498856979658663at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or9140604705432621368at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or250740698829186286at_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or2699333443382148811at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3808701207811398603at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1140352010380016476at_nat: ( nat > nat ) > set_nat_nat ).
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_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
fChoice_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > ( nat > nat ) > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fChoice_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > nat > nat > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member2991261302380110260at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat_nat5 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
member5318315686745620148at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat_nat4 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member4402528950554000163at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member1679187572556404771at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_na3764207730537033026at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member952132173341509300at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
member_nat_nat_int: ( ( nat > nat ) > int ) > set_nat_nat_int > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member2740455936716430260at_nat: ( nat > ( nat > nat ) > nat ) > set_nat_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member17114558718834868at_nat: ( nat > nat > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_v_L_H____,type,
l: nat > nat > nat ).
thf(sy_v_L____,type,
l2: nat > nat > nat ).
thf(sy_v_N_H____,type,
n: nat ).
thf(sy_v_N____,type,
n2: nat ).
thf(sy_v_S1____,type,
s1: ( nat > nat ) > nat > nat ).
thf(sy_v__092_060chi_062____,type,
chi: ( nat > nat ) > nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1267)
thf(fact_0_line__prop,axiom,
hales_is_line @ l @ n @ ( plus_plus_nat @ t @ one_one_nat ) ).
% line_prop
thf(fact_1_assms_I2_J,axiom,
! [R: nat] : ( hales_hj @ R @ t ) ).
% assms(2)
thf(fact_2_S1__def,axiom,
( s1
= ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( l @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% S1_def
thf(fact_3_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_4__092_060open_062is__subspace_A_I_092_060lambda_062y_092_060in_062cube_A1_At_O_AL_A_Iy_A0_J_J_A1_AN_H_At_092_060close_062,axiom,
( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( l2 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t ) )
@ one_one_nat
@ n
@ t ) ).
% \<open>is_subspace (\<lambda>y\<in>cube 1 t. L (y 0)) 1 N' t\<close>
thf(fact_5_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_6_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_7_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_8_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_9_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_10_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_11_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_12_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_13_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_14_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_15_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_16_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_17_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_18_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_19_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_20_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_21_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_22_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_23_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_24_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_25_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_26_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_27_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_28_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_29_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_30_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_31_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_32_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_33_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_34_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_35_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_36_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_37_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_38_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_39_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_40_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_41_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_42_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_43_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_44_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_45_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_46_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_47_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_48_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_49_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_50_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_51_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_52_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_53_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_54_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_55_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A @ ( collect_nat_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: nat > nat > nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A @ ( collect_nat_nat_nat2 @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_64_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_65_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_66_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_67_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_68_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_69_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_70_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_71_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_72_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_73_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_74_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_75_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_76_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_77_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_78_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_79_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_80_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_81_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_82_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_83_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_84_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_85_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_86_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_87_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_88_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_89_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_90_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_91_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_92_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_93_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_94_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_95_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_96_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_97_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_98_line__points__in__cube,axiom,
! [L2: nat > nat > nat,N: nat,T: nat,S: nat] :
( ( hales_is_line @ L2 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( member_nat_nat @ ( L2 @ S ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_99_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_100_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_101_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_102_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_103_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_104_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_105_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_106_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_107_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S2: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S2 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_108_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_109_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_110_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_111_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_112_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_113_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_114_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_115_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_116_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_117_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_118_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_119_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_120_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_121_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_122_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_123_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_124_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_125_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_126_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_127_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_128_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_129_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_130_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_131_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_132_cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X3 @ zero_zero_nat )
= S ) ) ) ).
% cube_props(1)
thf(fact_133_line__is__dim1__subspace__t__ge__1,axiom,
! [N: nat,T: nat,L2: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ T )
=> ( ( hales_is_line @ L2 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L2 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_134_line__is__dim1__subspace__t__1,axiom,
! [N: nat,L2: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( hales_is_line @ L2 @ N @ one_one_nat )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L2 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ one_one_nat ) )
@ one_one_nat
@ N
@ one_one_nat ) ) ) ).
% line_is_dim1_subspace_t_1
thf(fact_135_line__is__dim1__subspace,axiom,
! [N: nat,T: nat,L2: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_line @ L2 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L2 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_136_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_137_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_138_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_139_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_140_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_141_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_142_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_143_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_144_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_145_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_146_join__cubes,axiom,
! [F: nat > nat,N: nat,T: nat,G: nat > nat,M: nat] :
( ( member_nat_nat @ F @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( ( member_nat_nat @ G @ ( hales_cube @ M @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( member_nat_nat @ ( hales_join_nat @ F @ G @ N @ M ) @ ( hales_cube @ ( plus_plus_nat @ N @ M ) @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_147_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_148_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_149_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_150_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_151_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_152_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_153_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_154_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_155_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_156_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_157_add__less__zeroD,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_158_that_I1_J,axiom,
ord_less_eq_nat @ n2 @ n ).
% that(1)
thf(fact_159_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_160_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_161_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_162_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_163_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_164_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_165_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_166_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_167_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_168_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_169_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_170_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_171_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_172_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_173_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_174_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_175_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_176_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_177_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_178_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_179_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_180_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_181_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_182_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_183_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_184_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_185_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_186_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_187_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_188_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_189_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_190_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_191_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_192_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_193_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_194_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_195_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_196_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_197_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_198_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_199_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_200_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_201_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_202_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_203_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_204_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_205_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_206_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_207_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_208_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_209_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_210_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_211_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_212_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_213_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_214_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_215_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_216_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_217_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_218_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_219_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_220_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_221_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_222_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_223_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_224_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_225_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_226_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_227_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_228_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_229_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_230_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_231_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_232_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_233_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_234_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_235_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_236_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_237_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_238_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_239_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_240_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_241_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_242_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_243_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_244_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_245_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_246_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_247_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_248_add__nonneg__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_249_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_250_add__nonpos__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_251_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_252_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_253_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_254_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_255_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_256_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_257_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_258_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_259_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_260_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_261_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_262_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_263_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_264_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_265_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_266_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_267_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_268_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_269_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_270_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_271_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_272_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_273_linorder__neqE__linordered__idom,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_274_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_275_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_276_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_277_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_278_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_279_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_280_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_281_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_282_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_283_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_284_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_285_order__refl,axiom,
! [X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ X ) ).
% order_refl
thf(fact_286_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_287_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_288_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_289_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_290_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_291_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_292_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_293_order__le__imp__less__or__eq,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_less_set_nat_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_294_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_295_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_296_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_297_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_298_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_299_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_300_le__cases3,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_301_le__cases3,axiom,
! [X: int,Y2: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_302_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_303_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
& ( ord_le9059583361652607317at_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_304_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_305_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_306_ord__eq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_307_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_308_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_309_ord__le__eq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_310_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_311_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_312_order__antisym,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_313_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_314_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_315_order_Otrans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_316_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_317_order__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_318_order__trans,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_319_order__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_320_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_321_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_322_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_323_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_324_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_325_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_326_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_327_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_328_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_329_dual__order_Otrans,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_330_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_331_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_332_antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_333_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_334_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_335_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_336_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_337_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_338_order__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_339_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_340_order__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_341_order__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_342_order__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_343_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_344_order__subst1,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_345_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_346_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_347_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_348_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_349_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_350_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_351_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_352_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_353_order__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_354_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_355_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_356_order__eq__refl,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( X = Y2 )
=> ( ord_le9059583361652607317at_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_357_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_358_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_359_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_360_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_361_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_362_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_363_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_364_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_365_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_366_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_367_ord__eq__le__subst,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_368_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_369_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_370_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_371_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_372_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_373_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_374_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_375_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_376_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_377_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_378_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_379_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_380_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_381_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_382_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_383_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_384_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_385_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_386_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_387_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_388_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_389_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_390_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_391_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_392_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_393_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_394_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_395_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_396_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_397_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_398_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_399_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_400_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_401_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_402_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_403_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_404_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_405_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_406_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_407_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_408_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_409_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_410_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_411_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_412_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_413_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_414_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_415_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_416_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_417_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_418_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_419_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_420_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_421_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_422_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_423_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_424_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_425_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_426_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_427_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_428_order__less__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_429_order__less__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_430_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_431_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_432_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_433_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_434_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_435_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_436_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_437_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_438_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_439_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_440_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_441_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_442_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_443_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_444_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_445_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_446_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_447_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_448_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_449_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_450_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_451_order__less__imp__triv,axiom,
! [X: int,Y2: int,P: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_452_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_453_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_454_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_455_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_456_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_457_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_458_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_459_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_460_cube__subset,axiom,
! [N: nat,T: nat] : ( ord_le9059583361652607317at_nat @ ( hales_cube @ N @ T ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% cube_subset
thf(fact_461_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_462_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_463_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_464_leD,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ~ ( ord_less_set_nat_nat @ X @ Y2 ) ) ).
% leD
thf(fact_465_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_466_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_467_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_468_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_469_nless__le,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A @ B ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_470_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_471_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_472_antisym__conv1,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_473_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_474_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_475_antisym__conv2,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_set_nat_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_476_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_477_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_478_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
& ~ ( ord_le9059583361652607317at_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_479_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_480_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_481_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_482_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_483_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_484_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_485_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_486_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_487_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_488_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_489_order_Ostrict__trans1,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_490_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_491_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_492_order_Ostrict__trans2,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_493_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_494_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_495_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ~ ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_496_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_497_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_498_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_499_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_500_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_501_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_502_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_503_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_504_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_less_set_nat_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_505_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_506_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_507_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_508_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_509_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_510_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ~ ( ord_le9059583361652607317at_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_511_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_512_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_513_order_Ostrict__implies__order,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_514_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_515_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_516_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_517_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_518_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_519_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_520_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_521_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_522_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_523_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_524_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_525_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_526_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_527_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_528_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_529_order__less__imp__le,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y2 )
=> ( ord_le9059583361652607317at_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_530_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_531_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_532_order__le__neq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_533_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_534_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_535_order__neq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A != B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_536_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_537_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_538_order__le__less__trans,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_less_set_nat_nat @ Y2 @ Z )
=> ( ord_less_set_nat_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_539_order__le__less__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_540_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_541_order__less__le__trans,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_less_set_nat_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_542_order__less__le__trans,axiom,
! [X: int,Y2: int,Z: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_543_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_544_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_545_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_546_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_547_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_548_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_549_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_550_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_551_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_552_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_553_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_554_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_555_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_556_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_557_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_558_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_559_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_560_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_561_order__less__le__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_562_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_563_order__less__le__subst1,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_564_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_565_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_566_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_567_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_568_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_569_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_570_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_571_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_572_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_573_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_574_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_575_classes__subset__cube,axiom,
! [N: nat,T: nat,I: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_576_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_577_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_578_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_579_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_580_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_581_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_582_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_583_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_584_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_585_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_586_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_587_inv__into__cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% inv_into_cube_props(1)
thf(fact_588_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_589_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_590_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_591_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_592_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_593_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_594_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_595_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_596_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_597_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_598_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_599_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_600_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_601_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_602_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_603_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_604_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_605_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_606_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_607_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_608_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_609_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_610_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_611_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_612_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_613_inv__into__cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S
@ zero_zero_nat )
= S ) ) ).
% inv_into_cube_props(2)
thf(fact_614_psubsetI,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_615_subset__antisym,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_616_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_617_subsetI,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ X3 @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_618_subsetI,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ X3 @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_619_subsetI,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ X3 @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_620_some__inv__into__2,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into_2
thf(fact_621_some__inv__into,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into
thf(fact_622_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
@ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_623_psubsetD,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_624_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_625_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_626_psubsetD,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_627_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A6 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_628_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_629_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_630_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A6 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_631_verit__sko__forall__indirect2,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o,P4: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
= ( P4 @ X3 ) )
=> ( ( ! [X6: nat > nat] : ( P4 @ X6 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_632_verit__sko__forall__indirect,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) )
=> ( ( ! [X6: nat > nat] : ( P @ X6 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_633_verit__sko__ex__indirect2,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o,P4: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat @ P ) )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
= ( P4 @ X3 ) )
=> ( ( ? [X6: nat > nat] : ( P4 @ X6 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_634_verit__sko__ex__indirect,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat @ P ) )
=> ( ( ? [X6: nat > nat] : ( P @ X6 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_635_verit__sko__forall_H_H,axiom,
! [B3: nat > nat,A2: nat > nat,P: ( nat > nat ) > $o] :
( ( B3 = A2 )
=> ( ( ( fChoice_nat_nat @ P )
= A2 )
= ( ( fChoice_nat_nat @ P )
= B3 ) ) ) ).
% verit_sko_forall''
thf(fact_636_verit__sko__forall_H,axiom,
! [P: ( nat > nat ) > $o,A2: $o] :
( ( ( P
@ ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) )
= A2 )
=> ( ( ! [X6: nat > nat] : ( P @ X6 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_637_verit__sko__forall,axiom,
( ( ^ [P2: ( nat > nat ) > $o] :
! [X5: nat > nat] : ( P2 @ X5 ) )
= ( ^ [P3: ( nat > nat ) > $o] :
( P3
@ ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P3 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_638_verit__sko__ex_H,axiom,
! [P: ( nat > nat ) > $o,A2: $o] :
( ( ( P @ ( fChoice_nat_nat @ P ) )
= A2 )
=> ( ( ? [X6: nat > nat] : ( P @ X6 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_639_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_640_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_641_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_642_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_643_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_644_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_645_in__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,X: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_646_in__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,X: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B3 ) ) ) ).
% in_mono
thf(fact_647_in__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_648_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_649_subsetD,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_650_subsetD,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_651_subsetD,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_652_equalityE,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ~ ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_653_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_654_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A6 )
=> ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_655_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A6 )
=> ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_656_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A6 )
=> ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_657_equalityD1,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_658_equalityD2,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_659_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_660_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A6 )
=> ( member_nat_nat_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_661_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A6 )
=> ( member_nat_nat_nat2 @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_662_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A6 )
=> ( member_nat_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_663_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_664_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_665_subset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_666_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_667_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_668_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_less_set_nat_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_669_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat_nat @ B3 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_670_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ~ ( ord_le9059583361652607317at_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_671_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_672_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_673_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_674_psubsetE,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_675_cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) )
@ zero_zero_nat )
= S ) ) ).
% cube_props(2)
thf(fact_676_cube__props_I4_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) ) )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% cube_props(4)
thf(fact_677_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_678_Collect__subset,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_679_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_680_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_681_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_682_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_683_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A6 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_684_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A6 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_685_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S3: ( nat > nat ) > nat > nat,N: nat,R2: int,Chi: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S3 @ one_one_nat @ N @ T @ R2 @ Chi )
=> ! [S4: nat] :
( ( ord_less_nat @ S4 @ T )
=> ! [L3: nat] :
( ( ord_less_nat @ L3 @ T )
=> ( ( ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) ) )
= ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= L3 ) ) ) ) ) )
& ( ord_less_int
@ ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) ) )
@ R2 ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_686_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S3: ( nat > nat ) > nat > nat,N: nat,R2: nat,Chi: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ N @ T @ R2 @ Chi )
=> ! [S4: nat] :
( ( ord_less_nat @ S4 @ T )
=> ! [L3: nat] :
( ( ord_less_nat @ L3 @ T )
=> ( ( ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) ) )
= ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= L3 ) ) ) ) ) )
& ( ord_less_nat
@ ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) ) )
@ R2 ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_687_dim1__layered__subspace__as__line,axiom,
! [T: nat,S3: ( nat > nat ) > nat > nat,N: nat,R2: int,Chi: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S3 @ one_one_nat @ N @ T @ R2 @ Chi )
=> ? [C1: int,C22: int] :
( ( ord_less_int @ C1 @ R2 )
& ( ord_less_int @ C22 @ R2 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T )
=> ( ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) ) )
= C1 ) )
& ( ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_688_dim1__layered__subspace__as__line,axiom,
! [T: nat,S3: ( nat > nat ) > nat > nat,N: nat,R2: nat,Chi: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ N @ T @ R2 @ Chi )
=> ? [C1: nat,C22: nat] :
( ( ord_less_nat @ C1 @ R2 )
& ( ord_less_nat @ C22 @ R2 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T )
=> ( ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) ) )
= C1 ) )
& ( ( Chi
@ ( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_689_some__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ( P @ A )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( X3 = A ) )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_690_some__eq__trivial,axiom,
! [X: nat > nat] :
( ( fChoice_nat_nat
@ ^ [Y: nat > nat] : ( Y = X ) )
= X ) ).
% some_eq_trivial
thf(fact_691_some__sym__eq__trivial,axiom,
! [X: nat > nat] :
( ( fChoice_nat_nat
@ ( ^ [Y5: nat > nat,Z2: nat > nat] : ( Y5 = Z2 )
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_692_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restri4446420529079022766at_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_693_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_694_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_695_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_696_some__eq__imp,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat,B: nat > nat] :
( ( ( fChoice_nat_nat @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_697_tfl__some,axiom,
! [P6: ( nat > nat ) > $o,X4: nat > nat] :
( ( P6 @ X4 )
=> ( P6 @ ( fChoice_nat_nat @ P6 ) ) ) ).
% tfl_some
thf(fact_698_Eps__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice_nat_nat @ P )
= ( fChoice_nat_nat @ Q ) ) ) ).
% Eps_cong
thf(fact_699_someI,axiom,
! [P: ( nat > nat ) > $o,X: nat > nat] :
( ( P @ X )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI
thf(fact_700_some1__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ? [X4: nat > nat] :
( ( P @ X4 )
& ! [Y3: nat > nat] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_701_some__eq__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
% some_eq_ex
thf(fact_702_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat] :
( ( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_703_someI2__bex,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_704_someI2__bex,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_705_someI2__bex,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat > nat] :
( ( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_706_someI2__ex,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X_12: nat > nat] : ( P @ X_12 )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_707_someI__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ? [X_12: nat > nat] : ( P @ X_12 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI_ex
thf(fact_708_someI2,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat,Q: ( nat > nat ) > $o] :
( ( P @ A )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2
thf(fact_709_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_710_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_711_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= ( restrict_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_712_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= ( restrict_nat_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_713_pred__subset__eq,axiom,
! [R3: set_nat,S3: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S3 ) )
= ( ord_less_eq_set_nat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_714_pred__subset__eq,axiom,
! [R3: set_nat_nat_nat2,S3: set_nat_nat_nat2] :
( ( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ R3 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ S3 ) )
= ( ord_le5934964663421696068at_nat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_715_pred__subset__eq,axiom,
! [R3: set_nat_nat_nat,S3: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ R3 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ S3 ) )
= ( ord_le3211623285424100676at_nat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_716_pred__subset__eq,axiom,
! [R3: set_nat_nat,S3: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ R3 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ S3 ) )
= ( ord_le9059583361652607317at_nat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_717_dim1__subspace__is__line,axiom,
! [T: nat,S3: ( nat > nat ) > nat > nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_subspace @ S3 @ one_one_nat @ N @ T )
=> ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S5: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T ) )
@ N
@ T ) ) ) ).
% dim1_subspace_is_line
thf(fact_718_cube__props_I3_J,axiom,
! [S: nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat_nat2
@ ^ [S5: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat_nat2
@ ^ [S5: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_719_cube__props_I3_J,axiom,
! [S: nat,T: nat,S3: ( nat > nat ) > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat
@ ^ [S5: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat
@ ^ [S5: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_720_convex__bound__lt,axiom,
! [X: int,A: int,Y2: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y2 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_721_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_722_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_723_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_724_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_725_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_726_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_727_lessThan__eq__iff,axiom,
! [X: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_728_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_729_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_730_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_731_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_732_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_733_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_734_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_735_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_736_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_737_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_738_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_739_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_740_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_741_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_742_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_743_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_744_lessThan__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_745_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_746_lessThan__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I @ K ) ) ).
% lessThan_iff
thf(fact_747_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_748_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_749_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_750_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_751_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_752_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_753_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_754_lessThan__subset__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_755_lessThan__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_756_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_757_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_758_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_759_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_760_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_761_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_762_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_763_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_764_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_765_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_766_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_767_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_768_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_769_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_770_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_771_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_772_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_773_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_774_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_775_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_776_crossproduct__eq,axiom,
! [W: nat,Y2: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y2 ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_777_crossproduct__eq,axiom,
! [W: int,Y2: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y2 ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z ) ) ) ).
% crossproduct_eq
thf(fact_778_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_779_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_780_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_781_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_782_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_783_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_784_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_785_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_786_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_787_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_788_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_789_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_790_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_791_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_792_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_793_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_794_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_795_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_796_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_797_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_798_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_799_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_800_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_801_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_802_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_803_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_804_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_805_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_806_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_807_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_808_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_809_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_810_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_811_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_812_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_813_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_814_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_815_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_816_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_817_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_818_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_819_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_820_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_821_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_822_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_823_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_824_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_825_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_826_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_827_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_828_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_829_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_830_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_831_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_832_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_833_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_834_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_835_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_836_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_837_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_838_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_839_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_840_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_841_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_842_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_843_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_844_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_845_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_846_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_847_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_848_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_849_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_850_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_851_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_852_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_853_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_854_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_855_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_856_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_857_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_858_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_859_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_860_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_861_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_862_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_863_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_864_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_865_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_866_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_867_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_868_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_869_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_870_line__points__in__cube__unfolded,axiom,
! [L2: nat > nat > nat,N: nat,T: nat,S: nat,J: nat] :
( ( hales_is_line @ L2 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L2 @ S @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_871_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_872_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_873_cube__restrict,axiom,
! [J: nat,N: nat,Y2: nat > nat,T: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( member_nat_nat @ Y2 @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y2 @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T ) ) ) ) ).
% cube_restrict
thf(fact_874_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_875_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_876_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_877_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_878_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_879_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_880_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_881_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_882_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_883_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_884_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_885_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_886_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_887_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_888_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_889_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_890_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_891_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_892_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_893_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_894_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_895_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_896_sum__squares__ge__zero,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) ) ).
% sum_squares_ge_zero
thf(fact_897_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_898_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_899_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_900_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_901_mult__right__le__one__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_902_mult__left__le__one__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_903_not__sum__squares__lt__zero,axiom,
! [X: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_904_split__cube_I1_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_905_split__cube_I2_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y: nat] : ( X @ ( plus_plus_nat @ Y @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_906_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_907_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_908_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_909_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_910_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_911_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_912_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_913_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_914_convex__bound__le,axiom,
! [X: int,A: int,Y2: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y2 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_915_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_916_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_917_sum__squares__gt__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) )
= ( ( X != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_918_sum__squares__le__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_919_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y2 @ Z ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_920_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_921_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_922_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_923_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_924_mult__less__iff1,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y2 @ Z ) )
= ( ord_less_int @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_925_sum__squares__eq__zero__iff,axiom,
! [X: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_926_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_927_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_928_L__def,axiom,
( ( hales_is_line @ l2 @ n @ t )
& ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ l2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) @ X4 )
= C2 ) ) ) ) ).
% L_def
thf(fact_929_N__def,axiom,
( ( ord_less_nat @ zero_zero_nat @ n2 )
& ! [N4: nat] :
( ( ord_less_eq_nat @ n2 @ N4 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ t )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L4: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L4 @ N4 @ t )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi2 @ X4 )
= C2 ) ) ) ) ) ) ).
% N_def
thf(fact_930_that_I2_J,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% that(2)
thf(fact_931_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_932_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_933_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_934_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_935_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_936_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_937_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_938_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_939_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_940_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X: nat > nat > nat,A2: set_nat_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_941_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_942_image__eqI,axiom,
! [B: nat > nat > nat,F: ( nat > nat ) > nat > nat > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_3101123049818244468at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_943_image__ident,axiom,
! [Y6: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_944_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_945_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_946_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_947_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_948_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_949_image__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_950_image__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_951_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_952_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A2 @ B3 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_953_PiE__restrict,axiom,
! [F: nat > nat > nat,A2: set_nat,B3: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A2 @ B3 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_954_PiE__restrict,axiom,
! [F: nat > nat,A2: set_nat,B3: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B3 ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_955_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A2 @ B3 ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_956_image__add__0,axiom,
! [S3: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
= S3 ) ).
% image_add_0
thf(fact_957_image__add__0,axiom,
! [S3: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S3 )
= S3 ) ).
% image_add_0
thf(fact_958_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_959_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_960__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062N_O_A0_A_060_AN_A_092_060and_062_A_I_092_060forall_062N_H_092_060ge_062N_O_A_092_060forall_062_092_060chi_062_O_A_092_060chi_062_A_092_060in_062_Acube_AN_H_At_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_A_092_060longrightarrow_062_A_I_092_060exists_062L_Ac_O_Ac_A_060_Ar_A_092_060and_062_Ais__line_AL_AN_H_At_A_092_060and_062_A_I_092_060forall_062y_092_060in_062L_A_096_A_123_O_O_060t_125_O_A_092_060chi_062_Ay_A_061_Ac_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N5: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ N5 )
& ! [N4: nat] :
( ( ord_less_eq_nat @ N5 @ N4 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ t )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L4: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L4 @ N4 @ t )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi2 @ X4 )
= C2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>N. 0 < N \<and> (\<forall>N'\<ge>N. \<forall>\<chi>. \<chi> \<in> cube N' t \<rightarrow>\<^sub>E {..<r} \<longrightarrow> (\<exists>L c. c < r \<and> is_line L N' t \<and> (\<forall>y\<in>L ` {..<t}. \<chi> y = c))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_961__092_060open_062restrict_A_092_060chi_062_A_Icube_AN_H_At_J_A_092_060in_062_Acube_AN_H_At_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_092_060close_062,axiom,
( member_nat_nat_nat @ ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) )
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ t )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<open>restrict \<chi> (cube N' t) \<in> cube N' t \<rightarrow>\<^sub>E {..<r}\<close>
thf(fact_962_N_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ t )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L4: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L4 @ n @ t )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi2 @ X4 )
= C2 ) ) ) ) ) ).
% N'_props
thf(fact_963_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_964_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_965_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_966_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_967__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062L_O_Ais__line_AL_AN_H_At_A_092_060and_062_A_I_092_060exists_062c_060r_O_A_092_060forall_062y_092_060in_062L_A_096_A_123_O_O_060t_125_O_Arestrict_A_092_060chi_062_A_Icube_AN_H_At_J_Ay_A_061_Ac_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L4: nat > nat > nat] :
~ ( ( hales_is_line @ L4 @ n @ t )
& ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) @ X4 )
= C2 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>L. is_line L N' t \<and> (\<exists>c<r. \<forall>y\<in>L ` {..<t}. restrict \<chi> (cube N' t) y = c) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_968_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_969_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_970_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_971_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_972_power__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_973_power__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_974_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_975_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_976_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I5: set_nat_nat,X7: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I5 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ X7 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X7 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_977_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I5: set_nat,X7: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I5 ) @ ( piE_nat_nat_nat2 @ I5 @ X7 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X7 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_978_restrict__PiE__iff,axiom,
! [F: nat > nat,I5: set_nat,X7: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I5 ) @ ( piE_nat_nat @ I5 @ X7 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( member_nat @ ( F @ X2 ) @ ( X7 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_979_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I5: set_nat_nat,X7: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I5 ) @ ( piE_nat_nat_nat @ I5 @ X7 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat @ ( F @ X2 ) @ ( X7 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_980_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B3 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y4: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y4
@ ( piE_nat_nat_nat @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y4 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_981_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I4: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y4: nat > nat] :
( ( ( member_nat_nat @ Y4
@ ( piE_nat_nat @ A2
@ ^ [I4: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y4 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_982_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B3 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y4: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y4
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y4 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_983_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B3 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I4: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y4: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y4
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I4: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y4 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_984_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B3 ) ) ) ).
% image_mono
thf(fact_985_image__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_986_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_987_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B3: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_988_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B3: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_989_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_990_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_991_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B3: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_992_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B3: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_993_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_994_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_995_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_3101123049818244468at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_996_subset__imageE,axiom,
! [B3: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B3
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_997_subset__imageE,axiom,
! [B3: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B3
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_998_image__subset__iff,axiom,
! [F: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_999_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B3 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_1000_subset__image__iff,axiom,
! [B3: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1001_subset__image__iff,axiom,
! [B3: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B3
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1002_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1003_Compr__image__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1004_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1005_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1006_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_7809927846809980933at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_7809927846809980933at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1007_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_913610194320715013at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_913610194320715013at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1008_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1009_Compr__image__eq,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( image_6919068903512877573at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6919068903512877573at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1010_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_1262493855416953332at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1262493855416953332at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1011_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_1545173636400105204at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1545173636400105204at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1012_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1013_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1014_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1015_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1016_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1017_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1018_rev__image__eqI,axiom,
! [X: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1019_rev__image__eqI,axiom,
! [X: nat > nat > nat,A2: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1020_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1021_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat > nat > nat,F: ( nat > nat ) > nat > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat2 @ B @ ( image_3101123049818244468at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1022_image__image,axiom,
! [F: nat > nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1023_image__image,axiom,
! [F: nat > nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1024_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1025_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_3205354838064109189at_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1026_ball__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1027_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1028_image__cong,axiom,
! [M3: set_nat_nat,N6: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M3 = N6 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M3 )
= ( image_3205354838064109189at_nat @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_1029_image__cong,axiom,
! [M3: set_nat,N6: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M3 = N6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M3 )
= ( image_nat_nat_nat2 @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_1030_bex__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1031_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1032_image__iff,axiom,
! [Z: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ Z @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1033_image__iff,axiom,
! [Z: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1034_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1035_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1036_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_1037_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1038_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1039_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1040_imageI,axiom,
! [X: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1041_imageI,axiom,
! [X: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1042_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1043_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( image_3101123049818244468at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1044_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1045_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1046_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1047_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1048_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1049_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1050_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1051_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1052_imageE,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1053_imageE,axiom,
! [B: nat > nat,F: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat] :
( ( member_nat_nat @ B @ ( image_1545173636400105204at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1054_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B3: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I4: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1055_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B3: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1056_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B3: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I4: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1057_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B3: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1058_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B3 )
=> ? [X3: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X3
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I4: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1059_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B3: set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B3 )
=> ? [X3: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X3
@ ( piE_nat_nat_nat_nat5 @ A2
@ ^ [I4: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1060_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B3: set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X3
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I4: ( nat > nat ) > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1061_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B3: set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X3
@ ( piE_nat_nat_nat_nat2 @ A2
@ ^ [I4: nat > nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1062_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B3 )
=> ? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1063_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B3: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B3 )
=> ? [X3: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X3
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I4: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_1064_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1065_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1066_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1067_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1068_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1069_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1070_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1071_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1072_left__right__inverse__power,axiom,
! [X: nat,Y2: nat,N: nat] :
( ( ( times_times_nat @ X @ Y2 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1073_left__right__inverse__power,axiom,
! [X: int,Y2: int,N: nat] :
( ( ( times_times_int @ X @ Y2 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y2 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_1074_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1075_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1076_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_1077_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_1078_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1079_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B3 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_1080_PiE__mono,axiom,
! [A2: set_nat,B3: nat > set_nat,C4: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B3 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_1081_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B3 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_1082_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B3 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_1083_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat_nat,C4: ( nat > nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B3 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_1084_PiE__mono,axiom,
! [A2: set_nat,B3: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B3 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_1085_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1086_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1087_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1088_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1089_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1090_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1091_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1092_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1093_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1094_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1095_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1096_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1097_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1098_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_1099_power__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1100_power__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_1101_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1102_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1103_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1104_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1105_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1106_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1107_power__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1108_power__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1109_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1110_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1111_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1112_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1113_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1114_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1115_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1116_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1117_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1118_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1119_hj__def,axiom,
( hales_hj
= ( ^ [R4: nat,T2: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ T2 )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R4 ) ) )
=> ? [L5: nat > nat > nat,C3: nat] :
( ( ord_less_nat @ C3 @ R4 )
& ( hales_is_line @ L5 @ N8 @ T2 )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ L5 @ ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( Chi3 @ X2 )
= C3 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_1120_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1121_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1122_dim0__layered__subspace__ex,axiom,
! [Chi: ( nat > nat ) > nat,N: nat,T: nat,R2: nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S2: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S2 @ zero_zero_nat @ N @ T @ R2 @ Chi ) ) ).
% dim0_layered_subspace_ex
thf(fact_1123_lhj__def,axiom,
( hales_lhj
= ( ^ [R4: nat,T2: nat,K3: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R4 ) ) )
=> ? [S6: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S6 @ K3 @ N8 @ T2 @ R4 @ Chi3 ) ) ) ) ) ) ).
% lhj_def
thf(fact_1124_layered__subspace__def,axiom,
( hales_4259056829518216709ce_int
= ( ^ [S6: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T2: nat,R4: int,Chi3: ( nat > nat ) > int] :
( ( hales_is_subspace @ S6 @ K3 @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C3: int] :
( ( ord_less_int @ C3 @ R4 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K3 @ T2 @ X2 ) )
=> ( ( Chi3 @ ( S6 @ Y ) )
= C3 ) ) ) )
& ( member_nat_nat_int @ Chi3
@ ( piE_nat_nat_int @ ( hales_cube @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_int @ R4 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1125_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T2: nat,R4: nat,Chi3: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S6 @ K3 @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C3: nat] :
( ( ord_less_nat @ C3 @ R4 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K3 @ T2 @ X2 ) )
=> ( ( Chi3 @ ( S6 @ Y ) )
= C3 ) ) ) )
& ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R4 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1126_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat,B3: set_nat_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1127_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( collect_nat_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1128_all__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1129_all__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ A2 )
=> ( P @ ( image_3205354838064109189at_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1130_atMost__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or9140604705432621368at_nat @ K ) )
= ( ord_less_eq_nat_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1131_atMost__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or5033131092550408871at_nat @ K ) )
= ( ord_le2017632242545079438at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1132_atMost__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or6142498856979658663at_nat @ K ) )
= ( ord_le3127000006974329230at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1133_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1134_atMost__iff,axiom,
! [I: set_nat_nat,K: set_nat_nat] :
( ( member_set_nat_nat @ I @ ( set_or250740698829186286at_nat @ K ) )
= ( ord_le9059583361652607317at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1135_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_1136_atMost__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_1137_atMost__subset__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X ) @ ( set_or250740698829186286at_nat @ Y2 ) )
= ( ord_le9059583361652607317at_nat @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_1138_atMost__subset__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y2 ) )
= ( ord_less_eq_int @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_1139_atMost__subset__iff,axiom,
! [X: nat > nat,Y2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X ) @ ( set_or9140604705432621368at_nat @ Y2 ) )
= ( ord_less_eq_nat_nat @ X @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_1140_image__add__atMost,axiom,
! [C: int,A: int] :
( ( image_int_int @ ( plus_plus_int @ C ) @ ( set_ord_atMost_int @ A ) )
= ( set_ord_atMost_int @ ( plus_plus_int @ C @ A ) ) ) ).
% image_add_atMost
thf(fact_1141_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_1142_cube__def,axiom,
( hales_cube
= ( ^ [N3: nat,T2: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N3 )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T2 ) ) ) ) ).
% cube_def
thf(fact_1143_subspace__elems__embed,axiom,
! [S3: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S3 @ K @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S3 @ ( hales_cube @ K @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_1144_is__line__def,axiom,
( hales_is_line
= ( ^ [L5: nat > nat > nat,N3: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L5
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I4: nat] : ( hales_cube @ N3 @ T2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
=> ( ! [X2: nat] :
( ( ord_less_nat @ X2 @ T2 )
=> ! [Y: nat] :
( ( ord_less_nat @ Y @ T2 )
=> ( ( L5 @ X2 @ J3 )
= ( L5 @ Y @ J3 ) ) ) )
| ! [S5: nat] :
( ( ord_less_nat @ S5 @ T2 )
=> ( ( L5 @ S5 @ J3 )
= S5 ) ) ) )
& ? [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
& ! [S5: nat] :
( ( ord_less_nat @ S5 @ T2 )
=> ( ( L5 @ S5 @ J3 )
= S5 ) ) ) ) ) ) ).
% is_line_def
thf(fact_1145_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1146_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1147_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1148_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1149_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1150_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1151_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1152_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1153_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1154_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1155_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1156_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1157_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1158_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1159_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1160_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1161_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1162_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1163_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1164_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1165_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1166_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1167_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1168_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1169_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1170_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1171_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1172_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1173_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1174_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1175_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1176_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1177_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1178_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1179_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1180_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1181_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1182_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1183_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1184_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1185_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1186_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1187_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1188_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1189_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1190_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1191_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1192_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1193_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1194_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1195_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1196_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1197_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1198_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1199_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1200_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1201_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1202_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1203_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1204_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1205_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1206_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1207_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1208_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1209_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1210_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1211_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1212_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1213_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1214_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1215_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1216_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1217_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1218_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1219_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1220_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1221_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1222_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1223_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1224_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1225_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1226_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1227_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1228_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1229_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1230_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1231_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1232_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1233_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1234_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1235_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1236_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1237_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1238_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1239_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1240_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1241_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1242_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1243_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1244_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1245_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P4: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1246_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P4: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1247_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1248_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1249_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1250_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1251_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1252_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1253_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1254_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1255_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1256_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1257_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1258_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1259_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1260_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1261_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1262_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1263_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1264_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1265_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1266_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
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,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_T,axiom,
! [P: ( ( nat > nat ) > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat @ P ) )
= ( ? [X6: ( nat > nat ) > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( nat > nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat2 @ P ) )
= ( ? [X6: nat > nat > nat] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( l @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) )
@ one_one_nat
@ n
@ ( plus_plus_nat @ t @ one_one_nat ) ) ).
%------------------------------------------------------------------------------