TPTP Problem File: SLH0209^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Hales_Jewett/0002_Hales_Jewett/prob_00352_013618__5642834_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1482 ( 447 unt; 212 typ; 0 def)
% Number of atoms : 3839 (1191 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11175 ( 295 ~; 56 |; 232 &;8849 @)
% ( 0 <=>;1743 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 3693 (3693 >; 0 *; 0 +; 0 <<)
% Number of symbols : 193 ( 190 usr; 12 con; 0-6 aty)
% Number of variables : 4145 ( 363 ^;3630 !; 152 ?;4145 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:39:19.031
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_na862621915428039857at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na3160727226427813041at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_na6023386892149231537at_nat: $tType ).
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__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (190)
thf(sy_c_Fun_Oinj__on_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
inj_on8598928376616154831at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Oinj__on_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on484924104190628815at_nat: ( ( ( nat > nat ) > nat ) > nat > nat > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Oinj__on_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,
inj_on3244975737280743776at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Oinj__on_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
inj_on7066290451648512369at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
inj_on991820888481174479at_nat: ( ( nat > nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_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,
inj_on3527655518263895648at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
inj_on169972799159246449at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_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,
inj_on3974237167252785120at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on2461717442902640625at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
inj_on5066063743922159217at_nat: ( nat > ( nat > nat ) > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on6175431508351409009at_nat: ( nat > nat > nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on_nat_nat_nat2: ( nat > nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Othe__inv__into_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
the_in2738853937647574033at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
the_in3848221702076823825at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > nat > nat > nat ) > ( nat > nat > nat ) > ( nat > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_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,
the_in6455806401390066082at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > nat > nat ) > ( nat > nat ) > ( nat > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
the_in7568536272828005555at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > nat ) > nat > ( nat > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
the_in4355118486367369489at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_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,
the_in6738486182373217954at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > nat > nat ) > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
the_in672218620338739635at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_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,
the_in7185067831362107426at_nat: set_nat_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in2963963264082133811at_nat: set_nat_nat > ( ( nat > nat ) > nat > nat ) > ( nat > nat ) > nat > nat ).
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_Fun_Othe__inv__into_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
the_in5568309565101652403at_nat: set_nat > ( nat > ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
the_in6677677329530902195at_nat: set_nat > ( nat > nat > nat > nat ) > ( nat > nat > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in3844390324871770692at_nat: set_nat > ( nat > nat > nat ) > ( nat > nat ) > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_na7011197487990663704at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ) > set_na6023386892149231537at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na8120565252419913496at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat_nat ) > set_na3160727226427813041at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_na8627462036710459160at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat_nat_nat2 ) > set_na862621915428039857at_nat ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
restri3492029194894058636at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri4601396959323308428at_nat: ( ( ( nat > nat ) > nat ) > nat > nat > nat ) > set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_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,
restri5282449906285418141at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restri9050993537824894510at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
restri5108293743613854092at_nat: ( ( nat > nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat > ( nat > nat > nat ) > ( nat > 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_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_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
sgn_sgn_int: int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
uminus5164713059990985517at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
uminus2441371681993390125at_nat: set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
uminus4145589374814813630at_nat: set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
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_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_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_Oset__incr,type,
hales_set_incr: nat > set_nat > set_nat ).
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_Lattices__Big_Oord__class_Oarg__min_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
lattic4097496154029735450at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
lattic6702328201646789547at_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001t__Nat__Onat,type,
lattic8739620818006775868at_nat: ( nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_List_Ofolding__insort__key_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
foldin311176147066610597at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > set_nat_nat > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
foldin4490235068905269046at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > set_nat > ( nat > nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Int__Oint_001t__Int__Oint,type,
foldin9130795139525211007nt_int: ( int > int > $o ) > ( int > int > $o ) > set_int > ( int > int ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001t__Nat__Onat,type,
foldin8133931898133206727at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > set_nat > ( nat > nat ) > $o ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_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_Orderings_Oordering__top_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
orderi4315198199014340228at_nat: ( set_nat_nat > set_nat_nat > $o ) > ( set_nat_nat > set_nat_nat > $o ) > set_nat_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
top_to9043804989276028657_nat_o: ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
top_to4209272211376415217_nat_o: ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
top_top_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
top_to6379112975903909524at_nat: set_nat_nat_nat2 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
top_to3655771597906314132at_nat: set_nat_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_8393830757900314979at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2 > set_nat_nat_nat2 ).
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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_279826485474788963at_nat: ( ( ( nat > nat ) > nat ) > nat > nat > nat ) > set_nat_nat_nat2 > set_nat_nat_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_786723269765334627at_nat: ( ( nat > nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat > set_nat_nat_nat2 ).
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_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: 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_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member1506620086977839122at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_na6023386892149231537at_nat > $o ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8006650745835019538at_nat: ( ( ( nat > nat ) > nat ) > nat > nat > nat ) > set_na3160727226427813041at_nat > $o ).
thf(sy_c_member_001_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,type,
member4489290058226556451at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_na8843485148432118594at_nat > $o ).
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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member3122884635697634578at_nat: ( ( nat > nat > nat ) > ( nat > nat ) > nat ) > set_na862621915428039857at_nat > $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_k,type,
k: nat ).
% Relevant facts (1264)
thf(fact_0__092_060open_062inj__on_A_I_092_060lambda_062x_O_A_092_060lambda_062y_092_060in_062_123_O_O_0601_125_O_Ax_J_A_123_O_O_060k_125_092_060close_062,axiom,
( inj_on_nat_nat_nat2
@ ^ [X: nat] :
( restrict_nat_nat
@ ^ [Y: nat] : X
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ k ) ) ).
% \<open>inj_on (\<lambda>x. \<lambda>y\<in>{..<1}. x) {..<k}\<close>
thf(fact_1__C_K_C,axiom,
( ( image_nat_nat_nat2
@ ^ [X: nat] :
( restrict_nat_nat
@ ^ [Y: nat] : X
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ k ) )
= ( hales_cube @ one_one_nat @ k ) ) ).
% "*"
thf(fact_2_inj__on__restrict__eq,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A ) @ A )
= ( inj_on_nat_nat_nat2 @ F @ A ) ) ).
% inj_on_restrict_eq
thf(fact_3_inj__on__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ ( restri4446420529079022766at_nat @ F @ A ) @ A )
= ( inj_on2461717442902640625at_nat @ F @ A ) ) ).
% inj_on_restrict_eq
thf(fact_4_inj__on__restrict__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( restrict_nat_nat @ F @ A ) @ A )
= ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_restrict_eq
thf(fact_5_image__restrict__eq,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( image_786723269765334627at_nat @ ( restri5108293743613854092at_nat @ F @ A ) @ A )
= ( image_786723269765334627at_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_6_image__restrict__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( image_7809927846809980933at_nat @ ( restri9050993537824894510at_nat @ F @ A ) @ A )
= ( image_7809927846809980933at_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_7_image__restrict__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( image_1262493855416953332at_nat @ ( restri5282449906285418141at_nat @ F @ A ) @ A )
= ( image_1262493855416953332at_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_8_image__restrict__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( image_279826485474788963at_nat @ ( restri4601396959323308428at_nat @ F @ A ) @ A )
= ( image_279826485474788963at_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_9_image__restrict__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( image_8393830757900314979at_nat @ ( restri3492029194894058636at_nat @ F @ A ) @ A )
= ( image_8393830757900314979at_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_10_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F @ A ) @ A )
= ( image_3205354838064109189at_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_11_image__restrict__eq,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A ) @ A )
= ( image_nat_nat_nat2 @ F @ A ) ) ).
% image_restrict_eq
thf(fact_12_image__restrict__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A ) @ A )
= ( image_nat_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_13_image__ident,axiom,
! [Y2: set_nat_nat_nat2] :
( ( image_8393830757900314979at_nat
@ ^ [X: ( nat > nat ) > nat] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_14_image__ident,axiom,
! [Y2: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_15_lessThan__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_16_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_17_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_18_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_19_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_20_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_21_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_22_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X2: nat > nat > nat,A: set_nat_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat_nat2 @ X2 @ A )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_23_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X2: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_24_image__eqI,axiom,
! [B: nat > nat > nat,F: ( nat > nat ) > nat > nat > nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat_nat2 @ B @ ( image_3101123049818244468at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_25_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_26_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_27_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat_nat2 @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat_nat2 @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_28_inj__on__image__iff,axiom,
! [A: set_nat_nat,G: ( nat > nat ) > nat > nat,F: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( inj_on2461717442902640625at_nat @ G @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( inj_on2461717442902640625at_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_29_split__cube_I2_J,axiom,
! [X2: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y: nat] : ( X2 @ ( plus_plus_nat @ Y @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_30_split__cube_I1_J,axiom,
! [X2: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_31_restrict__ext,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat @ F @ A )
= ( restrict_nat_nat @ G @ A ) ) ) ).
% restrict_ext
thf(fact_32_restrict__ext,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A )
= ( restri4446420529079022766at_nat @ G @ A ) ) ) ).
% restrict_ext
thf(fact_33_restrict__ext,axiom,
! [A: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A )
= ( restrict_nat_nat_nat2 @ G @ A ) ) ) ).
% restrict_ext
thf(fact_34_restrict__apply_H,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( restrict_nat_nat @ F @ A @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_35_restrict__apply_H,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( restri4446420529079022766at_nat @ F @ A @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_36_restrict__apply_H,axiom,
! [X2: nat,A: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( restrict_nat_nat_nat2 @ F @ A @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_37_inj__on__id2,axiom,
! [A: set_nat_nat] :
( inj_on2461717442902640625at_nat
@ ^ [X: nat > nat] : X
@ A ) ).
% inj_on_id2
thf(fact_38_inj__on__id2,axiom,
! [A: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A ) ).
% inj_on_id2
thf(fact_39_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_40_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_41_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_42_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_43_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_44_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_45_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_46_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_47_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_48_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_49_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_50_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A2: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_51_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_52_add_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_53_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_54_add_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_55_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_56_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_57_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_58_add_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_59_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_60_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_61_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_62_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_63_inj__on__add,axiom,
! [A2: nat,A: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A2 ) @ A ) ).
% inj_on_add
thf(fact_64_inj__on__add,axiom,
! [A2: int,A: set_int] : ( inj_on_int_int @ ( plus_plus_int @ A2 ) @ A ) ).
% inj_on_add
thf(fact_65_inj__on__add_H,axiom,
! [A2: nat,A: set_nat] :
( inj_on_nat_nat
@ ^ [B3: nat] : ( plus_plus_nat @ B3 @ A2 )
@ A ) ).
% inj_on_add'
thf(fact_66_inj__on__add_H,axiom,
! [A2: int,A: set_int] :
( inj_on_int_int
@ ^ [B3: int] : ( plus_plus_int @ B3 @ A2 )
@ A ) ).
% inj_on_add'
thf(fact_67_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_68_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_69_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_70_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_71_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_72_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_73_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_74_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_75_rev__image__eqI,axiom,
! [X2: nat > nat > nat,A: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_76_rev__image__eqI,axiom,
! [X2: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_77_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: nat > nat > nat,F: ( nat > nat ) > nat > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_3101123049818244468at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_78_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_79_ball__imageD,axiom,
! [F: nat > nat > nat,A: set_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_80_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_81_ball__imageD,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ ( image_786723269765334627at_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_82_ball__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_7809927846809980933at_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_83_ball__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_1262493855416953332at_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_84_ball__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,P: ( nat > nat > nat ) > $o] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ ( image_279826485474788963at_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_85_ball__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ ( image_8393830757900314979at_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_86_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M )
= ( image_nat_nat_nat2 @ G @ N ) ) ) ) ).
% image_cong
thf(fact_87_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_88_image__cong,axiom,
! [M: set_nat_nat_nat,N: set_nat_nat_nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat,G: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( M = N )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_786723269765334627at_nat @ F @ M )
= ( image_786723269765334627at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_89_image__cong,axiom,
! [M: set_nat_nat_nat2,N: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,G: ( ( nat > nat ) > nat ) > nat] :
( ( M = N )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_7809927846809980933at_nat @ F @ M )
= ( image_7809927846809980933at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_90_image__cong,axiom,
! [M: set_nat_nat_nat2,N: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat,G: ( ( nat > nat ) > nat ) > nat > nat] :
( ( M = N )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1262493855416953332at_nat @ F @ M )
= ( image_1262493855416953332at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_91_image__cong,axiom,
! [M: set_nat_nat_nat2,N: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat > nat,G: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( M = N )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_279826485474788963at_nat @ F @ M )
= ( image_279826485474788963at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_92_image__cong,axiom,
! [M: set_nat_nat_nat2,N: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,G: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( M = N )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_8393830757900314979at_nat @ F @ M )
= ( image_8393830757900314979at_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_93_bex__imageD,axiom,
! [F: nat > nat > nat,A: set_nat,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_94_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_95_bex__imageD,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ ( image_786723269765334627at_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_96_bex__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_7809927846809980933at_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_97_bex__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_1262493855416953332at_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_98_bex__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,P: ( nat > nat > nat ) > $o] :
( ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ ( image_279826485474788963at_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_99_bex__imageD,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ ( image_8393830757900314979at_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_100_image__iff,axiom,
! [Z: nat > nat,F: nat > nat > nat,A: set_nat] :
( ( member_nat_nat @ Z @ ( image_nat_nat_nat2 @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_101_image__iff,axiom,
! [Z: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( member_nat_nat @ Z @ ( image_1262493855416953332at_nat @ F @ A ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_102_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_103_image__iff,axiom,
! [Z: nat,F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( member_nat @ Z @ ( image_7809927846809980933at_nat @ F @ A ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_104_image__iff,axiom,
! [Z: nat > nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( member_nat_nat_nat2 @ Z @ ( image_279826485474788963at_nat @ F @ A ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_105_image__iff,axiom,
! [Z: ( nat > nat ) > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( member_nat_nat_nat @ Z @ ( image_786723269765334627at_nat @ F @ A ) )
= ( ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_106_image__iff,axiom,
! [Z: ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ Z @ ( image_8393830757900314979at_nat @ F @ A ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_107_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_108_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_109_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_110_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_111_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( image_6919068903512877573at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_112_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( image_5809701139083627781at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_113_imageI,axiom,
! [X2: nat > nat > nat,A: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_913610194320715013at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_114_imageI,axiom,
! [X2: ( nat > nat ) > nat,A: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_115_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( image_3101123049818244468at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_116_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( image_1991755285388994676at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_117_inj__on__inverseI,axiom,
! [A: set_nat_nat,G: ( nat > nat ) > nat > nat,F: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on2461717442902640625at_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_118_inj__on__inverseI,axiom,
! [A: set_nat,G: ( nat > nat ) > nat,F: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat_nat2 @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_119_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_120_inj__on__contraD,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,X2: nat > nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( X2 != Y3 )
=> ( ( member_nat_nat @ X2 @ A )
=> ( ( member_nat_nat @ Y3 @ A )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_121_inj__on__contraD,axiom,
! [F: nat > nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( X2 != Y3 )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_122_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X2 != Y3 )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_123_inj__on__eq__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,X2: nat > nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( member_nat_nat @ X2 @ A )
=> ( ( member_nat_nat @ Y3 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_124_inj__on__eq__iff,axiom,
! [F: nat > nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_125_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_126_inj__on__cong,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [A4: nat > nat] :
( ( member_nat_nat @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on2461717442902640625at_nat @ F @ A )
= ( inj_on2461717442902640625at_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_127_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_nat_nat_nat2 @ F @ A )
= ( inj_on_nat_nat_nat2 @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_128_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_129_inj__on__def,axiom,
( inj_on_nat_nat_nat2
= ( ^ [F2: nat > nat > nat,A5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ! [Y: nat] :
( ( member_nat @ Y @ A5 )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_130_inj__on__def,axiom,
( inj_on2461717442902640625at_nat
= ( ^ [F2: ( nat > nat ) > nat > nat,A5: set_nat_nat] :
! [X: nat > nat] :
( ( member_nat_nat @ X @ A5 )
=> ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ A5 )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_131_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ! [Y: nat] :
( ( member_nat @ Y @ A5 )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_132_inj__onI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat,Y4: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( member_nat_nat @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on2461717442902640625at_nat @ F @ A ) ) ).
% inj_onI
thf(fact_133_inj__onI,axiom,
! [A: set_nat,F: nat > nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_nat_nat_nat2 @ F @ A ) ) ).
% inj_onI
thf(fact_134_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_135_inj__onD,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,X2: nat > nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( ( member_nat_nat @ Y3 @ A )
=> ( X2 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_136_inj__onD,axiom,
! [F: nat > nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( X2 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_137_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( X2 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_138_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_139_Compr__image__eq,axiom,
! [F: nat > nat > nat,A: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_nat_nat_nat2 @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_140_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_141_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_142_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat,A: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_913610194320715013at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_913610194320715013at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_143_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_7809927846809980933at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7809927846809980933at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_144_Compr__image__eq,axiom,
! [F: nat > nat > nat > nat,A: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ ( image_6919068903512877573at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_6919068903512877573at_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_145_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ ( image_5809701139083627781at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_146_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A: set_nat_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_1545173636400105204at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_1545173636400105204at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_147_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_1262493855416953332at_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_1262493855416953332at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_148_image__image,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_149_image__image,axiom,
! [F: ( nat > nat ) > nat,G: nat > nat > nat,A: set_nat] :
( ( image_nat_nat_nat @ F @ ( image_nat_nat_nat2 @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_150_image__image,axiom,
! [F: nat > nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat_nat2
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_151_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat,A: set_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ A ) )
= ( image_nat_nat_nat2
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_152_image__image,axiom,
! [F: nat > nat,G: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( image_nat_nat @ F @ ( image_7809927846809980933at_nat @ G @ A ) )
= ( image_7809927846809980933at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_153_image__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,G: nat > ( nat > nat ) > nat,A: set_nat] :
( ( image_7809927846809980933at_nat @ F @ ( image_5809701139083627781at_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_154_image__image,axiom,
! [F: ( nat > nat ) > nat,G: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( image_nat_nat_nat @ F @ ( image_1262493855416953332at_nat @ G @ A ) )
= ( image_7809927846809980933at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_155_image__image,axiom,
! [F: nat > nat > nat,G: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( image_nat_nat_nat2 @ F @ ( image_7809927846809980933at_nat @ G @ A ) )
= ( image_1262493855416953332at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_156_image__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,G: nat > ( nat > nat ) > nat,A: set_nat] :
( ( image_1262493855416953332at_nat @ F @ ( image_5809701139083627781at_nat @ G @ A ) )
= ( image_nat_nat_nat2
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_157_image__image,axiom,
! [F: nat > nat > nat > nat,G: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( image_6919068903512877573at_nat @ F @ ( image_7809927846809980933at_nat @ G @ A ) )
= ( image_279826485474788963at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_158_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A2: nat > nat > nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A2 @ ( collect_nat_nat_nat2 @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A2: ( nat > nat ) > nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A2 @ ( collect_nat_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_166_imageE,axiom,
! [B: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_167_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_168_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_169_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_170_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A ) ) ) ).
% imageE
thf(fact_171_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_172_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_173_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_174_imageE,axiom,
! [B: nat > nat,F: ( nat > nat > nat ) > nat > nat,A: set_nat_nat_nat] :
( ( member_nat_nat @ B @ ( image_1545173636400105204at_nat @ F @ A ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A ) ) ) ).
% imageE
thf(fact_175_imageE,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ A ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_176_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_177_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat_nat_nat2 > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( Sup
@ ( image_8393830757900314979at_nat
@ ^ [X: ( nat > nat ) > nat] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_178_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_179_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat_nat_nat2 > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( Inf
@ ( image_8393830757900314979at_nat
@ ^ [X: ( nat > nat ) > nat] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_180_cube__restrict,axiom,
! [J: nat,N2: nat,Y3: nat > nat,T: nat] :
( ( ord_less_nat @ J @ N2 )
=> ( ( member_nat_nat @ Y3 @ ( hales_cube @ N2 @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y3 @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T ) ) ) ) ).
% cube_restrict
thf(fact_181_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_182_the__inv__into__onto,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ A )
=> ( ( image_nat_nat_nat2 @ ( the_in5300466440149791684at_nat @ A @ F ) @ ( image_nat_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_183_the__inv__into__onto,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( image_nat_nat_nat @ ( the_in3844390324871770692at_nat @ A @ F ) @ ( image_nat_nat_nat2 @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_184_the__inv__into__onto,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ A )
=> ( ( image_5809701139083627781at_nat @ ( the_in7568536272828005555at_nat @ A @ F ) @ ( image_7809927846809980933at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_185_the__inv__into__onto,axiom,
! [F: nat > ( nat > nat ) > nat,A: set_nat] :
( ( inj_on5066063743922159217at_nat @ F @ A )
=> ( ( image_7809927846809980933at_nat @ ( the_in5568309565101652403at_nat @ A @ F ) @ ( image_5809701139083627781at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_186_the__inv__into__onto,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( image_3205354838064109189at_nat @ ( the_in2963963264082133811at_nat @ A @ F ) @ ( image_3205354838064109189at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_187_the__inv__into__onto,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ A )
=> ( ( image_1991755285388994676at_nat @ ( the_in6455806401390066082at_nat @ A @ F ) @ ( image_1262493855416953332at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_188_the__inv__into__onto,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat] :
( ( inj_on3974237167252785120at_nat @ F @ A )
=> ( ( image_1262493855416953332at_nat @ ( the_in7185067831362107426at_nat @ A @ F ) @ ( image_1991755285388994676at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_189_the__inv__into__onto,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ A )
=> ( ( image_786723269765334627at_nat @ ( the_in3848221702076823825at_nat @ A @ F ) @ ( image_279826485474788963at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_190_the__inv__into__onto,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ A )
=> ( ( image_279826485474788963at_nat @ ( the_in4355118486367369489at_nat @ A @ F ) @ ( image_786723269765334627at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_191_subspace__inj__on__cube,axiom,
! [S: ( nat > nat ) > nat > nat,K: nat,N2: nat,T: nat] :
( ( hales_is_subspace @ S @ K @ N2 @ T )
=> ( inj_on2461717442902640625at_nat @ S @ ( hales_cube @ K @ T ) ) ) ).
% subspace_inj_on_cube
thf(fact_192_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N3: nat] :
( image_nat_nat
@ ^ [A3: nat] : ( plus_plus_nat @ A3 @ N3 ) ) ) ) ).
% set_incr_def
thf(fact_193_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_194_cube__subset,axiom,
! [N2: nat,T: nat] : ( ord_le9059583361652607317at_nat @ ( hales_cube @ N2 @ T ) @ ( hales_cube @ N2 @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% cube_subset
thf(fact_195_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat > nat,D: nat > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat_nat2 @ C2 @ A ) )
= ( Sup @ ( image_nat_nat_nat2 @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_196_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_197_Sup_OSUP__cong,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat,C2: ( nat > nat > nat ) > ( nat > nat ) > nat,D: ( nat > nat > nat ) > ( nat > nat ) > nat,Sup: set_nat_nat_nat2 > ( nat > nat ) > nat] :
( ( A = B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_786723269765334627at_nat @ C2 @ A ) )
= ( Sup @ ( image_786723269765334627at_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_198_Sup_OSUP__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > nat,D: ( ( nat > nat ) > nat ) > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_7809927846809980933at_nat @ C2 @ A ) )
= ( Sup @ ( image_7809927846809980933at_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_199_Sup_OSUP__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > nat > nat,D: ( ( nat > nat ) > nat ) > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_1262493855416953332at_nat @ C2 @ A ) )
= ( Sup @ ( image_1262493855416953332at_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_200_Sup_OSUP__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > nat > nat > nat,D: ( ( nat > nat ) > nat ) > nat > nat > nat,Sup: set_nat_nat_nat > nat > nat > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_279826485474788963at_nat @ C2 @ A ) )
= ( Sup @ ( image_279826485474788963at_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_201_Sup_OSUP__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,D: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,Sup: set_nat_nat_nat2 > ( nat > nat ) > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_8393830757900314979at_nat @ C2 @ A ) )
= ( Sup @ ( image_8393830757900314979at_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_202_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat > nat,D: nat > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat_nat2 @ C2 @ A ) )
= ( Inf @ ( image_nat_nat_nat2 @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_203_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_204_Inf_OINF__cong,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat,C2: ( nat > nat > nat ) > ( nat > nat ) > nat,D: ( nat > nat > nat ) > ( nat > nat ) > nat,Inf: set_nat_nat_nat2 > ( nat > nat ) > nat] :
( ( A = B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_786723269765334627at_nat @ C2 @ A ) )
= ( Inf @ ( image_786723269765334627at_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_205_Inf_OINF__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > nat,D: ( ( nat > nat ) > nat ) > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_7809927846809980933at_nat @ C2 @ A ) )
= ( Inf @ ( image_7809927846809980933at_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_206_Inf_OINF__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > nat > nat,D: ( ( nat > nat ) > nat ) > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_1262493855416953332at_nat @ C2 @ A ) )
= ( Inf @ ( image_1262493855416953332at_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_207_Inf_OINF__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > nat > nat > nat,D: ( ( nat > nat ) > nat ) > nat > nat > nat,Inf: set_nat_nat_nat > nat > nat > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_279826485474788963at_nat @ C2 @ A ) )
= ( Inf @ ( image_279826485474788963at_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_208_Inf_OINF__cong,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C2: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,D: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,Inf: set_nat_nat_nat2 > ( nat > nat ) > nat] :
( ( A = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_8393830757900314979at_nat @ C2 @ A ) )
= ( Inf @ ( image_8393830757900314979at_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_209_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_210_surj__plus__right,axiom,
! [A2: int] :
( ( image_int_int
@ ^ [B3: int] : ( plus_plus_int @ B3 @ A2 )
@ top_top_set_int )
= top_top_set_int ) ).
% surj_plus_right
thf(fact_211_UNIV__I,axiom,
! [X2: nat > nat] : ( member_nat_nat @ X2 @ top_top_set_nat_nat ) ).
% UNIV_I
thf(fact_212_UNIV__I,axiom,
! [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ top_to3655771597906314132at_nat ) ).
% UNIV_I
thf(fact_213_UNIV__I,axiom,
! [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ top_to6379112975903909524at_nat ) ).
% UNIV_I
thf(fact_214_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_215_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_216_subsetI,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat_nat_nat2 @ X3 @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_217_subsetI,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
=> ( member_nat_nat_nat @ X3 @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_218_subsetI,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_219_subset__antisym,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_220_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_221_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_222_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_223_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_224_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_225_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_226_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_227_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_228_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_229_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_230_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_231_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_232_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_233_surj__plus,axiom,
! [A2: int] :
( ( image_int_int @ ( plus_plus_int @ A2 ) @ top_top_set_int )
= top_top_set_int ) ).
% surj_plus
thf(fact_234_lessThan__subset__iff,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y3 ) )
= ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_235_lessThan__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_236_in__mono,axiom,
! [A: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_237_in__mono,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat,X2: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( ( member_nat_nat_nat2 @ X2 @ A )
=> ( member_nat_nat_nat2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_238_in__mono,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,X2: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ( member_nat_nat_nat @ X2 @ A )
=> ( member_nat_nat_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_239_in__mono,axiom,
! [A: set_nat_nat,B2: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_240_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_241_subsetD,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_242_subsetD,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_243_subsetD,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_244_UNIV__eq__I,axiom,
! [A: set_nat_nat] :
( ! [X3: nat > nat] : ( member_nat_nat @ X3 @ A )
=> ( top_top_set_nat_nat = A ) ) ).
% UNIV_eq_I
thf(fact_245_UNIV__eq__I,axiom,
! [A: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A )
=> ( top_to3655771597906314132at_nat = A ) ) ).
% UNIV_eq_I
thf(fact_246_UNIV__eq__I,axiom,
! [A: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A )
=> ( top_to6379112975903909524at_nat = A ) ) ).
% UNIV_eq_I
thf(fact_247_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_248_equalityE,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( A = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_249_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_250_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A5 )
=> ( member_nat_nat_nat2 @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_251_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A5 )
=> ( member_nat_nat_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_252_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
! [X: nat > nat] :
( ( member_nat_nat @ X @ A5 )
=> ( member_nat_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_253_equalityD1,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( A = B2 )
=> ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_254_equalityD2,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( A = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_255_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_256_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A5 )
=> ( member_nat_nat_nat2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_257_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A5 )
=> ( member_nat_nat_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_258_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A5 )
=> ( member_nat_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_259_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_260_subset__UNIV,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ top_top_set_nat_nat ) ).
% subset_UNIV
thf(fact_261_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_262_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_263_UNIV__witness,axiom,
? [X3: nat > nat] : ( member_nat_nat @ X3 @ top_top_set_nat_nat ) ).
% UNIV_witness
thf(fact_264_UNIV__witness,axiom,
? [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ top_to3655771597906314132at_nat ) ).
% UNIV_witness
thf(fact_265_UNIV__witness,axiom,
? [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ top_to6379112975903909524at_nat ) ).
% UNIV_witness
thf(fact_266_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_267_subset__trans,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_268_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_269_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_270_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_271_add__less__le__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_272_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_273_add__le__less__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_274_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_275_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_276_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_277_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_278_range__subsetD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,B2: set_nat,I: ( nat > nat ) > nat] :
( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_279_range__subsetD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,B2: set_nat_nat_nat,I: ( nat > nat ) > nat] :
( ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat ) @ B2 )
=> ( member_nat_nat_nat2 @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_280_range__subsetD,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2,I: nat > nat > nat] :
( ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat ) @ B2 )
=> ( member_nat_nat_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_281_range__subsetD,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2,I: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat ) @ B2 )
=> ( member_nat_nat_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_282_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_283_range__subsetD,axiom,
! [F: nat > nat > nat > nat,B2: set_nat_nat_nat,I: nat] :
( ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat_nat_nat2 @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_284_range__subsetD,axiom,
! [F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2,I: nat] :
( ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat_nat_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_285_range__subsetD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,B2: set_nat_nat,I: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat ) @ B2 )
=> ( member_nat_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_286_range__subsetD,axiom,
! [F: nat > nat > nat,B2: set_nat_nat,I: nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_287_linorder__inj__onI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat,Y4: nat > nat] :
( ( ord_less_nat_nat @ X3 @ Y4 )
=> ( ( member_nat_nat @ X3 @ A )
=> ( ( member_nat_nat @ Y4 @ A )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X3: nat > nat,Y4: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( member_nat_nat @ Y4 @ A )
=> ( ( ord_less_eq_nat_nat @ X3 @ Y4 )
| ( ord_less_eq_nat_nat @ Y4 @ X3 ) ) ) )
=> ( inj_on2461717442902640625at_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_288_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ) )
=> ( inj_on_nat_nat_nat2 @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_289_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_290_linorder__injI,axiom,
! [F: nat > nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) )
=> ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_291_linorder__injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ( F @ X3 )
!= ( F @ Y4 ) ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_292_lessThan__strict__subset__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N2 ) )
= ( ord_less_int @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_293_lessThan__strict__subset__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_294_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_295_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_296_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_297_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_298_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_299_Collect__subset,axiom,
! [A: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_300_Collect__subset,axiom,
! [A: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_301_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_302_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_303_the__inv__f__f,axiom,
! [F: ( nat > nat ) > nat > nat,X2: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( the_in2963963264082133811at_nat @ top_top_set_nat_nat @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_304_the__inv__f__f,axiom,
! [F: nat > nat > nat,X2: nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( the_in3844390324871770692at_nat @ top_top_set_nat @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_305_the__inv__f__f,axiom,
! [F: nat > nat,X2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_306_inj__image__subset__iff,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ top_to3655771597906314132at_nat )
=> ( ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ A ) @ ( image_786723269765334627at_nat @ F @ B2 ) )
= ( ord_le3211623285424100676at_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_307_inj__image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ ( image_7809927846809980933at_nat @ F @ B2 ) )
= ( ord_le5934964663421696068at_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_308_inj__image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ A ) @ ( image_279826485474788963at_nat @ F @ B2 ) )
= ( ord_le5934964663421696068at_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_309_inj__image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on8598928376616154831at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ A ) @ ( image_8393830757900314979at_nat @ F @ B2 ) )
= ( ord_le5934964663421696068at_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_310_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_311_inj__image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ A ) @ ( image_1262493855416953332at_nat @ F @ B2 ) )
= ( ord_le5934964663421696068at_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_312_inj__image__subset__iff,axiom,
! [F: nat > nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_313_inj__image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B2 ) )
= ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_314_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_315_add__less__imp__less__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_316_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_317_add__less__imp__less__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_318_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_319_add__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_320_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_321_add__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_322_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_323_add__strict__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_324_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_325_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_326_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_327_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_328_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_329_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_330_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_331_add__le__imp__le__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_332_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_333_add__le__imp__le__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_334_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_335_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_336_add__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_337_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_338_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_339_add__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_340_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_341_add__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_342_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_343_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_344_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_345_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_346_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_347_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_348_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_349_image__mono,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ A ) @ ( image_786723269765334627at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_350_image__mono,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ ( image_7809927846809980933at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_351_image__mono,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ A ) @ ( image_279826485474788963at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_352_image__mono,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ A ) @ ( image_8393830757900314979at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_353_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ).
% image_mono
thf(fact_354_image__mono,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ A ) @ ( image_1262493855416953332at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_355_image__mono,axiom,
! [A: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_356_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_357_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_358_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_359_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_360_image__subsetI,axiom,
! [A: set_nat,F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_361_image__subsetI,axiom,
! [A: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B2: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_362_image__subsetI,axiom,
! [A: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B2: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_363_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_364_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_3101123049818244468at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_365_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_366_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B2
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_367_subset__imageE,axiom,
! [B2: set_nat_nat_nat2,F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( ord_le5934964663421696068at_nat @ B2 @ ( image_786723269765334627at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ C5 @ A )
=> ( B2
!= ( image_786723269765334627at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_368_subset__imageE,axiom,
! [B2: set_nat,F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( ord_less_eq_set_nat @ B2 @ ( image_7809927846809980933at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ C5 @ A )
=> ( B2
!= ( image_7809927846809980933at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_369_subset__imageE,axiom,
! [B2: set_nat_nat_nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( ord_le3211623285424100676at_nat @ B2 @ ( image_279826485474788963at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ C5 @ A )
=> ( B2
!= ( image_279826485474788963at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_370_subset__imageE,axiom,
! [B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B2 @ ( image_8393830757900314979at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ C5 @ A )
=> ( B2
!= ( image_8393830757900314979at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_371_subset__imageE,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B2
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_372_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_1262493855416953332at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ C5 @ A )
=> ( B2
!= ( image_1262493855416953332at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_373_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A )
=> ( B2
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_374_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_375_image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ B2 )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_376_image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ A ) @ B2 )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_377_image__subset__iff,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ A ) @ B2 )
= ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
=> ( member_nat_nat_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_378_image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ A ) @ B2 )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
=> ( member_nat_nat_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_379_image__subset__iff,axiom,
! [F: nat > nat > nat,A: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_380_image__subset__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ A ) @ B2 )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_381_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_382_subset__image__iff,axiom,
! [B2: set_nat_nat_nat2,F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( ord_le5934964663421696068at_nat @ B2 @ ( image_786723269765334627at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ AA @ A )
& ( B2
= ( image_786723269765334627at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_383_subset__image__iff,axiom,
! [B2: set_nat,F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( ord_less_eq_set_nat @ B2 @ ( image_7809927846809980933at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ AA @ A )
& ( B2
= ( image_7809927846809980933at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_384_subset__image__iff,axiom,
! [B2: set_nat_nat_nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( ord_le3211623285424100676at_nat @ B2 @ ( image_279826485474788963at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ AA @ A )
& ( B2
= ( image_279826485474788963at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_385_subset__image__iff,axiom,
! [B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B2 @ ( image_8393830757900314979at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ AA @ A )
& ( B2
= ( image_8393830757900314979at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_386_subset__image__iff,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_387_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_1262493855416953332at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ AA @ A )
& ( B2
= ( image_1262493855416953332at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_388_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A )
& ( B2
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_389_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X2: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_390_the__inv__into__into,axiom,
! [F: nat > nat > nat,A: set_nat,X2: nat > nat,B2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_in3844390324871770692at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_391_the__inv__into__into,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,X2: nat,B2: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_nat_nat_nat @ F @ A ) )
=> ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( member_nat_nat @ ( the_in5300466440149791684at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_392_the__inv__into__into,axiom,
! [F: ( nat > nat > nat ) > nat,A: set_nat_nat_nat,X2: nat,B2: set_nat_nat_nat] :
( ( inj_on169972799159246449at_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_913610194320715013at_nat @ F @ A ) )
=> ( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( member_nat_nat_nat2 @ ( the_in672218620338739635at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_393_the__inv__into__into,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,X2: nat,B2: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ A )
=> ( ( member_nat @ X2 @ ( image_7809927846809980933at_nat @ F @ A ) )
=> ( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( member_nat_nat_nat @ ( the_in7568536272828005555at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_394_the__inv__into__into,axiom,
! [F: nat > nat > nat > nat,A: set_nat,X2: nat > nat > nat,B2: set_nat] :
( ( inj_on6175431508351409009at_nat @ F @ A )
=> ( ( member_nat_nat_nat2 @ X2 @ ( image_6919068903512877573at_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_in6677677329530902195at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_395_the__inv__into__into,axiom,
! [F: nat > ( nat > nat ) > nat,A: set_nat,X2: ( nat > nat ) > nat,B2: set_nat] :
( ( inj_on5066063743922159217at_nat @ F @ A )
=> ( ( member_nat_nat_nat @ X2 @ ( image_5809701139083627781at_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_in5568309565101652403at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_396_the__inv__into__into,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,X2: nat > nat,B2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( member_nat_nat @ ( the_in2963963264082133811at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_397_the__inv__into__into,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A: set_nat_nat_nat,X2: nat > nat,B2: set_nat_nat_nat] :
( ( inj_on3527655518263895648at_nat @ F @ A )
=> ( ( member_nat_nat @ X2 @ ( image_1545173636400105204at_nat @ F @ A ) )
=> ( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( member_nat_nat_nat2 @ ( the_in6738486182373217954at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_398_the__inv__into__into,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,X2: nat > nat,B2: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ A )
=> ( ( member_nat_nat @ X2 @ ( image_1262493855416953332at_nat @ F @ A ) )
=> ( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( member_nat_nat_nat @ ( the_in6455806401390066082at_nat @ A @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_399_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat > nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A )
=> ( ( member_nat @ J2 @ A )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat_nat2 @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_400_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A )
=> ( ( member_nat @ J2 @ A )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_401_surjD,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,Y3: ( nat > nat ) > nat] :
( ( ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat )
= top_to6379112975903909524at_nat )
=> ? [X3: nat > nat > nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_402_surjD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,Y3: nat > nat] :
( ( ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat )
= top_top_set_nat_nat )
=> ? [X3: ( nat > nat ) > nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_403_surjD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,Y3: nat > nat > nat] :
( ( ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat )
= top_to3655771597906314132at_nat )
=> ? [X3: ( nat > nat ) > nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_404_surjD,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,Y3: ( nat > nat ) > nat] :
( ( ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat )
= top_to6379112975903909524at_nat )
=> ? [X3: ( nat > nat ) > nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_405_surjD,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,Y3: nat] :
( ( ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat )
= top_top_set_nat )
=> ? [X3: ( nat > nat ) > nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_406_surjD,axiom,
! [F: nat > nat > nat,Y3: nat > nat] :
( ( ( image_nat_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat_nat )
=> ? [X3: nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_407_surjD,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_408_surjE,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,Y3: ( nat > nat ) > nat] :
( ( ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat )
= top_to6379112975903909524at_nat )
=> ~ ! [X3: nat > nat > nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_409_surjE,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,Y3: nat > nat] :
( ( ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat )
= top_top_set_nat_nat )
=> ~ ! [X3: ( nat > nat ) > nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_410_surjE,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,Y3: nat > nat > nat] :
( ( ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat )
= top_to3655771597906314132at_nat )
=> ~ ! [X3: ( nat > nat ) > nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_411_surjE,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,Y3: ( nat > nat ) > nat] :
( ( ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat )
= top_to6379112975903909524at_nat )
=> ~ ! [X3: ( nat > nat ) > nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_412_surjE,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,Y3: nat] :
( ( ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat )
= top_top_set_nat )
=> ~ ! [X3: ( nat > nat ) > nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_413_surjE,axiom,
! [F: nat > nat > nat,Y3: nat > nat] :
( ( ( image_nat_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat_nat )
=> ~ ! [X3: nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_414_surjE,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_415_surjI,axiom,
! [G: ( nat > nat > nat ) > ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_786723269765334627at_nat @ G @ top_to3655771597906314132at_nat )
= top_to6379112975903909524at_nat ) ) ).
% surjI
thf(fact_416_surjI,axiom,
! [G: ( ( nat > nat ) > nat ) > nat > nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_1262493855416953332at_nat @ G @ top_to6379112975903909524at_nat )
= top_top_set_nat_nat ) ) ).
% surjI
thf(fact_417_surjI,axiom,
! [G: ( ( nat > nat ) > nat ) > nat > nat > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ! [X3: nat > nat > nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_279826485474788963at_nat @ G @ top_to6379112975903909524at_nat )
= top_to3655771597906314132at_nat ) ) ).
% surjI
thf(fact_418_surjI,axiom,
! [G: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_8393830757900314979at_nat @ G @ top_to6379112975903909524at_nat )
= top_to6379112975903909524at_nat ) ) ).
% surjI
thf(fact_419_surjI,axiom,
! [G: ( ( nat > nat ) > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_7809927846809980933at_nat @ G @ top_to6379112975903909524at_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_420_surjI,axiom,
! [G: nat > nat > nat,F: ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat_nat2 @ G @ top_top_set_nat )
= top_top_set_nat_nat ) ) ).
% surjI
thf(fact_421_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_422_rangeI,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,X2: ( nat > nat ) > nat] : ( member_nat_nat @ ( F @ X2 ) @ ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat ) ) ).
% rangeI
thf(fact_423_rangeI,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,X2: ( nat > nat ) > nat] : ( member_nat @ ( F @ X2 ) @ ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat ) ) ).
% rangeI
thf(fact_424_rangeI,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,X2: ( nat > nat ) > nat] : ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat ) ) ).
% rangeI
thf(fact_425_rangeI,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,X2: nat > nat > nat] : ( member_nat_nat_nat @ ( F @ X2 ) @ ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat ) ) ).
% rangeI
thf(fact_426_rangeI,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ ( F @ X2 ) @ ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat ) ) ).
% rangeI
thf(fact_427_rangeI,axiom,
! [F: nat > nat > nat,X2: nat] : ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_428_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_429_rangeI,axiom,
! [F: nat > nat > nat > nat,X2: nat] : ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( image_6919068903512877573at_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_430_rangeI,axiom,
! [F: nat > ( nat > nat ) > nat,X2: nat] : ( member_nat_nat_nat @ ( F @ X2 ) @ ( image_5809701139083627781at_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_431_surj__def,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat )
= top_to6379112975903909524at_nat )
= ( ! [Y: ( nat > nat ) > nat] :
? [X: nat > nat > nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_432_surj__def,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat] :
( ( ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat )
= top_top_set_nat_nat )
= ( ! [Y: nat > nat] :
? [X: ( nat > nat ) > nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_433_surj__def,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat )
= top_to3655771597906314132at_nat )
= ( ! [Y: nat > nat > nat] :
? [X: ( nat > nat ) > nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_434_surj__def,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat )
= top_to6379112975903909524at_nat )
= ( ! [Y: ( nat > nat ) > nat] :
? [X: ( nat > nat ) > nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_435_surj__def,axiom,
! [F: ( ( nat > nat ) > nat ) > nat] :
( ( ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: ( nat > nat ) > nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_436_surj__def,axiom,
! [F: nat > nat > nat] :
( ( ( image_nat_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat_nat )
= ( ! [Y: nat > nat] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_437_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_438_range__eqI,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,X2: ( nat > nat ) > nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat ) ) ) ).
% range_eqI
thf(fact_439_range__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X2: ( nat > nat ) > nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat ) ) ) ).
% range_eqI
thf(fact_440_range__eqI,axiom,
! [B: nat > nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat,X2: ( nat > nat ) > nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat ) ) ) ).
% range_eqI
thf(fact_441_range__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat,X2: nat > nat > nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat @ B @ ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat ) ) ) ).
% range_eqI
thf(fact_442_range__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,X2: ( nat > nat ) > nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat @ B @ ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat ) ) ) ).
% range_eqI
thf(fact_443_range__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_444_range__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_445_range__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_446_range__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_447_inj__on__subset,axiom,
! [F: nat > nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( inj_on_nat_nat_nat2 @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_448_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( inj_on_nat_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_449_inj__on__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( inj_on2461717442902640625at_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_450_subset__inj__on,axiom,
! [F: nat > nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( inj_on_nat_nat_nat2 @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_451_subset__inj__on,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_452_subset__inj__on,axiom,
! [F: ( nat > nat ) > nat > nat,B2: set_nat_nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( inj_on2461717442902640625at_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_453_injD,axiom,
! [F: ( nat > nat ) > nat > nat,X2: nat > nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% injD
thf(fact_454_injD,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% injD
thf(fact_455_injD,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% injD
thf(fact_456_injI,axiom,
! [F: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat,Y4: nat > nat] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat ) ) ).
% injI
thf(fact_457_injI,axiom,
! [F: nat > nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_458_injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_459_inj__eq,axiom,
! [F: ( nat > nat ) > nat > nat,X2: nat > nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% inj_eq
thf(fact_460_inj__eq,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% inj_eq
thf(fact_461_inj__eq,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% inj_eq
thf(fact_462_inj__def,axiom,
! [F: ( nat > nat ) > nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
= ( ! [X: nat > nat,Y: nat > nat] :
( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ) ).
% inj_def
thf(fact_463_inj__def,axiom,
! [F: nat > nat > nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
= ( ! [X: nat,Y: nat] :
( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ) ).
% inj_def
thf(fact_464_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X: nat,Y: nat] :
( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ) ).
% inj_def
thf(fact_465_restrict__UNIV,axiom,
! [F: ( nat > nat ) > nat > nat] :
( ( restri4446420529079022766at_nat @ F @ top_top_set_nat_nat )
= F ) ).
% restrict_UNIV
thf(fact_466_restrict__UNIV,axiom,
! [F: nat > nat] :
( ( restrict_nat_nat @ F @ top_top_set_nat )
= F ) ).
% restrict_UNIV
thf(fact_467_restrict__UNIV,axiom,
! [F: nat > nat > nat] :
( ( restrict_nat_nat_nat2 @ F @ top_top_set_nat )
= F ) ).
% restrict_UNIV
thf(fact_468_rangeE,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_469_rangeE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_470_rangeE,axiom,
! [B: nat > nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat2 @ B @ ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_471_rangeE,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ B @ ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat ) )
=> ~ ! [X3: nat > nat > nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_472_rangeE,axiom,
! [B: ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ B @ ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_473_rangeE,axiom,
! [B: nat > nat,F: nat > nat > nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_474_rangeE,axiom,
! [B: nat,F: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_475_rangeE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_476_rangeE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_477_range__composition,axiom,
! [F: nat > nat,G: nat > nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_478_range__composition,axiom,
! [F: nat > nat > nat,G: nat > nat] :
( ( image_nat_nat_nat2
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_479_range__composition,axiom,
! [F: ( nat > nat ) > nat,G: nat > nat > nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_nat_nat @ F @ ( image_nat_nat_nat2 @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_480_range__composition,axiom,
! [F: nat > nat,G: ( ( nat > nat ) > nat ) > nat] :
( ( image_7809927846809980933at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ top_to6379112975903909524at_nat )
= ( image_nat_nat @ F @ ( image_7809927846809980933at_nat @ G @ top_to6379112975903909524at_nat ) ) ) ).
% range_composition
thf(fact_481_range__composition,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat] :
( ( image_nat_nat_nat2
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_482_range__composition,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,G: nat > ( nat > nat ) > nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_7809927846809980933at_nat @ F @ ( image_5809701139083627781at_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_483_range__composition,axiom,
! [F: ( nat > nat ) > nat,G: ( ( nat > nat ) > nat ) > nat > nat] :
( ( image_7809927846809980933at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ top_to6379112975903909524at_nat )
= ( image_nat_nat_nat @ F @ ( image_1262493855416953332at_nat @ G @ top_to6379112975903909524at_nat ) ) ) ).
% range_composition
thf(fact_484_range__composition,axiom,
! [F: nat > nat > nat,G: ( ( nat > nat ) > nat ) > nat] :
( ( image_1262493855416953332at_nat
@ ^ [X: ( nat > nat ) > nat] : ( F @ ( G @ X ) )
@ top_to6379112975903909524at_nat )
= ( image_nat_nat_nat2 @ F @ ( image_7809927846809980933at_nat @ G @ top_to6379112975903909524at_nat ) ) ) ).
% range_composition
thf(fact_485_range__composition,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,G: nat > ( nat > nat ) > nat] :
( ( image_nat_nat_nat2
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_1262493855416953332at_nat @ F @ ( image_5809701139083627781at_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_486_range__composition,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,G: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( image_913610194320715013at_nat
@ ^ [X: nat > nat > nat] : ( F @ ( G @ X ) )
@ top_to3655771597906314132at_nat )
= ( image_7809927846809980933at_nat @ F @ ( image_786723269765334627at_nat @ G @ top_to3655771597906314132at_nat ) ) ) ).
% range_composition
thf(fact_487_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X: int] : ( ord_less_int @ X @ U ) ) ) ) ).
% lessThan_def
thf(fact_488_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ X @ U ) ) ) ) ).
% lessThan_def
thf(fact_489_the__inv__into__f__f,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,X2: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( member_nat_nat @ X2 @ A )
=> ( ( the_in2963963264082133811at_nat @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_490_the__inv__into__f__f,axiom,
! [F: nat > nat > nat,A: set_nat,X2: nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( the_in3844390324871770692at_nat @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_491_the__inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_492_the__inv__into__f__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,X2: nat > nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( ( F @ X2 )
= Y3 )
=> ( ( member_nat_nat @ X2 @ A )
=> ( ( the_in2963963264082133811at_nat @ A @ F @ Y3 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_493_the__inv__into__f__eq,axiom,
! [F: nat > nat > nat,A: set_nat,X2: nat,Y3: nat > nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( ( F @ X2 )
= Y3 )
=> ( ( member_nat @ X2 @ A )
=> ( ( the_in3844390324871770692at_nat @ A @ F @ Y3 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_494_the__inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X2 )
= Y3 )
=> ( ( member_nat @ X2 @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ Y3 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_495_inj__on__image__eq__iff,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,C2: set_nat_nat_nat,A: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ C2 )
=> ( ( ord_le3211623285424100676at_nat @ A @ C2 )
=> ( ( ord_le3211623285424100676at_nat @ B2 @ C2 )
=> ( ( ( image_786723269765334627at_nat @ F @ A )
= ( image_786723269765334627at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_496_inj__on__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,C2: set_nat_nat_nat2,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ A @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ C2 )
=> ( ( ( image_7809927846809980933at_nat @ F @ A )
= ( image_7809927846809980933at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_497_inj__on__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,C2: set_nat_nat_nat2,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ A @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ C2 )
=> ( ( ( image_1262493855416953332at_nat @ F @ A )
= ( image_1262493855416953332at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_498_inj__on__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,C2: set_nat_nat_nat2,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ A @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ C2 )
=> ( ( ( image_279826485474788963at_nat @ F @ A )
= ( image_279826485474788963at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_499_inj__on__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,C2: set_nat_nat_nat2,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on8598928376616154831at_nat @ F @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ A @ C2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ C2 )
=> ( ( ( image_8393830757900314979at_nat @ F @ A )
= ( image_8393830757900314979at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_500_inj__on__image__eq__iff,axiom,
! [F: nat > nat > nat,C2: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( ( image_nat_nat_nat2 @ F @ A )
= ( image_nat_nat_nat2 @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_501_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C2: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_502_inj__on__image__eq__iff,axiom,
! [F: ( nat > nat ) > nat > nat,C2: set_nat_nat,A: set_nat_nat,B2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ( ( image_3205354838064109189at_nat @ F @ A )
= ( image_3205354838064109189at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_503_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_504_inj__on__image__mem__iff,axiom,
! [F: nat > nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_nat_nat_nat2 @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_505_inj__on__image__mem__iff,axiom,
! [F: ( nat > nat ) > nat,B2: set_nat_nat,A2: nat > nat,A: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ B2 )
=> ( ( member_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat_nat @ F @ A ) )
= ( member_nat_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_506_inj__on__image__mem__iff,axiom,
! [F: nat > nat > nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on6175431508351409009at_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat_nat_nat2 @ ( F @ A2 ) @ ( image_6919068903512877573at_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_507_inj__on__image__mem__iff,axiom,
! [F: nat > ( nat > nat ) > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on5066063743922159217at_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat_nat_nat @ ( F @ A2 ) @ ( image_5809701139083627781at_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_508_inj__on__image__mem__iff,axiom,
! [F: ( nat > nat > nat ) > nat,B2: set_nat_nat_nat,A2: nat > nat > nat,A: set_nat_nat_nat] :
( ( inj_on169972799159246449at_nat @ F @ B2 )
=> ( ( member_nat_nat_nat2 @ A2 @ B2 )
=> ( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_913610194320715013at_nat @ F @ A ) )
= ( member_nat_nat_nat2 @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_509_inj__on__image__mem__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,B2: set_nat_nat_nat2,A2: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ B2 )
=> ( ( member_nat_nat_nat @ A2 @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_7809927846809980933at_nat @ F @ A ) )
= ( member_nat_nat_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_510_inj__on__image__mem__iff,axiom,
! [F: ( nat > nat ) > nat > nat,B2: set_nat_nat,A2: nat > nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ B2 )
=> ( ( member_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( member_nat_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_511_inj__on__image__mem__iff,axiom,
! [F: ( nat > nat > nat ) > nat > nat,B2: set_nat_nat_nat,A2: nat > nat > nat,A: set_nat_nat_nat] :
( ( inj_on3527655518263895648at_nat @ F @ B2 )
=> ( ( member_nat_nat_nat2 @ A2 @ B2 )
=> ( ( ord_le3211623285424100676at_nat @ A @ B2 )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_1545173636400105204at_nat @ F @ A ) )
= ( member_nat_nat_nat2 @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_512_inj__on__image__mem__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,B2: set_nat_nat_nat2,A2: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ B2 )
=> ( ( member_nat_nat_nat @ A2 @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ A @ B2 )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_1262493855416953332at_nat @ F @ A ) )
= ( member_nat_nat_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_513_range__ex1__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,B: nat > nat] :
( ( inj_on3244975737280743776at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ top_to6379112975903909524at_nat ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
& ! [Y: ( nat > nat ) > nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_514_range__ex1__eq,axiom,
! [F: ( nat > nat ) > nat > nat,B: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ top_top_set_nat_nat ) )
= ( ? [X: nat > nat] :
( ( B
= ( F @ X ) )
& ! [Y: nat > nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_515_range__ex1__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,B: nat] :
( ( inj_on7066290451648512369at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ top_to6379112975903909524at_nat ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
& ! [Y: ( nat > nat ) > nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_516_range__ex1__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,B: nat > nat > nat] :
( ( inj_on484924104190628815at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( member_nat_nat_nat2 @ B @ ( image_279826485474788963at_nat @ F @ top_to6379112975903909524at_nat ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
& ! [Y: ( nat > nat ) > nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_517_range__ex1__eq,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,B: ( nat > nat ) > nat] :
( ( inj_on991820888481174479at_nat @ F @ top_to3655771597906314132at_nat )
=> ( ( member_nat_nat_nat @ B @ ( image_786723269765334627at_nat @ F @ top_to3655771597906314132at_nat ) )
= ( ? [X: nat > nat > nat] :
( ( B
= ( F @ X ) )
& ! [Y: nat > nat > nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_518_range__ex1__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,B: ( nat > nat ) > nat] :
( ( inj_on8598928376616154831at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( member_nat_nat_nat @ B @ ( image_8393830757900314979at_nat @ F @ top_to6379112975903909524at_nat ) )
= ( ? [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
& ! [Y: ( nat > nat ) > nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_519_range__ex1__eq,axiom,
! [F: nat > nat > nat,B: nat > nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_520_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_521_range__ex1__eq,axiom,
! [F: nat > nat > nat > nat,B: nat > nat > nat] :
( ( inj_on6175431508351409009at_nat @ F @ top_top_set_nat )
=> ( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_522_range__ex1__eq,axiom,
! [F: nat > ( nat > nat ) > nat,B: ( nat > nat ) > nat] :
( ( inj_on5066063743922159217at_nat @ F @ top_top_set_nat )
=> ( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F @ X ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_523_inj__image__eq__iff,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ top_to3655771597906314132at_nat )
=> ( ( ( image_786723269765334627at_nat @ F @ A )
= ( image_786723269765334627at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_524_inj__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ( image_7809927846809980933at_nat @ F @ A )
= ( image_7809927846809980933at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_525_inj__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ( image_1262493855416953332at_nat @ F @ A )
= ( image_1262493855416953332at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_526_inj__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ( image_279826485474788963at_nat @ F @ A )
= ( image_279826485474788963at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_527_inj__image__eq__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( inj_on8598928376616154831at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( ( image_8393830757900314979at_nat @ F @ A )
= ( image_8393830757900314979at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_528_inj__image__eq__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( ( image_3205354838064109189at_nat @ F @ A )
= ( image_3205354838064109189at_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_529_inj__image__eq__iff,axiom,
! [F: nat > nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat_nat2 @ F @ A )
= ( image_nat_nat_nat2 @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_530_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_531_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_532_inj__image__mem__iff,axiom,
! [F: ( nat > nat ) > nat,A2: nat > nat,A: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ top_top_set_nat_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat_nat @ F @ A ) )
= ( member_nat_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_533_inj__image__mem__iff,axiom,
! [F: nat > nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_nat_nat_nat2 @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_534_inj__image__mem__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: nat > nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( member_nat_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_535_inj__image__mem__iff,axiom,
! [F: ( nat > nat > nat ) > nat,A2: nat > nat > nat,A: set_nat_nat_nat] :
( ( inj_on169972799159246449at_nat @ F @ top_to3655771597906314132at_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_913610194320715013at_nat @ F @ A ) )
= ( member_nat_nat_nat2 @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_536_inj__image__mem__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_7809927846809980933at_nat @ F @ A ) )
= ( member_nat_nat_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_537_inj__image__mem__iff,axiom,
! [F: nat > nat > nat > nat,A2: nat,A: set_nat] :
( ( inj_on6175431508351409009at_nat @ F @ top_top_set_nat )
=> ( ( member_nat_nat_nat2 @ ( F @ A2 ) @ ( image_6919068903512877573at_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_538_inj__image__mem__iff,axiom,
! [F: nat > ( nat > nat ) > nat,A2: nat,A: set_nat] :
( ( inj_on5066063743922159217at_nat @ F @ top_top_set_nat )
=> ( ( member_nat_nat_nat @ ( F @ A2 ) @ ( image_5809701139083627781at_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_539_inj__image__mem__iff,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A2: nat > nat > nat,A: set_nat_nat_nat] :
( ( inj_on3527655518263895648at_nat @ F @ top_to3655771597906314132at_nat )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_1545173636400105204at_nat @ F @ A ) )
= ( member_nat_nat_nat2 @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_540_inj__image__mem__iff,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ( member_nat_nat @ ( F @ A2 ) @ ( image_1262493855416953332at_nat @ F @ A ) )
= ( member_nat_nat_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_541_inj__on__restrict__iff,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( inj_on_nat_nat @ ( restrict_nat_nat @ F @ B2 ) @ A )
= ( inj_on_nat_nat @ F @ A ) ) ) ).
% inj_on_restrict_iff
thf(fact_542_inj__on__restrict__iff,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( inj_on_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ B2 ) @ A )
= ( inj_on_nat_nat_nat2 @ F @ A ) ) ) ).
% inj_on_restrict_iff
thf(fact_543_inj__on__restrict__iff,axiom,
! [A: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( inj_on2461717442902640625at_nat @ ( restri4446420529079022766at_nat @ F @ B2 ) @ A )
= ( inj_on2461717442902640625at_nat @ F @ A ) ) ) ).
% inj_on_restrict_iff
thf(fact_544_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N3: nat] : ( image_nat_nat @ ( plus_plus_nat @ N3 ) ) ) ) ).
% set_incr_altdef
thf(fact_545_f__the__inv__into__f,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,Y3: nat > nat] :
( ( inj_on3244975737280743776at_nat @ F @ A )
=> ( ( member_nat_nat @ Y3 @ ( image_1262493855416953332at_nat @ F @ A ) )
=> ( ( F @ ( the_in6455806401390066082at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_546_f__the__inv__into__f,axiom,
! [F: nat > nat > nat,A: set_nat,Y3: nat > nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( ( member_nat_nat @ Y3 @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( ( F @ ( the_in3844390324871770692at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_547_f__the__inv__into__f,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,Y3: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( ( member_nat_nat @ Y3 @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ( ( F @ ( the_in2963963264082133811at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_548_f__the__inv__into__f,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,Y3: nat] :
( ( inj_on7066290451648512369at_nat @ F @ A )
=> ( ( member_nat @ Y3 @ ( image_7809927846809980933at_nat @ F @ A ) )
=> ( ( F @ ( the_in7568536272828005555at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_549_f__the__inv__into__f,axiom,
! [F: nat > nat,A: set_nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_550_f__the__inv__into__f,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,Y3: nat > nat > nat] :
( ( inj_on484924104190628815at_nat @ F @ A )
=> ( ( member_nat_nat_nat2 @ Y3 @ ( image_279826485474788963at_nat @ F @ A ) )
=> ( ( F @ ( the_in3848221702076823825at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_551_f__the__inv__into__f,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,Y3: ( nat > nat ) > nat] :
( ( inj_on991820888481174479at_nat @ F @ A )
=> ( ( member_nat_nat_nat @ Y3 @ ( image_786723269765334627at_nat @ F @ A ) )
=> ( ( F @ ( the_in4355118486367369489at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_552_f__the__inv__into__f,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,Y3: ( nat > nat ) > nat] :
( ( inj_on8598928376616154831at_nat @ F @ A )
=> ( ( member_nat_nat_nat @ Y3 @ ( image_8393830757900314979at_nat @ F @ A ) )
=> ( ( F @ ( the_in2738853937647574033at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_553_inj__on__the__inv__into,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ A )
=> ( inj_on484924104190628815at_nat @ ( the_in4355118486367369489at_nat @ A @ F ) @ ( image_786723269765334627at_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_554_inj__on__the__inv__into,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ A )
=> ( inj_on5066063743922159217at_nat @ ( the_in7568536272828005555at_nat @ A @ F ) @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_555_inj__on__the__inv__into,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ A )
=> ( inj_on3974237167252785120at_nat @ ( the_in6455806401390066082at_nat @ A @ F ) @ ( image_1262493855416953332at_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_556_inj__on__the__inv__into,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ A )
=> ( inj_on991820888481174479at_nat @ ( the_in3848221702076823825at_nat @ A @ F ) @ ( image_279826485474788963at_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_557_inj__on__the__inv__into,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on8598928376616154831at_nat @ F @ A )
=> ( inj_on8598928376616154831at_nat @ ( the_in2738853937647574033at_nat @ A @ F ) @ ( image_8393830757900314979at_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_558_inj__on__the__inv__into,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ A )
=> ( inj_on_nat_nat_nat2 @ ( the_in5300466440149791684at_nat @ A @ F ) @ ( image_nat_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_559_inj__on__the__inv__into,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ A )
=> ( inj_on_nat_nat_nat @ ( the_in3844390324871770692at_nat @ A @ F ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_560_inj__on__the__inv__into,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ A )
=> ( inj_on2461717442902640625at_nat @ ( the_in2963963264082133811at_nat @ A @ F ) @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_561_inj__on__the__inv__into,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_562_inj__fun,axiom,
! [F: ( nat > nat ) > nat] :
( ( inj_on_nat_nat_nat @ F @ top_top_set_nat_nat )
=> ( inj_on2461717442902640625at_nat
@ ^ [X: nat > nat,Y: nat] : ( F @ X )
@ top_top_set_nat_nat ) ) ).
% inj_fun
thf(fact_563_inj__fun,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on_nat_nat_nat2
@ ^ [X: nat,Y: nat] : ( F @ X )
@ top_top_set_nat ) ) ).
% inj_fun
thf(fact_564_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_565_classes__subset__cube,axiom,
! [N2: nat,T: nat,I: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N2 @ T @ I ) @ ( hales_cube @ N2 @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_566_subset__image__inj,axiom,
! [S: set_nat_nat_nat2,F: ( nat > nat > nat ) > ( nat > nat ) > nat,T3: set_nat_nat_nat] :
( ( ord_le5934964663421696068at_nat @ S @ ( image_786723269765334627at_nat @ F @ T3 ) )
= ( ? [U2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ U2 @ T3 )
& ( inj_on991820888481174479at_nat @ F @ U2 )
& ( S
= ( image_786723269765334627at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_567_subset__image__inj,axiom,
! [S: set_nat,F: ( ( nat > nat ) > nat ) > nat,T3: set_nat_nat_nat2] :
( ( ord_less_eq_set_nat @ S @ ( image_7809927846809980933at_nat @ F @ T3 ) )
= ( ? [U2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ U2 @ T3 )
& ( inj_on7066290451648512369at_nat @ F @ U2 )
& ( S
= ( image_7809927846809980933at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_568_subset__image__inj,axiom,
! [S: set_nat_nat_nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat,T3: set_nat_nat_nat2] :
( ( ord_le3211623285424100676at_nat @ S @ ( image_279826485474788963at_nat @ F @ T3 ) )
= ( ? [U2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ U2 @ T3 )
& ( inj_on484924104190628815at_nat @ F @ U2 )
& ( S
= ( image_279826485474788963at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_569_subset__image__inj,axiom,
! [S: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,T3: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ S @ ( image_8393830757900314979at_nat @ F @ T3 ) )
= ( ? [U2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ U2 @ T3 )
& ( inj_on8598928376616154831at_nat @ F @ U2 )
& ( S
= ( image_8393830757900314979at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_570_subset__image__inj,axiom,
! [S: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_nat @ F @ U2 )
& ( S
= ( image_nat_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_571_subset__image__inj,axiom,
! [S: set_nat_nat,F: ( ( nat > nat ) > nat ) > nat > nat,T3: set_nat_nat_nat2] :
( ( ord_le9059583361652607317at_nat @ S @ ( image_1262493855416953332at_nat @ F @ T3 ) )
= ( ? [U2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ U2 @ T3 )
& ( inj_on3244975737280743776at_nat @ F @ U2 )
& ( S
= ( image_1262493855416953332at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_572_subset__image__inj,axiom,
! [S: set_nat_nat,F: nat > nat > nat,T3: set_nat] :
( ( ord_le9059583361652607317at_nat @ S @ ( image_nat_nat_nat2 @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_nat_nat2 @ F @ U2 )
& ( S
= ( image_nat_nat_nat2 @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_573_subset__image__inj,axiom,
! [S: set_nat_nat,F: ( nat > nat ) > nat > nat,T3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ ( image_3205354838064109189at_nat @ F @ T3 ) )
= ( ? [U2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ U2 @ T3 )
& ( inj_on2461717442902640625at_nat @ F @ U2 )
& ( S
= ( image_3205354838064109189at_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_574_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_575_add__mono1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_576_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_577_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_578_iso__tuple__UNIV__I,axiom,
! [X2: nat > nat] : ( member_nat_nat @ X2 @ top_top_set_nat_nat ) ).
% iso_tuple_UNIV_I
thf(fact_579_iso__tuple__UNIV__I,axiom,
! [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ top_to3655771597906314132at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_580_iso__tuple__UNIV__I,axiom,
! [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ top_to6379112975903909524at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_581_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_582_sorted__list__of__set_Oinj__on,axiom,
( inj_on_nat_nat
@ ^ [X: nat] : X
@ top_top_set_nat ) ).
% sorted_list_of_set.inj_on
thf(fact_583_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_584_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_585_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_586_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_587_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_588_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_589_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_590_image__Collect__subsetI,axiom,
! [P: ( ( nat > nat ) > nat ) > $o,F: ( ( nat > nat ) > nat ) > nat,B2: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ ( collect_nat_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_591_image__Collect__subsetI,axiom,
! [P: ( ( nat > nat ) > nat ) > $o,F: ( ( nat > nat ) > nat ) > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( P @ X3 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ ( collect_nat_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_592_image__Collect__subsetI,axiom,
! [P: ( nat > nat > nat ) > $o,F: ( nat > nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: nat > nat > nat] :
( ( P @ X3 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ ( collect_nat_nat_nat2 @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_593_image__Collect__subsetI,axiom,
! [P: ( ( nat > nat ) > nat ) > $o,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( P @ X3 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ ( collect_nat_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_594_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_595_image__Collect__subsetI,axiom,
! [P: ( ( nat > nat ) > nat ) > $o,F: ( ( nat > nat ) > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( P @ X3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ ( collect_nat_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_596_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_597_psubsetI,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_598_psubsetD,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A @ B2 )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_599_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_600_psubsetD,axiom,
! [A: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_601_psubsetD,axiom,
! [A: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A @ B2 )
=> ( ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_602_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A5 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_603_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_604_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A5: set_nat_nat_nat,B4: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A5 )
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_605_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A5: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A5 )
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_606_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_607_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_608_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_609_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_610_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_611_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_612_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_613_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_614_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_615_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_616_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_617_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_618_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_619_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_620_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_621_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_622_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_623_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_624_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_625_subset__psubset__trans,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C2 )
=> ( ord_less_set_nat_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_626_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B4 )
& ~ ( ord_le9059583361652607317at_nat @ B4 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_627_psubset__subset__trans,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_less_set_nat_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_628_psubset__imp__subset,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_629_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_630_psubsetE,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_631_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_632_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_633_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B4: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A5 )
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_634_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A5 )
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_635_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B4: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A5 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_636_image__strict__mono,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat,A: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ B2 )
=> ( ( ord_le6871433888996735800at_nat @ A @ B2 )
=> ( ord_le371403230139555384at_nat @ ( image_786723269765334627at_nat @ F @ A ) @ ( image_786723269765334627at_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_637_image__strict__mono,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,B2: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ B2 )
=> ( ( ord_le371403230139555384at_nat @ A @ B2 )
=> ( ord_less_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ ( image_7809927846809980933at_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_638_image__strict__mono,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,B2: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ B2 )
=> ( ( ord_le371403230139555384at_nat @ A @ B2 )
=> ( ord_less_set_nat_nat @ ( image_1262493855416953332at_nat @ F @ A ) @ ( image_1262493855416953332at_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_639_image__strict__mono,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,B2: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ B2 )
=> ( ( ord_le371403230139555384at_nat @ A @ B2 )
=> ( ord_le6871433888996735800at_nat @ ( image_279826485474788963at_nat @ F @ A ) @ ( image_279826485474788963at_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_640_image__strict__mono,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( inj_on8598928376616154831at_nat @ F @ B2 )
=> ( ( ord_le371403230139555384at_nat @ A @ B2 )
=> ( ord_le371403230139555384at_nat @ ( image_8393830757900314979at_nat @ F @ A ) @ ( image_8393830757900314979at_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_641_image__strict__mono,axiom,
! [F: nat > nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_nat_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_642_image__strict__mono,axiom,
! [F: ( nat > nat ) > nat > nat,B2: set_nat_nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ B2 )
=> ( ( ord_less_set_nat_nat @ A @ B2 )
=> ( ord_less_set_nat_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_643_image__strict__mono,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_644_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_645_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_646_le__cases3,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_647_le__cases3,axiom,
! [X2: int,Y3: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y3 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_648_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
& ( ord_le9059583361652607317at_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_649_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_650_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_651_ord__eq__le__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A2 = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_652_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_653_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_654_ord__le__eq__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_655_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_656_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_657_order__antisym,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_658_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_659_order__antisym,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_660_order_Otrans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_661_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_662_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_663_order__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ Z )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_664_order__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_665_order__trans,axiom,
! [X2: int,Y3: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_666_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_667_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: int,B5: int] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_668_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_669_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_670_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_671_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_672_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_673_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_674_dual__order_Otrans,axiom,
! [B: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_675_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_676_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_677_antisym,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_678_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_679_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_680_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_681_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_682_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_683_order__subst1,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_684_order__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_685_order__subst1,axiom,
! [A2: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_686_order__subst1,axiom,
! [A2: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_688_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_689_order__subst1,axiom,
! [A2: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_690_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_691_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_692_order__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_693_order__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_694_order__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_695_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_696_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_697_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_698_order__subst2,axiom,
! [A2: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_699_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_700_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_701_order__eq__refl,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( X2 = Y3 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_702_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_703_order__eq__refl,axiom,
! [X2: int,Y3: int] :
( ( X2 = Y3 )
=> ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_704_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_705_linorder__linear,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
| ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_706_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_707_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_708_ord__eq__le__subst,axiom,
! [A2: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_709_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_710_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_711_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_712_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_713_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_714_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_715_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_716_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_717_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_718_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_719_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_720_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_721_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_724_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_725_linorder__le__cases,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_726_order__antisym__conv,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_727_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_728_order__antisym__conv,axiom,
! [Y3: int,X2: int] :
( ( ord_less_eq_int @ Y3 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_729_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_730_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_731_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_732_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_733_less__imp__neq,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_734_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_735_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_736_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_737_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_738_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_739_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_740_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_741_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_742_antisym__conv3,axiom,
! [Y3: int,X2: int] :
( ~ ( ord_less_int @ Y3 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_743_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_744_linorder__cases,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_745_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_746_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_747_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_748_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_749_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_750_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_751_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B5: int] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_752_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_753_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_754_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_755_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y3: int] :
( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( ( ord_less_int @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_756_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_757_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_758_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_759_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_760_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_761_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_762_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_763_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_764_linorder__neqE,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_765_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_766_order__less__asym,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_767_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_768_linorder__neq__iff,axiom,
! [X2: int,Y3: int] :
( ( X2 != Y3 )
= ( ( ord_less_int @ X2 @ Y3 )
| ( ord_less_int @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_769_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_770_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_771_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_772_order__less__trans,axiom,
! [X2: int,Y3: int,Z: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_773_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_774_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_775_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_776_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_777_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_778_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_779_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_780_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_781_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_782_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_783_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_784_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_785_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_786_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_787_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_788_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_789_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_790_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_791_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_792_order__less__not__sym,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_793_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_794_order__less__imp__triv,axiom,
! [X2: int,Y3: int,P: $o] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_int @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_795_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_796_linorder__less__linear,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_int @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_797_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_798_order__less__imp__not__eq,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_799_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_800_order__less__imp__not__eq2,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_801_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_802_order__less__imp__not__less,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_803_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_804_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_805_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_806_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_807_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_808_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_809_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_810_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_811_prop__restrict,axiom,
! [X2: nat,Z3: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_812_prop__restrict,axiom,
! [X2: nat > nat > nat,Z3: set_nat_nat_nat,X6: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ X2 @ Z3 )
=> ( ( ord_le3211623285424100676at_nat @ Z3
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_813_prop__restrict,axiom,
! [X2: ( nat > nat ) > nat,Z3: set_nat_nat_nat2,X6: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ X2 @ Z3 )
=> ( ( ord_le5934964663421696068at_nat @ Z3
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_814_prop__restrict,axiom,
! [X2: nat > nat,Z3: set_nat_nat,X6: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X2 @ Z3 )
=> ( ( ord_le9059583361652607317at_nat @ Z3
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_815_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_816_Collect__restrict,axiom,
! [X6: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_817_Collect__restrict,axiom,
! [X6: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_818_Collect__restrict,axiom,
! [X6: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_819_leD,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_nat_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_820_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_821_leD,axiom,
! [Y3: int,X2: int] :
( ( ord_less_eq_int @ Y3 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y3 ) ) ).
% leD
thf(fact_822_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_823_leI,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% leI
thf(fact_824_nless__le,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A2 @ B ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_825_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_826_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_827_antisym__conv1,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_828_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_829_antisym__conv1,axiom,
! [X2: int,Y3: int] :
( ~ ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_eq_int @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_830_antisym__conv2,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_nat_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_831_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_832_antisym__conv2,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_833_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
& ~ ( ord_le9059583361652607317at_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_834_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_835_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_836_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_837_not__le__imp__less,axiom,
! [Y3: int,X2: int] :
( ~ ( ord_less_eq_int @ Y3 @ X2 )
=> ( ord_less_int @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_838_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_839_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_840_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_841_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_842_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_843_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_844_order_Ostrict__trans1,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_845_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_846_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_847_order_Ostrict__trans2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_848_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_849_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_850_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_851_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_852_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_853_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_854_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_855_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_856_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_857_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_858_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_859_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( ord_less_set_nat_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_860_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_861_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_862_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_863_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_864_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_865_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
& ~ ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_866_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_867_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_868_order_Ostrict__implies__order,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_869_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_870_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_871_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A2 )
=> ( ord_le9059583361652607317at_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_872_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_873_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_874_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_875_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_876_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_877_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_878_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_879_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_880_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_881_linorder__not__le,axiom,
! [X2: int,Y3: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
= ( ord_less_int @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_882_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_883_linorder__not__less,axiom,
! [X2: int,Y3: int] :
( ( ~ ( ord_less_int @ X2 @ Y3 ) )
= ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_884_order__less__imp__le,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y3 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_885_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_886_order__less__imp__le,axiom,
! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_887_order__le__neq__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_888_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_889_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_890_order__neq__le__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 != B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ord_less_set_nat_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_891_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_892_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_893_order__le__less__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat_nat @ Y3 @ Z )
=> ( ord_less_set_nat_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_894_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_895_order__le__less__trans,axiom,
! [X2: int,Y3: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_896_order__less__le__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ Z )
=> ( ord_less_set_nat_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_897_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_898_order__less__le__trans,axiom,
! [X2: int,Y3: int,Z: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_899_order__le__less__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_900_order__le__less__subst1,axiom,
! [A2: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_901_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_902_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_903_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_904_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_905_order__le__less__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_906_order__le__less__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_907_order__le__less__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_908_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_909_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_910_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_911_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_912_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_913_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_914_order__less__le__subst1,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_915_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_916_order__less__le__subst1,axiom,
! [A2: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_917_order__less__le__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_less_set_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_918_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_919_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_920_order__less__le__subst1,axiom,
! [A2: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_less_set_nat_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_921_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_922_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_923_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_924_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_925_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_926_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_927_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_928_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_929_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_930_linorder__le__less__linear,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
| ( ord_less_int @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_931_order__le__imp__less__or__eq,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_932_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_933_order__le__imp__less__or__eq,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_934_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_935_top__greatest,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ top_top_set_nat_nat ) ).
% top_greatest
thf(fact_936_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_937_top_Oextremum__unique,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ top_top_set_nat_nat @ A2 )
= ( A2 = top_top_set_nat_nat ) ) ).
% top.extremum_unique
thf(fact_938_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_939_top_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ top_top_set_nat_nat @ A2 )
=> ( A2 = top_top_set_nat_nat ) ) ).
% top.extremum_uniqueI
thf(fact_940_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_941_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_942_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_943_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_944_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_945_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_946_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_947_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_948_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_949_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_950_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_951_all__subset__image,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,P: set_nat_nat_nat2 > $o] :
( ( ! [B4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B4 @ ( image_786723269765334627at_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ B4 @ A )
=> ( P @ ( image_786723269765334627at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_952_all__subset__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_7809927846809980933at_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B4 @ A )
=> ( P @ ( image_7809927846809980933at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_953_all__subset__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,P: set_nat_nat_nat > $o] :
( ( ! [B4: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ B4 @ ( image_279826485474788963at_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B4 @ A )
=> ( P @ ( image_279826485474788963at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_954_all__subset__image,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,P: set_nat_nat_nat2 > $o] :
( ( ! [B4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B4 @ ( image_8393830757900314979at_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B4 @ A )
=> ( P @ ( image_8393830757900314979at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_955_all__subset__image,axiom,
! [F: nat > nat > nat,A: set_nat,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat_nat2 @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_956_all__subset__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ ( image_1262493855416953332at_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ B4 @ A )
=> ( P @ ( image_1262493855416953332at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_957_all__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A )
=> ( P @ ( image_3205354838064109189at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_958_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_959_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_960_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_961_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_962_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_963_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_964_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_965_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_966_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_967_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_968_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_969_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: int] :
( ( ord_less_eq_int @ A2 @ C4 )
& ( ord_less_eq_int @ C4 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A2 @ X4 )
& ( ord_less_int @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_970_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_971_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_972_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_973_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_974_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_975_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_976_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_977_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_978_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_979_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_980_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_981_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_982_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_983_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_984_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_985_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_986_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_987_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_988_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_989_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_990_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_991_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_992_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_993_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_994_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_995_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_996_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_997_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_998_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_999_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1000_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1001_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1002_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1003_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1004_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_1005_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_1006_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_1007_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_1008_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1009_pred__subset__eq,axiom,
! [R: set_nat_nat_nat,S: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ R )
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ S ) )
= ( ord_le3211623285424100676at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1010_pred__subset__eq,axiom,
! [R: set_nat_nat_nat2,S: set_nat_nat_nat2] :
( ( ord_le996020443555834177_nat_o
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ R )
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ S ) )
= ( ord_le5934964663421696068at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1011_pred__subset__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
= ( ord_le9059583361652607317at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1012_top__empty__eq,axiom,
( top_top_nat_nat_o
= ( ^ [X: nat > nat] : ( member_nat_nat @ X @ top_top_set_nat_nat ) ) ) ).
% top_empty_eq
thf(fact_1013_top__empty__eq,axiom,
( top_to4209272211376415217_nat_o
= ( ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ top_to3655771597906314132at_nat ) ) ) ).
% top_empty_eq
thf(fact_1014_top__empty__eq,axiom,
( top_to9043804989276028657_nat_o
= ( ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ top_to6379112975903909524at_nat ) ) ) ).
% top_empty_eq
thf(fact_1015_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_1016_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1017_line__points__in__cube,axiom,
! [L2: nat > nat > nat,N2: nat,T: nat,S2: nat] :
( ( hales_is_line @ L2 @ N2 @ T )
=> ( ( ord_less_nat @ S2 @ T )
=> ( member_nat_nat @ ( L2 @ S2 ) @ ( hales_cube @ N2 @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_1018_line__points__in__cube__unfolded,axiom,
! [L2: nat > nat > nat,N2: nat,T: nat,S2: nat,J: nat] :
( ( hales_is_line @ L2 @ N2 @ T )
=> ( ( ord_less_nat @ S2 @ T )
=> ( ( ord_less_nat @ J @ N2 )
=> ( member_nat @ ( L2 @ S2 @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_1019_arg__min__inj__eq,axiom,
! [F: nat > nat > nat,P: nat > $o,A2: nat] :
( ( inj_on_nat_nat_nat2 @ F @ ( collect_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ ( F @ Y4 ) ) )
=> ( ( lattic6702328201646789547at_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_1020_arg__min__inj__eq,axiom,
! [F: ( nat > nat ) > nat > nat,P: ( nat > nat ) > $o,A2: nat > nat] :
( ( inj_on2461717442902640625at_nat @ F @ ( collect_nat_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y4: nat > nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ ( F @ Y4 ) ) )
=> ( ( lattic4097496154029735450at_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_1021_arg__min__inj__eq,axiom,
! [F: nat > nat,P: nat > $o,A2: nat] :
( ( inj_on_nat_nat @ F @ ( collect_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ Y4 ) ) )
=> ( ( lattic8739620818006775868at_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_1022_is__line__def,axiom,
( hales_is_line
= ( ^ [L3: nat > nat > nat,N3: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L3
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I4: nat] : ( hales_cube @ N3 @ T2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
=> ( ! [X: nat] :
( ( ord_less_nat @ X @ T2 )
=> ! [Y: nat] :
( ( ord_less_nat @ Y @ T2 )
=> ( ( L3 @ X @ J3 )
= ( L3 @ Y @ J3 ) ) ) )
| ! [S3: nat] :
( ( ord_less_nat @ S3 @ T2 )
=> ( ( L3 @ S3 @ J3 )
= S3 ) ) ) )
& ? [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
& ! [S3: nat] :
( ( ord_less_nat @ S3 @ T2 )
=> ( ( L3 @ S3 @ J3 )
= S3 ) ) ) ) ) ) ).
% is_line_def
thf(fact_1023_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_1024_top_Oordering__top__axioms,axiom,
orderi4315198199014340228at_nat @ ord_le9059583361652607317at_nat @ ord_less_set_nat_nat @ top_top_set_nat_nat ).
% top.ordering_top_axioms
thf(fact_1025_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A @ B2 ) )
=> ( ( restrict_nat_nat_nat @ F @ A )
= F ) ) ).
% PiE_restrict
thf(fact_1026_PiE__restrict,axiom,
! [F: nat > nat,A: set_nat,B2: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A @ B2 ) )
=> ( ( restrict_nat_nat @ F @ A )
= F ) ) ).
% PiE_restrict
thf(fact_1027_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A @ B2 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A )
= F ) ) ).
% PiE_restrict
thf(fact_1028_PiE__restrict,axiom,
! [F: nat > nat > nat,A: set_nat,B2: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A @ B2 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A )
= F ) ) ).
% PiE_restrict
thf(fact_1029_PiE__UNIV,axiom,
( ( piE_nat_nat_nat @ top_top_set_nat_nat
@ ^ [I4: nat > nat] : top_top_set_nat )
= top_to6379112975903909524at_nat ) ).
% PiE_UNIV
thf(fact_1030_PiE__UNIV,axiom,
( ( piE_nat_nat_nat2 @ top_top_set_nat
@ ^ [I4: nat] : top_top_set_nat_nat )
= top_to3655771597906314132at_nat ) ).
% PiE_UNIV
thf(fact_1031_PiE__UNIV,axiom,
( ( piE_nat_nat @ top_top_set_nat
@ ^ [I4: nat] : top_top_set_nat )
= top_top_set_nat_nat ) ).
% PiE_UNIV
thf(fact_1032_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A2 @ Top ) ) ).
% ordering_top.extremum
thf(fact_1033_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A2 ) ) ).
% ordering_top.extremum_strict
thf(fact_1034_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
= ( A2 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_1035_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A2 != Top )
= ( Less @ A2 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_1036_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
=> ( A2 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_1037_fun__ex,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A
@ ^ [I4: nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1038_fun__ex,axiom,
! [A2: nat,A: set_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A
@ ^ [I4: nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1039_fun__ex,axiom,
! [A2: nat > nat,A: set_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A
@ ^ [I4: nat > nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1040_fun__ex,axiom,
! [A2: nat > nat,A: set_nat_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A
@ ^ [I4: nat > nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1041_fun__ex,axiom,
! [A2: nat,A: set_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X3: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X3
@ ( piE_nat_nat_nat_nat5 @ A
@ ^ [I4: nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1042_fun__ex,axiom,
! [A2: nat,A: set_nat,B: ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat_nat_nat @ B @ B2 )
=> ? [X3: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X3
@ ( piE_nat_nat_nat_nat4 @ A
@ ^ [I4: nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1043_fun__ex,axiom,
! [A2: nat > nat > nat,A: set_nat_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat_nat2 @ A2 @ A )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X3
@ ( piE_nat_nat_nat_nat2 @ A
@ ^ [I4: nat > nat > nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1044_fun__ex,axiom,
! [A2: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: nat,B2: set_nat] :
( ( member_nat_nat_nat @ A2 @ A )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X3
@ ( piE_nat_nat_nat_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1045_fun__ex,axiom,
! [A2: nat > nat,A: set_nat_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X3
@ ( piE_na8678869062391380393at_nat @ A
@ ^ [I4: nat > nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1046_fun__ex,axiom,
! [A2: nat > nat,A: set_nat_nat,B: ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ( member_nat_nat @ A2 @ A )
=> ( ( member_nat_nat_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3
@ ( piE_na7569501297962130601at_nat @ A
@ ^ [I4: nat > nat] : B2 ) )
& ( ( X3 @ A2 )
= B ) ) ) ) ).
% fun_ex
thf(fact_1047_PiE__ext,axiom,
! [X2: nat > nat > nat,K: set_nat,S2: nat > set_nat_nat,Y3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( piE_nat_nat_nat2 @ K @ S2 ) )
=> ( ( member_nat_nat_nat2 @ Y3 @ ( piE_nat_nat_nat2 @ K @ S2 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ K )
=> ( ( X2 @ I2 )
= ( Y3 @ I2 ) ) )
=> ( X2 = Y3 ) ) ) ) ).
% PiE_ext
thf(fact_1048_PiE__ext,axiom,
! [X2: nat > nat,K: set_nat,S2: nat > set_nat,Y3: nat > nat] :
( ( member_nat_nat @ X2 @ ( piE_nat_nat @ K @ S2 ) )
=> ( ( member_nat_nat @ Y3 @ ( piE_nat_nat @ K @ S2 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ K )
=> ( ( X2 @ I2 )
= ( Y3 @ I2 ) ) )
=> ( X2 = Y3 ) ) ) ) ).
% PiE_ext
thf(fact_1049_PiE__ext,axiom,
! [X2: ( nat > nat ) > nat,K: set_nat_nat,S2: ( nat > nat ) > set_nat,Y3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( piE_nat_nat_nat @ K @ S2 ) )
=> ( ( member_nat_nat_nat @ Y3 @ ( piE_nat_nat_nat @ K @ S2 ) )
=> ( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ K )
=> ( ( X2 @ I2 )
= ( Y3 @ I2 ) ) )
=> ( X2 = Y3 ) ) ) ) ).
% PiE_ext
thf(fact_1050_PiE__mem,axiom,
! [F: nat > nat,S: set_nat,T3: nat > set_nat,X2: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S @ T3 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1051_PiE__mem,axiom,
! [F: nat > nat > nat,S: set_nat,T3: nat > set_nat_nat,X2: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ S @ T3 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1052_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat,X2: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1053_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat_nat,X2: nat > nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ S @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S )
=> ( member_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1054_PiE__mem,axiom,
! [F: nat > nat > nat > nat,S: set_nat,T3: nat > set_nat_nat_nat,X2: nat] :
( ( member17114558718834868at_nat @ F @ ( piE_nat_nat_nat_nat5 @ S @ T3 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1055_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat,S: set_nat,T3: nat > set_nat_nat_nat2,X2: nat] :
( ( member2740455936716430260at_nat @ F @ ( piE_nat_nat_nat_nat4 @ S @ T3 ) )
=> ( ( member_nat @ X2 @ S )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1056_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat,S: set_nat_nat_nat,T3: ( nat > nat > nat ) > set_nat,X2: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F @ ( piE_nat_nat_nat_nat2 @ S @ T3 ) )
=> ( ( member_nat_nat_nat2 @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1057_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,S: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat,X2: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F @ ( piE_nat_nat_nat_nat @ S @ T3 ) )
=> ( ( member_nat_nat_nat @ X2 @ S )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1058_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat_nat_nat,X2: nat > nat] :
( ( member1679187572556404771at_nat @ F @ ( piE_na8678869062391380393at_nat @ S @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1059_PiE__mem,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat_nat_nat2,X2: nat > nat] :
( ( member4402528950554000163at_nat @ F @ ( piE_na7569501297962130601at_nat @ S @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_1060_PiE__cong,axiom,
! [I5: set_nat,A: nat > set_nat_nat,B2: nat > set_nat_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat_nat2 @ I5 @ A )
= ( piE_nat_nat_nat2 @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_1061_PiE__cong,axiom,
! [I5: set_nat,A: nat > set_nat,B2: nat > set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat @ I5 @ A )
= ( piE_nat_nat @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_1062_PiE__cong,axiom,
! [I5: set_nat_nat,A: ( nat > nat ) > set_nat,B2: ( nat > nat ) > set_nat] :
( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ I5 )
=> ( ( A @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat_nat @ I5 @ A )
= ( piE_nat_nat_nat @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_1063_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I5: set_nat_nat,X6: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I5 ) @ ( piE_nat_nat_nat @ I5 @ X6 ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ I5 )
=> ( member_nat @ ( F @ X ) @ ( X6 @ X ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_1064_restrict__PiE__iff,axiom,
! [F: nat > nat,I5: set_nat,X6: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I5 ) @ ( piE_nat_nat @ I5 @ X6 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( member_nat @ ( F @ X ) @ ( X6 @ X ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_1065_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I5: set_nat_nat,X6: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I5 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ X6 ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ I5 )
=> ( member_nat_nat @ ( F @ X ) @ ( X6 @ X ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_1066_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I5: set_nat,X6: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I5 ) @ ( piE_nat_nat_nat2 @ I5 @ X6 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( member_nat_nat @ ( F @ X ) @ ( X6 @ X ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_1067_PiE__uniqueness,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ A ) @ B2 )
=> ? [X3: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( member3122884635697634578at_nat @ X3
@ ( piE_na8627462036710459160at_nat @ A
@ ^ [I4: nat > nat > nat] : B2 ) )
& ! [Xa2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( ( member3122884635697634578at_nat @ Y6
@ ( piE_na8627462036710459160at_nat @ A
@ ^ [I4: nat > nat > nat] : B2 ) )
& ! [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1068_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X3
@ ( piE_nat_nat_nat_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: ( ( nat > nat ) > nat ) > nat] :
( ( ( member2991261302380110260at_nat @ Y6
@ ( piE_nat_nat_nat_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1069_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ A ) @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member8006650745835019538at_nat @ X3
@ ( piE_na8120565252419913496at_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( ( member8006650745835019538at_nat @ Y6
@ ( piE_na8120565252419913496at_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1070_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ A ) @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member1506620086977839122at_nat @ X3
@ ( piE_na7011197487990663704at_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( ( member1506620086977839122at_nat @ Y6
@ ( piE_na7011197487990663704at_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1071_PiE__uniqueness,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A
@ ^ [I4: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: nat > nat] :
( ( ( member_nat_nat @ Y6
@ ( piE_nat_nat @ A
@ ^ [I4: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1072_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A ) @ B2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A
@ ^ [I4: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y6
@ ( piE_nat_nat_nat @ A
@ ^ [I4: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1073_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ A ) @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member4489290058226556451at_nat @ X3
@ ( piE_na6840239867990089257at_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: ( ( nat > nat ) > nat ) > nat > nat] :
( ( ( member4489290058226556451at_nat @ Y6
@ ( piE_na6840239867990089257at_nat @ A
@ ^ [I4: ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1074_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A
@ ^ [I4: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [Y6: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y6
@ ( piE_nat_nat_nat2 @ A
@ ^ [I4: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( Y6 @ Xa )
= ( F @ Xa ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1075_PiE__mono,axiom,
! [A: set_nat_nat,B2: ( nat > nat ) > set_nat,C2: ( nat > nat ) > set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C2 @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A @ B2 ) @ ( piE_nat_nat_nat @ A @ C2 ) ) ) ).
% PiE_mono
thf(fact_1076_PiE__mono,axiom,
! [A: set_nat,B2: nat > set_nat,C2: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C2 @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A @ B2 ) @ ( piE_nat_nat @ A @ C2 ) ) ) ).
% PiE_mono
thf(fact_1077_PiE__mono,axiom,
! [A: set_nat_nat,B2: ( nat > nat ) > set_nat_nat,C2: ( nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C2 @ X3 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A @ B2 ) @ ( piE_nat_nat_nat_nat3 @ A @ C2 ) ) ) ).
% PiE_mono
thf(fact_1078_PiE__mono,axiom,
! [A: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat_nat,C2: ( nat > nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C2 @ X3 ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A @ B2 ) @ ( piE_na7122919648973241129at_nat @ A @ C2 ) ) ) ).
% PiE_mono
thf(fact_1079_PiE__mono,axiom,
! [A: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat_nat,C2: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C2 @ X3 ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A @ B2 ) @ ( piE_na6840239867990089257at_nat @ A @ C2 ) ) ) ).
% PiE_mono
thf(fact_1080_PiE__mono,axiom,
! [A: set_nat,B2: nat > set_nat_nat,C2: nat > set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C2 @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A @ B2 ) @ ( piE_nat_nat_nat2 @ A @ C2 ) ) ) ).
% PiE_mono
thf(fact_1081_subset__Collect__iff,axiom,
! [B2: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1082_subset__Collect__iff,axiom,
! [B2: set_nat_nat_nat,A: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ B2 @ A )
=> ( ( ord_le3211623285424100676at_nat @ B2
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1083_subset__Collect__iff,axiom,
! [B2: set_nat_nat_nat2,A: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le5934964663421696068at_nat @ B2 @ A )
=> ( ( ord_le5934964663421696068at_nat @ B2
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1084_subset__Collect__iff,axiom,
! [B2: set_nat_nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ B2
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1085_subset__CollectI,axiom,
! [B2: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1086_subset__CollectI,axiom,
! [B2: set_nat_nat_nat,A: set_nat_nat_nat,Q: ( nat > nat > nat ) > $o,P: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ B2 @ A )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1087_subset__CollectI,axiom,
! [B2: set_nat_nat_nat2,A: set_nat_nat_nat2,Q: ( ( nat > nat ) > nat ) > $o,P: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le5934964663421696068at_nat @ B2 @ A )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1088_subset__CollectI,axiom,
! [B2: set_nat_nat,A: set_nat_nat,Q: ( nat > nat ) > $o,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1089_conj__subset__def,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le9059583361652607317at_nat @ A @ ( collect_nat_nat @ P ) )
& ( ord_le9059583361652607317at_nat @ A @ ( collect_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1090_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
( foldin8133931898133206727at_nat @ ord_less_eq_nat @ ord_less_nat @ top_top_set_nat
@ ^ [X: nat] : X ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_1091_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
( foldin9130795139525211007nt_int @ ord_less_eq_int @ ord_less_int @ top_top_set_int
@ ^ [X: int] : X ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_1092_folding__insort__key_Oinj__on,axiom,
! [Less_eq: ( nat > nat ) > ( nat > nat ) > $o,Less: ( nat > nat ) > ( nat > nat ) > $o,S: set_nat,F: nat > nat > nat] :
( ( foldin4490235068905269046at_nat @ Less_eq @ Less @ S @ F )
=> ( inj_on_nat_nat_nat2 @ F @ S ) ) ).
% folding_insort_key.inj_on
thf(fact_1093_folding__insort__key_Oinj__on,axiom,
! [Less_eq: ( nat > nat ) > ( nat > nat ) > $o,Less: ( nat > nat ) > ( nat > nat ) > $o,S: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( foldin311176147066610597at_nat @ Less_eq @ Less @ S @ F )
=> ( inj_on2461717442902640625at_nat @ F @ S ) ) ).
% folding_insort_key.inj_on
thf(fact_1094_folding__insort__key_Oinj__on,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F: nat > nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F )
=> ( inj_on_nat_nat @ F @ S ) ) ).
% folding_insort_key.inj_on
thf(fact_1095_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_1096_layered__subspace__def,axiom,
( hales_4259056829518216709ce_int
= ( ^ [S4: ( nat > nat ) > nat > nat,K2: nat,N3: nat,T2: nat,R2: int,Chi: ( nat > nat ) > int] :
( ( hales_is_subspace @ S4 @ K2 @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X: nat] :
( ( member_nat @ X @ ( set_ord_atMost_nat @ K2 ) )
=> ? [C3: int] :
( ( ord_less_int @ C3 @ R2 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K2 @ T2 @ X ) )
=> ( ( Chi @ ( S4 @ Y ) )
= C3 ) ) ) )
& ( member_nat_nat_int @ Chi
@ ( piE_nat_nat_int @ ( hales_cube @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_int @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1097_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S4: ( nat > nat ) > nat > nat,K2: nat,N3: nat,T2: nat,R2: nat,Chi: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S4 @ K2 @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X: nat] :
( ( member_nat @ X @ ( set_ord_atMost_nat @ K2 ) )
=> ? [C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K2 @ T2 @ X ) )
=> ( ( Chi @ ( S4 @ Y ) )
= C3 ) ) ) )
& ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1098_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1099_inj__image__Compl__subset,axiom,
! [F: ( nat > nat > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat] :
( ( inj_on991820888481174479at_nat @ F @ top_to3655771597906314132at_nat )
=> ( ord_le5934964663421696068at_nat @ ( image_786723269765334627at_nat @ F @ ( uminus2441371681993390125at_nat @ A ) ) @ ( uminus5164713059990985517at_nat @ ( image_786723269765334627at_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1100_inj__image__Compl__subset,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on7066290451648512369at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ ( uminus5164713059990985517at_nat @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1101_inj__image__Compl__subset,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat > nat,A: set_nat_nat_nat2] :
( ( inj_on484924104190628815at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ord_le3211623285424100676at_nat @ ( image_279826485474788963at_nat @ F @ ( uminus5164713059990985517at_nat @ A ) ) @ ( uminus2441371681993390125at_nat @ ( image_279826485474788963at_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1102_inj__image__Compl__subset,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( inj_on8598928376616154831at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ord_le5934964663421696068at_nat @ ( image_8393830757900314979at_nat @ F @ ( uminus5164713059990985517at_nat @ A ) ) @ ( uminus5164713059990985517at_nat @ ( image_8393830757900314979at_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1103_inj__image__Compl__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1104_inj__image__Compl__subset,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A: set_nat_nat_nat2] :
( ( inj_on3244975737280743776at_nat @ F @ top_to6379112975903909524at_nat )
=> ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ ( uminus5164713059990985517at_nat @ A ) ) @ ( uminus4145589374814813630at_nat @ ( image_1262493855416953332at_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1105_inj__image__Compl__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ top_top_set_nat_nat )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( uminus4145589374814813630at_nat @ A ) ) @ ( uminus4145589374814813630at_nat @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1106_inj__image__Compl__subset,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ top_top_set_nat )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus4145589374814813630at_nat @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1107_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_1108_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_1109_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_1110_ComplI,axiom,
! [C: nat > nat,A: set_nat_nat] :
( ~ ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ ( uminus4145589374814813630at_nat @ A ) ) ) ).
% ComplI
thf(fact_1111_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_1112_ComplI,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat] :
( ~ ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ ( uminus2441371681993390125at_nat @ A ) ) ) ).
% ComplI
thf(fact_1113_ComplI,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ~ ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ ( uminus5164713059990985517at_nat @ A ) ) ) ).
% ComplI
thf(fact_1114_Compl__iff,axiom,
! [C: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ C @ ( uminus4145589374814813630at_nat @ A ) )
= ( ~ ( member_nat_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1115_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1116_Compl__iff,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( uminus2441371681993390125at_nat @ A ) )
= ( ~ ( member_nat_nat_nat2 @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1117_Compl__iff,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( uminus5164713059990985517at_nat @ A ) )
= ( ~ ( member_nat_nat_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1118_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_1119_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_1120_minus__add__distrib,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_1121_minus__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1122_add__minus__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1123_inj__uminus,axiom,
! [A: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A ) ).
% inj_uminus
thf(fact_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_Compl__anti__mono,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( uminus4145589374814813630at_nat @ B2 ) @ ( uminus4145589374814813630at_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_1131_Compl__subset__Compl__iff,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( uminus4145589374814813630at_nat @ A ) @ ( uminus4145589374814813630at_nat @ B2 ) )
= ( ord_le9059583361652607317at_nat @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1132_surj__uminus,axiom,
( ( image_int_int @ uminus_uminus_int @ top_top_set_int )
= top_top_set_int ) ).
% surj_uminus
thf(fact_1133_atMost__subset__iff,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X2 ) @ ( set_or250740698829186286at_nat @ Y3 ) )
= ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_1134_atMost__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_1135_atMost__subset__iff,axiom,
! [X2: int,Y3: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X2 ) @ ( set_ord_atMost_int @ Y3 ) )
= ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_1136_atMost__subset__iff,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X2 ) @ ( set_or9140604705432621368at_nat @ Y3 ) )
= ( ord_less_eq_nat_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_1137_image__add__atMost,axiom,
! [C: int,A2: int] :
( ( image_int_int @ ( plus_plus_int @ C ) @ ( set_ord_atMost_int @ A2 ) )
= ( set_ord_atMost_int @ ( plus_plus_int @ C @ A2 ) ) ) ).
% image_add_atMost
thf(fact_1138_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_1139_le__imp__neg__le,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1140_minus__le__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1141_le__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1142_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1143_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1144_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1145_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1146_add_Oinverse__distrib__swap,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1147_is__num__normalize_I8_J,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1148_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1149_Compl__eq,axiom,
( uminus4145589374814813630at_nat
= ( ^ [A5: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
~ ( member_nat_nat @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1150_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A5: set_nat] :
( collect_nat
@ ^ [X: nat] :
~ ( member_nat @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1151_Compl__eq,axiom,
( uminus2441371681993390125at_nat
= ( ^ [A5: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1152_Compl__eq,axiom,
( uminus5164713059990985517at_nat
= ( ^ [A5: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_1153_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1154_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1155_ComplD,axiom,
! [C: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ C @ ( uminus4145589374814813630at_nat @ A ) )
=> ~ ( member_nat_nat @ C @ A ) ) ).
% ComplD
thf(fact_1156_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_1157_ComplD,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( uminus2441371681993390125at_nat @ A ) )
=> ~ ( member_nat_nat_nat2 @ C @ A ) ) ).
% ComplD
thf(fact_1158_ComplD,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( uminus5164713059990985517at_nat @ A ) )
=> ~ ( member_nat_nat_nat @ C @ A ) ) ).
% ComplD
thf(fact_1159_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B: int] :
( ( A2 = B )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1160_not__UNIV__eq__Iic,axiom,
! [H: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_eq_Iic
thf(fact_1161_layered__eq__classes,axiom,
! [S: ( nat > nat ) > nat > nat,K: nat,N2: nat,T: nat,R3: nat,Chi2: ( nat > nat ) > nat] :
( ( hales_4261547300027266985ce_nat @ S @ K @ N2 @ T @ R3 @ Chi2 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ ( set_ord_atMost_nat @ K ) )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ ( hales_classes @ K @ T @ X4 ) )
=> ! [Xb: nat > nat] :
( ( member_nat_nat @ Xb @ ( hales_classes @ K @ T @ X4 ) )
=> ( ( Chi2 @ ( S @ Xa2 ) )
= ( Chi2 @ ( S @ Xb ) ) ) ) ) ) ) ).
% layered_eq_classes
thf(fact_1162_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X: int] : ( ord_less_eq_int @ X @ U ) ) ) ) ).
% atMost_def
thf(fact_1163_dim0__layered__subspace__ex,axiom,
! [Chi2: ( nat > nat ) > nat,N2: nat,T: nat,R3: nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N2 @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ zero_zero_nat @ N2 @ T @ R3 @ Chi2 ) ) ).
% dim0_layered_subspace_ex
thf(fact_1164_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_1165_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1166_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1167_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1168_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1169_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1170_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1171_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1172_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1173_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1174_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1175_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1176_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1177_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1178_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1179_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1180_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1181_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1182_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1183_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1184_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1185_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1186_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1187_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_1188_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1189_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1190_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1191_dim0__subspace__ex,axiom,
! [T: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S5 @ zero_zero_nat @ N2 @ T ) ) ).
% dim0_subspace_ex
thf(fact_1192_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
= ( P @ B5 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1193_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1194_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1195_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1196_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1197_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1198_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1199_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1200_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1201_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1202_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1203_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1204_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1205_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_1206_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_1207_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_1208_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1209_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1210_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1211_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1212_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_1213_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_1214_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1215_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1216_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1217_left__add__mult__distrib,axiom,
! [I: nat,U3: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U3 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U3 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1218_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1219_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1220_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1221_nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1222_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1223_mult__inc,axiom,
! [X2: num,Y3: num] :
( ( times_times_num @ X2 @ ( inc @ Y3 ) )
= ( plus_plus_num @ ( times_times_num @ X2 @ Y3 ) @ X2 ) ) ).
% mult_inc
thf(fact_1224_add__inc,axiom,
! [X2: num,Y3: num] :
( ( plus_plus_num @ X2 @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus_num @ X2 @ Y3 ) ) ) ).
% add_inc
thf(fact_1225_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1226_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1227_int__cases4,axiom,
! [M2: int] :
( ! [N4: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1228_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1229_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1230_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1231_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_1232_zadd__int__left,axiom,
! [M2: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1233_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_1234_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_1235_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1236_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1237_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1238_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1239_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1240_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1241_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1242_int__ops_I3_J,axiom,
! [N2: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
= ( numeral_numeral_int @ N2 ) ) ).
% int_ops(3)
thf(fact_1243_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_1244_not__int__zless__negative,axiom,
! [N2: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1245_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1246_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1247_int__ops_I8_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1248_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_1249_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_1250_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_1251_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_1252_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_1253_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_1254_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1255_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_1256_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_1257_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_1258_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1259_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1260_verit__less__mono__div__int2,axiom,
! [A: int,B2: int,N2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N2 ) @ ( divide_divide_int @ A @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1261_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_1262_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1263_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $true @ X2 @ Y3 )
= X2 ) ).
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,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( inj_on_nat_nat_nat2
@ ^ [X: nat] :
( restrict_nat_nat
@ ^ [Y: nat] : X
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ k ) )
& ( ( image_nat_nat_nat2
@ ^ [X: nat] :
( restrict_nat_nat
@ ^ [Y: nat] : X
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ k ) )
= ( hales_cube @ one_one_nat @ k ) ) ) ).
%------------------------------------------------------------------------------