TPTP Problem File: SLH0718^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Hales_Jewett/0002_Hales_Jewett/prob_01285_056575__5864744_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1545 ( 553 unt; 270 typ; 0 def)
% Number of atoms : 3646 (1161 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11562 ( 194 ~; 21 |; 308 &;9504 @)
% ( 0 <=>;1535 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 3803 (3803 >; 0 *; 0 +; 0 <<)
% Number of symbols : 247 ( 244 usr; 19 con; 0-6 aty)
% Number of variables : 3926 ( 461 ^;3322 !; 143 ?;3926 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:31.695
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__Set__Oset_I_062_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_M_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_J,type,
set_na988703028659548781at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na2061094069944666878at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_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_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na2687664174320723471at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_na5550323840042141967at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na6273678875609698720at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na7233567106578532785at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na7223334286561397169at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8175506400003264433at_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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (244)
thf(sy_c_Finite__Set_Ocard_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
finite1794908990118856198at_nat: set_nat_nat_nat2 > nat ).
thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_card_nat_nat: set_nat_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Fun_Obij__betw_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,
bij_be3563731812766147924at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Obij__betw_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,
bij_be4581752835692700517at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_be1059735840858801910at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_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,
bij_be4864432616675852389at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
bij_be3386790225224311798at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bij_be1775884438337700052at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Fun_Obij__betw_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,
bij_be5311014265664741861at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Obij__betw_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,
bij_be6420382030093991653at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_be5678534868967705974at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_betw_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Int__Oint,type,
bij_betw_int_int: ( int > int ) > set_int > set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
bij_be8282881169987224566at_nat: ( nat > ( nat > nat ) > nat ) > set_nat > set_nat_nat_nat2 > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bij_be168876897561698550at_nat: ( nat > nat > nat > nat ) > set_nat > set_nat_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_betw_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_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_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_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_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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_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,
piE_na799184809307736020at_nat: set_na6626867396258451522at_nat > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat ) > set_na988703028659548781at_nat ).
thf(sy_c_FuncSet_OPiE_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_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na8009433643268206821at_nat: set_na6626867396258451522at_nat > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat_nat3 ) > set_na2061094069944666878at_nat ).
thf(sy_c_FuncSet_OPiE_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_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_na3061559539522535286at_nat: set_na6626867396258451522at_nat > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2 ) > set_na5550323840042141967at_nat ).
thf(sy_c_FuncSet_OPiE_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_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na4170927303951785078at_nat: set_na6626867396258451522at_nat > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat ) > set_na2687664174320723471at_nat ).
thf(sy_c_FuncSet_OPiE_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na5629913657871898759at_nat: set_na6626867396258451522at_nat > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat ) > set_na6273678875609698720at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6564615839001774232at_nat: set_nat_nat_nat_nat3 > ( ( ( nat > nat ) > nat > nat ) > set_nat_nat ) > set_na8175506400003264433at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6840239867990089257at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat ) > set_na8843485148432118594at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat_nat5 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na513457764284933144at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat_nat_nat ) > set_na7223334286561397169at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na5223350113562215832at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat_nat3 ) > set_na7233567106578532785at_nat ).
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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri395706236947657049at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_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_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
restri3376681761679556074at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_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_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri4486049526108805866at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restri3045778531447280891at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat ) > set_na6626867396258451522at_nat > ( ( nat > 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_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_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri1704181820465610764at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat_nat > ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_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,
restri6011711336257459485at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_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_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_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_M_Eo_J,type,
minus_6692596912184789802_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_7158188067284919257_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
minus_2851842960567056136_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_7240682219522218504_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
minus_167519014754328503_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_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,
minus_5225787954611647771at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_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,
minus_4646100876039749548at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
minus_1221035652888719293at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_7721066311745899709at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_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_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hales__Jewett_Oclasses,type,
hales_classes: nat > nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ohj,type,
hales_hj: nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
hales_114318738418697479at_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > ( ( nat > nat ) > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
hales_4783935871306402712at_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > 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_Olhj,type,
hales_lhj: nat > nat > nat > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_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_M_Eo_J,type,
ord_le6599672692516096367_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le4961065272816086430_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
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_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le7877100967975825120at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le4629963735342356977at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le2785809691299232406at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_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_le6177938698872215975at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le319988079983864419_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5430825838364970130_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $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_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le3015115239550301420at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le747776305331315197at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
ord_le6328358011536291439at_nat: set_na2687664174320723471at_nat > set_na2687664174320723471at_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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4724818764771537408at_nat: set_na6273678875609698720at_nat > set_na6273678875609698720at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le9041126503034175505at_nat: set_na8175506400003264433at_nat > set_na8175506400003264433at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le973658574027395234at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_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_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
collec2410089373097230945at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na6626867396258451522at_nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec3567154360959927026at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
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_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_Oimage_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_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,
image_323718453976782111at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Set_Oimage_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_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6137295791034124976at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat ) > set_na6626867396258451522at_nat > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_Oimage_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_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_4065942021260649921at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_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_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_5175309785689899713at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ) > set_na6626867396258451522at_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_722231358656203602at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat ) > set_na6626867396258451522at_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_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_001t__Nat__Onat,type,
image_3521005150465447523at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat ) > set_na6626867396258451522at_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_8194121248528334964at_nat: ( ( ( nat > nat ) > nat > nat ) > nat ) > set_nat_nat_nat_nat3 > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1262493855416953332at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_7809927846809980933at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1545173636400105204at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_913610194320715013at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6605983383471867107at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat_nat_nat3 ).
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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
image_3941236537129881699at_nat: ( nat > ( nat > nat ) > ( nat > nat ) > nat ) > set_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6393715451659844596at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_nat > set_nat_nat_nat_nat3 ).
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_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_or7738058496500099077at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3591701359631937174at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
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_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_or6177432841829679145at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or7562748684798938298at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or2699333443382148811at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3808701207811398603at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1140352010380016476at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
fChoic2516396905127217208at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( nat > nat ) > ( nat > nat ) > nat ).
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fChoic52552927678224201at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( nat > nat ) > nat > nat ).
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fChoice_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > ( nat > nat ) > nat ).
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thf(sy_c_fChoice_001t__Nat__Onat,type,
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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_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member399543257700039186at_nat: ( ( nat > nat > nat ) > nat > nat > nat ) > set_na7223334286561397169at_nat > $o ).
thf(sy_c_member_001_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,type,
member4771969839209708323at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_na8778986904112484418at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8881365325514865170at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_na7233567106578532785at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member4402528950554000163at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member1679187572556404771at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_na3764207730537033026at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member952132173341509300at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
member_nat_nat_int: ( ( nat > nat ) > int ) > set_nat_nat_int > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member2740455936716430260at_nat: ( nat > ( nat > nat ) > nat ) > set_nat_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member17114558718834868at_nat: ( nat > nat > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_v_L____,type,
l: ( nat > nat ) > nat > nat ).
thf(sy_v_L__line____,type,
l_line: nat > nat > nat ).
thf(sy_v_M_H____,type,
m: nat ).
thf(sy_v_S____,type,
s: ( nat > nat ) > nat > nat ).
thf(sy_v_T_H____,type,
t: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_v_Tset____,type,
tset: set_nat_nat ).
thf(sy_v__092_060chi_062L____,type,
chi_L: ( nat > nat ) > ( nat > nat ) > nat ).
thf(sy_v__092_060chi_062L__s____,type,
chi_L_s: ( nat > nat ) > nat ).
thf(sy_v__092_060chi_062S____,type,
chi_S: ( nat > nat ) > nat ).
thf(sy_v__092_060chi_062____,type,
chi: ( nat > nat ) > nat ).
thf(sy_v__092_060phi_062____,type,
phi: ( ( nat > nat ) > nat ) > nat ).
thf(sy_v_d____,type,
d: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_m____,type,
m2: nat ).
thf(sy_v_n_H____,type,
n: nat ).
thf(sy_v_n____,type,
n2: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_t,type,
t2: nat ).
thf(sy_v_x____,type,
x: nat > nat ).
% Relevant facts (1265)
thf(fact_0_a,axiom,
member_nat_nat @ x @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ).
% a
thf(fact_1_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_2_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_3__092_060open_062n_A_L_Am_A_061_AM_H_092_060close_062,axiom,
( ( plus_plus_nat @ n2 @ m2 )
= m ) ).
% \<open>n + m = M'\<close>
thf(fact_4_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_5_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_6_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_7_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_8_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_9_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X: ( nat > nat > nat ) > nat > nat] :
( ( member4771969839209708323at_nat @ X
@ ( piE_na7122919648973241129at_nat @ A2
@ ^ [I: nat > nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_10_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member4489290058226556451at_nat @ X
@ ( piE_na6840239867990089257at_nat @ A2
@ ^ [I: ( nat > nat ) > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_11_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B2 )
=> ? [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_12_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat3] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ B2 )
=> ? [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X
@ ( piE_na5223350113562215832at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_13_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X: ( nat > nat > nat ) > nat > nat > nat] :
( ( member399543257700039186at_nat @ X
@ ( piE_na513457764284933144at_nat @ A2
@ ^ [I: nat > nat > nat] : B2 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_14_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_15_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_16_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_17_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_18_join__cubes,axiom,
! [F: nat > nat,N: nat,T: nat,G: nat > nat,M: nat] :
( ( member_nat_nat @ F @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( ( member_nat_nat @ G @ ( hales_cube @ M @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( member_nat_nat @ ( hales_join_nat @ F @ G @ N @ M ) @ ( hales_cube @ ( plus_plus_nat @ N @ M ) @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_19_L__line__base__prop,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
=> ( member_nat_nat @ ( l_line @ X2 ) @ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% L_line_base_prop
thf(fact_20_assms_I4_J,axiom,
! [K: nat,R: nat] :
( ( ord_less_eq_nat @ K @ k )
=> ( hales_lhj @ R @ t2 @ K ) ) ).
% assms(4)
thf(fact_21__092_060chi_062L__s__def,axiom,
( chi_L_s
= ( restrict_nat_nat_nat
@ ^ [X3: nat > nat] : ( phi @ ( chi_L @ X3 ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% \<chi>L_s_def
thf(fact_22_T_H__def,axiom,
( t
= ( restri1704181820465610764at_nat
@ ^ [X3: nat > nat] :
( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( hales_join_nat @ ( l_line @ ( X3 @ zero_zero_nat ) ) @ ( s @ Y ) @ n2 @ m2 )
@ ( hales_cube @ k @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% T'_def
thf(fact_23_PiE__ext,axiom,
! [X4: ( nat > nat ) > nat > nat,K2: set_nat_nat,S: ( nat > nat ) > set_nat_nat,Y2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ ( piE_nat_nat_nat_nat3 @ K2 @ S ) )
=> ( ( member952132173341509300at_nat @ Y2 @ ( piE_nat_nat_nat_nat3 @ K2 @ S ) )
=> ( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_24_PiE__ext,axiom,
! [X4: nat > nat,K2: set_nat,S: nat > set_nat,Y2: nat > nat] :
( ( member_nat_nat @ X4 @ ( piE_nat_nat @ K2 @ S ) )
=> ( ( member_nat_nat @ Y2 @ ( piE_nat_nat @ K2 @ S ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_25_PiE__ext,axiom,
! [X4: ( nat > nat ) > nat,K2: set_nat_nat,S: ( nat > nat ) > set_nat,Y2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ ( piE_nat_nat_nat @ K2 @ S ) )
=> ( ( member_nat_nat_nat @ Y2 @ ( piE_nat_nat_nat @ K2 @ S ) )
=> ( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_26_PiE__ext,axiom,
! [X4: ( nat > nat ) > ( nat > nat ) > nat,K2: set_nat_nat,S: ( nat > nat ) > set_nat_nat_nat2,Y2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X4 @ ( piE_na7569501297962130601at_nat @ K2 @ S ) )
=> ( ( member4402528950554000163at_nat @ Y2 @ ( piE_na7569501297962130601at_nat @ K2 @ S ) )
=> ( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_27_PiE__ext,axiom,
! [X4: nat > nat > nat,K2: set_nat,S: nat > set_nat_nat,Y2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ ( piE_nat_nat_nat2 @ K2 @ S ) )
=> ( ( member_nat_nat_nat2 @ Y2 @ ( piE_nat_nat_nat2 @ K2 @ S ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_28_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,K2: set_na6626867396258451522at_nat,S: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat] :
( ( member1174580258192983937at_nat @ X4 @ ( piE_na5629913657871898759at_nat @ K2 @ S ) )
=> ( ( member1174580258192983937at_nat @ Y2 @ ( piE_na5629913657871898759at_nat @ K2 @ S ) )
=> ( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_29_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,K2: set_na6626867396258451522at_nat,S: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member3693257457796161904at_nat @ X4 @ ( piE_na4170927303951785078at_nat @ K2 @ S ) )
=> ( ( member3693257457796161904at_nat @ Y2 @ ( piE_na4170927303951785078at_nat @ K2 @ S ) )
=> ( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_30_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,K2: set_na6626867396258451522at_nat,S: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member6416598835793757296at_nat @ X4 @ ( piE_na3061559539522535286at_nat @ K2 @ S ) )
=> ( ( member6416598835793757296at_nat @ Y2 @ ( piE_na3061559539522535286at_nat @ K2 @ S ) )
=> ( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_31_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat,K2: set_na6626867396258451522at_nat,S: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat_nat3,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat] :
( ( member5171902557168503775at_nat @ X4 @ ( piE_na8009433643268206821at_nat @ K2 @ S ) )
=> ( ( member5171902557168503775at_nat @ Y2 @ ( piE_na8009433643268206821at_nat @ K2 @ S ) )
=> ( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_32_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat,K2: set_na6626867396258451522at_nat,S: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat] :
( ( member6105598001968527566at_nat @ X4 @ ( piE_na799184809307736020at_nat @ K2 @ S ) )
=> ( ( member6105598001968527566at_nat @ Y2 @ ( piE_na799184809307736020at_nat @ K2 @ S ) )
=> ( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ K2 )
=> ( ( X4 @ I2 )
= ( Y2 @ I2 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_33_PiE__mem,axiom,
! [F: nat > nat,S2: set_nat,T2: nat > set_nat,X4: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S2 @ T2 ) )
=> ( ( member_nat @ X4 @ S2 )
=> ( member_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_34_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S2: set_nat_nat,T2: ( nat > nat ) > set_nat,X4: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S2 @ T2 ) )
=> ( ( member_nat_nat @ X4 @ S2 )
=> ( member_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_35_PiE__mem,axiom,
! [F: nat > nat > nat,S2: set_nat,T2: nat > set_nat_nat,X4: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ S2 @ T2 ) )
=> ( ( member_nat @ X4 @ S2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_36_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat,S2: set_nat_nat_nat,T2: ( nat > nat > nat ) > set_nat,X4: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F @ ( piE_nat_nat_nat_nat2 @ S2 @ T2 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ S2 )
=> ( member_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_37_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,S2: set_nat_nat_nat2,T2: ( ( nat > nat ) > nat ) > set_nat,X4: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F @ ( piE_nat_nat_nat_nat @ S2 @ T2 ) )
=> ( ( member_nat_nat_nat @ X4 @ S2 )
=> ( member_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_38_PiE__mem,axiom,
! [F: nat > nat > nat > nat,S2: set_nat,T2: nat > set_nat_nat_nat,X4: nat] :
( ( member17114558718834868at_nat @ F @ ( piE_nat_nat_nat_nat5 @ S2 @ T2 ) )
=> ( ( member_nat @ X4 @ S2 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_39_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat,S2: set_nat,T2: nat > set_nat_nat_nat2,X4: nat] :
( ( member2740455936716430260at_nat @ F @ ( piE_nat_nat_nat_nat4 @ S2 @ T2 ) )
=> ( ( member_nat @ X4 @ S2 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_40_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat,S2: set_nat_nat,T2: ( nat > nat ) > set_nat_nat,X4: nat > nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ S2 @ T2 ) )
=> ( ( member_nat_nat @ X4 @ S2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_41_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat > nat,S2: set_nat_nat,T2: ( nat > nat ) > set_nat_nat_nat,X4: nat > nat] :
( ( member1679187572556404771at_nat @ F @ ( piE_na8678869062391380393at_nat @ S2 @ T2 ) )
=> ( ( member_nat_nat @ X4 @ S2 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_42_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat > nat,S2: set_nat_nat_nat,T2: ( nat > nat > nat ) > set_nat_nat,X4: nat > nat > nat] :
( ( member4771969839209708323at_nat @ F @ ( piE_na7122919648973241129at_nat @ S2 @ T2 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ S2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( T2 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_43_PiE__cong,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat_nat_nat3 @ I3 @ A2 )
= ( piE_nat_nat_nat_nat3 @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_44_PiE__cong,axiom,
! [I3: set_nat,A2: nat > set_nat,B2: nat > set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat @ I3 @ A2 )
= ( piE_nat_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_45_PiE__cong,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat,B2: ( nat > nat ) > set_nat] :
( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat_nat @ I3 @ A2 )
= ( piE_nat_nat_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_46_PiE__cong,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat_nat_nat2,B2: ( nat > nat ) > set_nat_nat_nat2] :
( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_na7569501297962130601at_nat @ I3 @ A2 )
= ( piE_na7569501297962130601at_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_47_PiE__cong,axiom,
! [I3: set_nat,A2: nat > set_nat_nat,B2: nat > set_nat_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_nat_nat_nat2 @ I3 @ A2 )
= ( piE_nat_nat_nat2 @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_48_PiE__cong,axiom,
! [I3: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_na5629913657871898759at_nat @ I3 @ A2 )
= ( piE_na5629913657871898759at_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_49_PiE__cong,axiom,
! [I3: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_na4170927303951785078at_nat @ I3 @ A2 )
= ( piE_na4170927303951785078at_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_50_PiE__cong,axiom,
! [I3: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2] :
( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_na3061559539522535286at_nat @ I3 @ A2 )
= ( piE_na3061559539522535286at_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_51_PiE__cong,axiom,
! [I3: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat_nat3] :
( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_na8009433643268206821at_nat @ I3 @ A2 )
= ( piE_na8009433643268206821at_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_52_PiE__cong,axiom,
! [I3: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat] :
( ! [I2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I2 @ I3 )
=> ( ( A2 @ I2 )
= ( B2 @ I2 ) ) )
=> ( ( piE_na799184809307736020at_nat @ I3 @ A2 )
= ( piE_na799184809307736020at_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_53_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_54_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_55__092_060open_062n_H_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ n @ n2 ).
% \<open>n' \<le> n\<close>
thf(fact_56_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_57_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_58_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_59_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_60_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_61_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_62_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_63_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y2 ) )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_64_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y2: nat] :
( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_65_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_66_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_67_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_68_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_69_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_70_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_71_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_72_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_73_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_74_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_75_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_76_PiE__restrict,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_77_PiE__restrict,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_78_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_79_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_80_PiE__restrict,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat_nat2] :
( ( member4402528950554000163at_nat @ F @ ( piE_na7569501297962130601at_nat @ A2 @ B2 ) )
=> ( ( restri6011711336257459485at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_81_PiE__restrict,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat_nat_nat3] :
( ( member8881365325514865170at_nat @ F @ ( piE_na5223350113562215832at_nat @ A2 @ B2 ) )
=> ( ( restri1704181820465610764at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_82_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ( member1174580258192983937at_nat @ F @ ( piE_na5629913657871898759at_nat @ A2 @ B2 ) )
=> ( ( restri3045778531447280891at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_83_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ( member3693257457796161904at_nat @ F @ ( piE_na4170927303951785078at_nat @ A2 @ B2 ) )
=> ( ( restri4486049526108805866at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_84_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2] :
( ( member6416598835793757296at_nat @ F @ ( piE_na3061559539522535286at_nat @ A2 @ B2 ) )
=> ( ( restri3376681761679556074at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_85_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat_nat3] :
( ( member5171902557168503775at_nat @ F @ ( piE_na8009433643268206821at_nat @ A2 @ B2 ) )
=> ( ( restri395706236947657049at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_86_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t2 ).
% assms(1)
thf(fact_87_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_88_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_89_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_90_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_91_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_92_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_93_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_94_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_95_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_96_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_97_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_98_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_99_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_100_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_101_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_102_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_103_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_104_mem__Collect__eq,axiom,
! [A: nat > nat > nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A @ ( collect_nat_nat_nat2 @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A @ ( collect_nat_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ A @ ( collec3567154360959927026at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( member4402528950554000163at_nat @ A @ ( collec2410089373097230945at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( collec2410089373097230945at_nat
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_116_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_117_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_118_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_119_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_120_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_121_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_122_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_123_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_124_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_125_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_126_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_127_add__nonneg__eq__0__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X4 @ Y2 )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_128_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_129_add__nonpos__eq__0__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X4 @ Y2 )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_130_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_131_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= ( restrict_nat_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_132_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_133_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat,G: ( nat > nat ) > ( nat > nat ) > nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restri6011711336257459485at_nat @ F @ A2 )
= ( restri6011711336257459485at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_134_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat > nat,G: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restri1704181820465610764at_nat @ F @ A2 )
= ( restri1704181820465610764at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_135_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_136_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_137_restrict__apply_H,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_138_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restrict_nat_nat_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_139_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restri4446420529079022766at_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_140_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restri6011711336257459485at_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_141_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restri1704181820465610764at_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_142_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_143_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_144_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_145_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_146_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C2: nat] :
( B3
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_147_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_148_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_149_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_150_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_151_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_152_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_153_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_154_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_155_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_156_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_157_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( I4 = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_158_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_159_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_160_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_161_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_162_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_163_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_164_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_165_cube__def,axiom,
( hales_cube
= ( ^ [N2: nat,T3: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N2 )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T3 ) ) ) ) ).
% cube_def
thf(fact_166_restrict__PiE__iff,axiom,
! [F: nat > nat,I3: set_nat,X5: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I3 ) @ ( piE_nat_nat @ I3 @ X5 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I3 )
=> ( member_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_167_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I3: set_nat,X5: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I3 ) @ ( piE_nat_nat_nat2 @ I3 @ X5 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I3 )
=> ( member_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_168_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I3: set_nat_nat,X5: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I3 ) @ ( piE_nat_nat_nat @ I3 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I3 )
=> ( member_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_169_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I3: set_nat_nat,X5: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I3 ) @ ( piE_nat_nat_nat_nat3 @ I3 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I3 )
=> ( member_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_170_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,I3: set_nat_nat,X5: ( nat > nat ) > set_nat_nat_nat2] :
( ( member4402528950554000163at_nat @ ( restri6011711336257459485at_nat @ F @ I3 ) @ ( piE_na7569501297962130601at_nat @ I3 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I3 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_171_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,I3: set_nat_nat,X5: ( nat > nat ) > set_nat_nat_nat_nat3] :
( ( member8881365325514865170at_nat @ ( restri1704181820465610764at_nat @ F @ I3 ) @ ( piE_na5223350113562215832at_nat @ I3 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I3 )
=> ( member952132173341509300at_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_172_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,I3: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ( member1174580258192983937at_nat @ ( restri3045778531447280891at_nat @ F @ I3 ) @ ( piE_na5629913657871898759at_nat @ I3 @ X5 ) )
= ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ I3 )
=> ( member_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_173_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,I3: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ( member3693257457796161904at_nat @ ( restri4486049526108805866at_nat @ F @ I3 ) @ ( piE_na4170927303951785078at_nat @ I3 @ X5 ) )
= ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ I3 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_174_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,I3: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2] :
( ( member6416598835793757296at_nat @ ( restri3376681761679556074at_nat @ F @ I3 ) @ ( piE_na3061559539522535286at_nat @ I3 @ X5 ) )
= ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ I3 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_175_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat,I3: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat_nat3] :
( ( member5171902557168503775at_nat @ ( restri395706236947657049at_nat @ F @ I3 ) @ ( piE_na8009433643268206821at_nat @ I3 @ X5 ) )
= ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ I3 )
=> ( member952132173341509300at_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_176_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_177_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_178_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_179_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_180_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_181_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_182_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_183_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_184_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_185_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_186_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_187_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_188_group__cancel_Oadd2,axiom,
! [B2: nat,K2: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_189_group__cancel_Oadd2,axiom,
! [B2: int,K2: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_190_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_191_group__cancel_Oadd1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_192_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( I4 = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I4 @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_193_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( I4 = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I4 @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_194_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_195_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_196_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_197_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_198__092_060chi_062S__def,axiom,
( chi_S
= ( restrict_nat_nat_nat
@ ^ [Y: nat > nat] : ( chi @ ( hales_join_nat @ ( l_line @ zero_zero_nat ) @ Y @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% \<chi>S_def
thf(fact_199__092_060open_062_092_060chi_062L__s_A_092_060in_062_Acube_An_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060s_125_092_060close_062,axiom,
( member_nat_nat_nat @ chi_L_s
@ ( piE_nat_nat_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) ) ).
% \<open>\<chi>L_s \<in> cube n (t + 1) \<rightarrow>\<^sub>E {..<s}\<close>
thf(fact_200__092_060chi_062L__def,axiom,
( chi_L
= ( restri6011711336257459485at_nat
@ ^ [X3: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y: nat > nat] : ( chi @ ( hales_join_nat @ X3 @ Y @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% \<chi>L_def
thf(fact_201__092_060chi_062L__prop,axiom,
( member4402528950554000163at_nat @ chi_L
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] :
( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [J2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ) ).
% \<chi>L_prop
thf(fact_202_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_203_lessThan__subset__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X4 ) @ ( set_ord_lessThan_int @ Y2 ) )
= ( ord_less_eq_int @ X4 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_204_lessThan__subset__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_205_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_206_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_207_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_208_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_209_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_210__092_060chi_062__prop,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ m @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<chi>_prop
thf(fact_211_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_212__092_060open_0620_A_060_As_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ s2 ).
% \<open>0 < s\<close>
thf(fact_213_lessThan__eq__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X4 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X4 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_214_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_215_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_216_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_217_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_218_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_219_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_220_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_221_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_222_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_223_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_224_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_225_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_226_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_227_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_228_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_229_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_230_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_231_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_232_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_233_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_234_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_235_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_236_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_237_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_238_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_239_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_240_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_241_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_242_lessThan__iff,axiom,
! [I4: nat > nat,K2: nat > nat] :
( ( member_nat_nat @ I4 @ ( set_or1140352010380016476at_nat @ K2 ) )
= ( ord_less_nat_nat @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_243_lessThan__iff,axiom,
! [I4: nat > nat > nat,K2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ ( set_or3808701207811398603at_nat @ K2 ) )
= ( ord_less_nat_nat_nat2 @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_244_lessThan__iff,axiom,
! [I4: ( nat > nat ) > nat,K2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ ( set_or2699333443382148811at_nat @ K2 ) )
= ( ord_less_nat_nat_nat @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_245_lessThan__iff,axiom,
! [I4: ( nat > nat ) > nat > nat,K2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ ( set_or7562748684798938298at_nat @ K2 ) )
= ( ord_le4629963735342356977at_nat @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_246_lessThan__iff,axiom,
! [I4: ( nat > nat ) > ( nat > nat ) > nat,K2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ ( set_or6177432841829679145at_nat @ K2 ) )
= ( ord_le7877100967975825120at_nat @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_247_lessThan__iff,axiom,
! [I4: int,K2: int] :
( ( member_int @ I4 @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_248_lessThan__iff,axiom,
! [I4: nat,K2: nat] :
( ( member_nat @ I4 @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I4 @ K2 ) ) ).
% lessThan_iff
thf(fact_249_diff__diff__cancel,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_250_diff__diff__left,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K2 )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_251_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_252_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_253_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_254_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_255_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_256_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_257_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_258_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_259_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_260_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_261_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_262_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_263_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_264_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_265_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_266_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_267_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_268_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_269_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_270_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_271_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_272__092_060open_062_092_060chi_062S_A_092_060in_062_Acube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_092_060close_062,axiom,
( member_nat_nat_nat @ chi_S
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<open>\<chi>S \<in> cube m (t + 1) \<rightarrow>\<^sub>E {..<r}\<close>
thf(fact_273_A,axiom,
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
=> ( member_nat @ ( chi @ ( hales_join_nat @ X2 @ Xa @ n2 @ m2 ) ) @ ( set_ord_lessThan_nat @ r ) ) ) ) ).
% A
thf(fact_274_S__prop,axiom,
hales_4261547300027266985ce_nat @ s @ k @ m2 @ t2 @ r @ chi_S ).
% S_prop
thf(fact_275_n_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [N3: nat] :
( ( ord_less_eq_nat @ n @ N3 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ N3 @ t2 @ s2 @ Chi ) ) ) ) ).
% n'_props
thf(fact_276_m__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ m2 )
& ! [M2: nat] :
( ( ord_less_eq_nat @ m2 @ M2 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ M2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ k @ M2 @ t2 @ r @ Chi ) ) ) ) ).
% m_props
thf(fact_277_s__def,axiom,
( s2
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) @ m2 ) ) ) ).
% s_def
thf(fact_278_L__prop,axiom,
hales_4261547300027266985ce_nat @ l @ one_one_nat @ n2 @ t2 @ s2 @ chi_L_s ).
% L_prop
thf(fact_279_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_280_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_281_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_282_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_283_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_284_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_285_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_286_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_287_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_288_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_289_less__diff__conv,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_290_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_291_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_292_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_293_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_294_linorder__neqE__nat,axiom,
! [X4: nat,Y2: nat] :
( ( X4 != Y2 )
=> ( ~ ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ Y2 @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_295_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_296_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_297_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_298_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_299_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_300_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_301_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_302_diff__commute,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_303_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_304_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_305_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_306_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_307_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat,C4: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_308_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_309_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B2 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_310_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_311_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat_nat2,C4: ( nat > nat ) > set_nat_nat_nat2] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le5934964663421696068at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le973658574027395234at_nat @ ( piE_na7569501297962130601at_nat @ A2 @ B2 ) @ ( piE_na7569501297962130601at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_312_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat_nat,C4: ( nat > nat > nat ) > set_nat_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B2 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_313_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B2 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_314_PiE__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le9041126503034175505at_nat @ ( piE_na6564615839001774232at_nat @ A2 @ B2 ) @ ( piE_na6564615839001774232at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_315_PiE__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le4724818764771537408at_nat @ ( piE_na5629913657871898759at_nat @ A2 @ B2 ) @ ( piE_na5629913657871898759at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_316_PiE__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat,C4: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A2 )
=> ( ord_le3211623285424100676at_nat @ ( B2 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le6328358011536291439at_nat @ ( piE_na4170927303951785078at_nat @ A2 @ B2 ) @ ( piE_na4170927303951785078at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_317_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_318_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_319_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_320_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_321_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_322_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_323_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_324_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_325_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_326_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_327_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_328_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_329_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_330_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_331_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_332_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_333_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_334_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_335_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_336_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_337_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_338_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_339_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_340_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_341_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_342_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_343_add__lessD1,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ K2 )
=> ( ord_less_nat @ I4 @ K2 ) ) ).
% add_lessD1
thf(fact_344_add__less__mono,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_345_not__add__less1,axiom,
! [I4: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J ) @ I4 ) ).
% not_add_less1
thf(fact_346_not__add__less2,axiom,
! [J: nat,I4: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I4 ) @ I4 ) ).
% not_add_less2
thf(fact_347_add__less__mono1,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_348_trans__less__add1,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_349_trans__less__add2,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_350_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_351_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_352_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_353_less__diff__conv2,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I4 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I4 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_354_lessThan__def,axiom,
( set_or1140352010380016476at_nat
= ( ^ [U: nat > nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] : ( ord_less_nat_nat @ X3 @ U ) ) ) ) ).
% lessThan_def
thf(fact_355_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_int @ X3 @ U ) ) ) ) ).
% lessThan_def
thf(fact_356_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U ) ) ) ) ).
% lessThan_def
thf(fact_357_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_358_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_359_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_360_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_361_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_362_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_363_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_364_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_365_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_366_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_367_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_368_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_369_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_370_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_371_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_372_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_373_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_374_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_375_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_376_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_377_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_378_add__mono__thms__linordered__field_I5_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_379_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( I4 = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_380_add__mono__thms__linordered__field_I2_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( I4 = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_381_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_382_add__mono__thms__linordered__field_I1_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_383_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_384_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_385_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_386_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_387_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_388_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_389_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_390_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_391_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_392_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_393_cube__restrict,axiom,
! [J: nat,N: nat,Y2: nat > nat,T: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( member_nat_nat @ Y2 @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y2 @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T ) ) ) ) ).
% cube_restrict
thf(fact_394_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_395_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ( minus_minus_nat @ J @ I4 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_396_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I4 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_397_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J ) @ K2 )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_398_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_399_le__diff__conv,axiom,
! [J: nat,K2: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I4 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I4 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_400_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_401_less__imp__add__positive,axiom,
! [I4: nat,J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I4 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_402_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_403_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_404_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_405_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_406_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_407_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_408_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_409_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_410_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_411_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_412_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_413_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_414_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_415_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_416_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_417_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_418_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_419_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_420_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I4 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_421_add__mono__thms__linordered__field_I3_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I4 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_422_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I4 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_423_add__mono__thms__linordered__field_I4_J,axiom,
! [I4: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I4 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I4 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_424_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_425_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_426_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_427_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_428_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_429_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_430_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_431_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_432_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_433_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_434_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_435_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_436_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_437_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_438_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_439_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_440_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_441_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_442_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_443_cube__subset,axiom,
! [N: nat,T: nat] : ( ord_le9059583361652607317at_nat @ ( hales_cube @ N @ T ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% cube_subset
thf(fact_444_cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X @ zero_zero_nat )
= S ) ) ) ).
% cube_props(1)
thf(fact_445_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M6: nat] :
( ( P @ X4 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M6 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_446_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_447_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_448_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_449_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_450_le__trans,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I4 @ K2 ) ) ) ).
% le_trans
thf(fact_451_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_452_split__cube_I2_J,axiom,
! [X4: nat > nat,K2: nat,T: nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ ( plus_plus_nat @ K2 @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y: nat] : ( X4 @ ( plus_plus_nat @ Y @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K2 ) )
@ ( hales_cube @ K2 @ T ) ) ) ).
% split_cube(2)
thf(fact_453_split__cube_I1_J,axiom,
! [X4: nat > nat,K2: nat,T: nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ ( plus_plus_nat @ K2 @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X4 @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_454_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_455_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_456_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_457_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_458_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_459_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_460_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_461_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_462_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_463_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_464_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_465_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K2 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_466_add__le__mono,axiom,
! [I4: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_467_add__le__mono1,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_468_trans__le__add1,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_469_trans__le__add2,axiom,
! [I4: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_470_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_471__092_060phi_062__prop,axiom,
( bij_be1059735840858801910at_nat @ phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s2 ) ) ).
% \<phi>_prop
thf(fact_472__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A0_A_060_Am_A_092_060and_062_A_I_092_060forall_062M_H_092_060ge_062m_O_A_092_060forall_062_092_060chi_062_O_A_092_060chi_062_A_092_060in_062_Acube_AM_H_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_A_092_060longrightarrow_062_A_I_092_060exists_062S_O_Alayered__subspace_AS_Ak_AM_H_At_Ar_A_092_060chi_062_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M5: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ M5 )
& ! [M2: nat] :
( ( ord_less_eq_nat @ M5 @ M2 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ M2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ k @ M2 @ t2 @ r @ Chi ) ) ) ) ).
% \<open>\<And>thesis. (\<And>m. 0 < m \<and> (\<forall>M'\<ge>m. \<forall>\<chi>. \<chi> \<in> cube M' (t + 1) \<rightarrow>\<^sub>E {..<r} \<longrightarrow> (\<exists>S. layered_subspace S k M' t r \<chi>)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_473__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_O_A0_A_060_An_H_A_092_060and_062_A_I_092_060forall_062N_092_060ge_062n_H_O_A_092_060forall_062_092_060chi_062_O_A_092_060chi_062_A_092_060in_062_Acube_AN_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060s_125_A_092_060longrightarrow_062_A_I_092_060exists_062S_O_Alayered__subspace_AS_A1_AN_At_As_A_092_060chi_062_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N5: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ N5 )
& ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ N3 @ t2 @ s2 @ Chi ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n'. 0 < n' \<and> (\<forall>N\<ge>n'. \<forall>\<chi>. \<chi> \<in> cube N (t + 1) \<rightarrow>\<^sub>E {..<s} \<longrightarrow> (\<exists>S. layered_subspace S 1 N t s \<chi>)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_474__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_O_Alayered__subspace_AS_Ak_Am_At_Ar_A_092_060chi_062S_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S3: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ S3 @ k @ m2 @ t2 @ r @ chi_S ) ).
% \<open>\<And>thesis. (\<And>S. layered_subspace S k m t r \<chi>S \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_475__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062L_O_Alayered__subspace_AL_A1_An_At_As_A_092_060chi_062L__s_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L2: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ L2 @ one_one_nat @ n2 @ t2 @ s2 @ chi_L_s ) ).
% \<open>\<And>thesis. (\<And>L. layered_subspace L 1 n t s \<chi>L_s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_476_L__line__def,axiom,
( l_line
= ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( l
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% L_line_def
thf(fact_477_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_478_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_479_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_480_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_481_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_482_s__coloured,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= s2 ) ).
% s_coloured
thf(fact_483_bij__betw__restrict__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ ( restri9050993537824894510at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_484_bij__betw__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_betw_nat_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_485_bij__betw__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 @ B2 )
= ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_486_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_betw_nat_nat_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_487_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_488_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( bij_be5311014265664741861at_nat @ ( restri6011711336257459485at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be5311014265664741861at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_489_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat_nat_nat3] :
( ( bij_be1775884438337700052at_nat @ ( restri1704181820465610764at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be1775884438337700052at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_490__092_060open_062card_A_Icube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_J_A_061_Ar_A_094_A_It_A_L_A1_J_A_094_Am_092_060close_062,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) @ m2 ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = r ^ (t + 1) ^ m\<close>
thf(fact_491__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060phi_062_O_Abij__betw_A_092_060phi_062_A_Icube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_J_A_123_O_O_060s_125_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Phi: ( ( nat > nat ) > nat ) > nat] :
~ ( bij_be1059735840858801910at_nat @ Phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s2 ) ) ).
% \<open>\<And>thesis. (\<And>\<phi>. bij_betw \<phi> (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) {..<s} \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_492_line__subspace__s,axiom,
! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] :
( ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ n2 @ t2 @ s2 @ Chi2 )
& ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
@ n2
@ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
% line_subspace_s
thf(fact_493_ex__bij__betw__nat__finite__2,axiom,
! [A2: set_nat_nat_nat2,N: nat] :
( ( ( finite1794908990118856198at_nat @ A2 )
= N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [F2: ( ( nat > nat ) > nat ) > nat] : ( bij_be1059735840858801910at_nat @ F2 @ A2 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% ex_bij_betw_nat_finite_2
thf(fact_494_ex__bij__betw__nat__finite__2,axiom,
! [A2: set_nat,N: nat] :
( ( ( finite_card_nat @ A2 )
= N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [F2: nat > nat] : ( bij_betw_nat_nat @ F2 @ A2 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% ex_bij_betw_nat_finite_2
thf(fact_495_ex__bij__betw__nat__finite__2,axiom,
! [A2: set_nat_nat,N: nat] :
( ( ( finite_card_nat_nat @ A2 )
= N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [F2: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ F2 @ A2 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% ex_bij_betw_nat_finite_2
thf(fact_496_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y2: int] :
( ( X4 != Y2 )
=> ( ~ ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ Y2 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_497_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_498_cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S ) )
@ zero_zero_nat )
= S ) ) ).
% cube_props(2)
thf(fact_499_cube__props_I4_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S ) ) )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% cube_props(4)
thf(fact_500_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: int,Chi2: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S2 @ one_one_nat @ N @ T @ R @ Chi2 )
=> ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ! [L3: nat] :
( ( ord_less_nat @ L3 @ T )
=> ( ( ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= L3 ) ) ) ) ) )
& ( ord_less_int
@ ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
@ R ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_501_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: nat,Chi2: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S2 @ one_one_nat @ N @ T @ R @ Chi2 )
=> ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ! [L3: nat] :
( ( ord_less_nat @ L3 @ T )
=> ( ( ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= L3 ) ) ) ) ) )
& ( ord_less_nat
@ ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
@ R ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_502_dim1__layered__subspace__as__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: int,Chi2: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S2 @ one_one_nat @ N @ T @ R @ Chi2 )
=> ? [C1: int,C22: int] :
( ( ord_less_int @ C1 @ R )
& ( ord_less_int @ C22 @ R )
& ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ( ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= C1 ) )
& ( ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_503_dim1__layered__subspace__as__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: nat,Chi2: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S2 @ one_one_nat @ N @ T @ R @ Chi2 )
=> ? [C1: nat,C22: nat] :
( ( ord_less_nat @ C1 @ R )
& ( ord_less_nat @ C22 @ R )
& ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ( ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= C1 ) )
& ( ( Chi2
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_504_cube__props_I3_J,axiom,
! [S: nat,T: nat,S2: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_505_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_506_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_507_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_508_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_509_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_510_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_511_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_512_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_513_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_514_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_515_dim0__layered__subspace__ex,axiom,
! [Chi2: ( nat > nat ) > nat,N: nat,T: nat,R: nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ zero_zero_nat @ N @ T @ R @ Chi2 ) ) ).
% dim0_layered_subspace_ex
thf(fact_516_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_517_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_518_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_519_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_520_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_521_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_522_add__less__zeroD,axiom,
! [X4: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X4 @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X4 @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_523_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_524_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_525_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_526_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_527_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_528_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_529_add__le__imp__le__diff,axiom,
! [I4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ N )
=> ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_530_add__le__imp__le__diff,axiom,
! [I4: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K2 ) @ N )
=> ( ord_less_eq_int @ I4 @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_531_add__le__add__imp__diff__le,axiom,
! [I4: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_532_add__le__add__imp__diff__le,axiom,
! [I4: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I4 @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_533_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_534_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_535_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_536_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_537_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_538_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_539_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_540_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_541_lhj__def,axiom,
( hales_lhj
= ( ^ [R2: nat,T3: nat,K4: nat] :
? [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
& ! [N7: nat] :
( ( ord_less_eq_nat @ N6 @ N7 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N7 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S6: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S6 @ K4 @ N7 @ T3 @ R2 @ Chi3 ) ) ) ) ) ) ).
% lhj_def
thf(fact_542_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_543_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_544_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_545_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_546_power__increasing__iff,axiom,
! [B: nat,X4: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_547_power__increasing__iff,axiom,
! [B: int,X4: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_548_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_549_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_550_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_551_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_552_power__strict__increasing__iff,axiom,
! [B: nat,X4: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X4 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_553_power__strict__increasing__iff,axiom,
! [B: int,X4: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X4 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_554_card__lessThan,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U2 ) )
= U2 ) ).
% card_lessThan
thf(fact_555_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_556_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_557_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_558__092_060open_062r_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_A_061_Ar_A_094_A_It_A_L_A1_J_A_094_Am_092_060close_062,axiom,
( ( power_power_nat @ r @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) )
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) @ m2 ) ) ) ).
% \<open>r ^ card (cube m (t + 1)) = r ^ (t + 1) ^ m\<close>
thf(fact_559__092_060open_062card_A_123_O_O_060r_125_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_A_061_Ar_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_092_060close_062,axiom,
( ( power_power_nat @ ( finite_card_nat @ ( set_ord_lessThan_nat @ r ) ) @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) )
= ( power_power_nat @ r @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ) ).
% \<open>card {..<r} ^ card (cube m (t + 1)) = r ^ card (cube m (t + 1))\<close>
thf(fact_560_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_561_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_562_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_563__092_060open_062card_A_Icube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_J_A_061_Acard_A_123_O_O_060r_125_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_092_060close_062,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= ( power_power_nat @ ( finite_card_nat @ ( set_ord_lessThan_nat @ r ) ) @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = card {..<r} ^ card (cube m (t + 1))\<close>
thf(fact_564_cube__card,axiom,
! [N: nat,T: nat] :
( ( finite_card_nat_nat
@ ( piE_nat_nat @ ( set_ord_lessThan_nat @ N )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T ) ) )
= ( power_power_nat @ T @ N ) ) ).
% cube_card
thf(fact_565_line__points__in__cube__unfolded,axiom,
! [L4: nat > nat > nat,N: nat,T: nat,S: nat,J: nat] :
( ( hales_is_line @ L4 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L4 @ S @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_566_line__points__in__cube,axiom,
! [L4: nat > nat > nat,N: nat,T: nat,S: nat] :
( ( hales_is_line @ L4 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( member_nat_nat @ ( L4 @ S ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_567_one__dim__cube__eq__nat__set,axiom,
! [K2: nat] :
( bij_betw_nat_nat_nat
@ ^ [F3: nat > nat] : ( F3 @ zero_zero_nat )
@ ( hales_cube @ one_one_nat @ K2 )
@ ( set_ord_lessThan_nat @ K2 ) ) ).
% one_dim_cube_eq_nat_set
thf(fact_568_nat__set__eq__one__dim__cube,axiom,
! [K2: nat] :
( bij_betw_nat_nat_nat2
@ ^ [X3: nat] :
( restrict_nat_nat
@ ^ [Y: nat] : X3
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K2 )
@ ( hales_cube @ one_one_nat @ K2 ) ) ).
% nat_set_eq_one_dim_cube
thf(fact_569_is__line__def,axiom,
( hales_is_line
= ( ^ [L5: nat > nat > nat,N2: nat,T3: nat] :
( ( member_nat_nat_nat2 @ L5
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T3 )
@ ^ [I: nat] : ( hales_cube @ N2 @ T3 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ! [X3: nat] :
( ( ord_less_nat @ X3 @ T3 )
=> ! [Y: nat] :
( ( ord_less_nat @ Y @ T3 )
=> ( ( L5 @ X3 @ J2 )
= ( L5 @ Y @ J2 ) ) ) )
| ! [S4: nat] :
( ( ord_less_nat @ S4 @ T3 )
=> ( ( L5 @ S4 @ J2 )
= S4 ) ) ) )
& ? [J2: nat] :
( ( ord_less_nat @ J2 @ N2 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T3 )
=> ( ( L5 @ S4 @ J2 )
= S4 ) ) ) ) ) ) ).
% is_line_def
thf(fact_570_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_571_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_572_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_573_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_574_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_575_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_576_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_577_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_578_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_579_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_580_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_581_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_582_nat__power__less__imp__less,axiom,
! [I4: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I4 )
=> ( ( ord_less_nat @ ( power_power_nat @ I4 @ M ) @ ( power_power_nat @ I4 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_583_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_584_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_585_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_586_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_587_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_588_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_589_power__increasing,axiom,
! [N: nat,N8: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N8 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N8 ) ) ) ) ).
% power_increasing
thf(fact_590_power__increasing,axiom,
! [N: nat,N8: nat,A: int] :
( ( ord_less_eq_nat @ N @ N8 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N8 ) ) ) ) ).
% power_increasing
thf(fact_591_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_592_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_593_power__strict__increasing,axiom,
! [N: nat,N8: nat,A: nat] :
( ( ord_less_nat @ N @ N8 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N8 ) ) ) ) ).
% power_strict_increasing
thf(fact_594_power__strict__increasing,axiom,
! [N: nat,N8: nat,A: int] :
( ( ord_less_nat @ N @ N8 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N8 ) ) ) ) ).
% power_strict_increasing
thf(fact_595_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_596_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_597_power__decreasing,axiom,
! [N: nat,N8: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N8 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N8 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_598_power__decreasing,axiom,
! [N: nat,N8: nat,A: int] :
( ( ord_less_eq_nat @ N @ N8 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N8 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_599_power__strict__decreasing,axiom,
! [N: nat,N8: nat,A: nat] :
( ( ord_less_nat @ N @ N8 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N8 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_600_power__strict__decreasing,axiom,
! [N: nat,N8: nat,A: int] :
( ( ord_less_nat @ N @ N8 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N8 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_601_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_602_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_603_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_604_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_605_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_606_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_607_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_608_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_609_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_610_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_611_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_612_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_613_hj__imp__lhj__base,axiom,
! [T: nat,R: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ! [R3: nat] : ( hales_hj @ R3 @ T )
=> ( hales_lhj @ R @ T @ one_one_nat ) ) ) ).
% hj_imp_lhj_base
thf(fact_614_psubsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_615_dim1__subspace__is__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_subspace @ S2 @ one_one_nat @ N @ T )
=> ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T ) )
@ N
@ T ) ) ) ).
% dim1_subspace_is_line
thf(fact_616_some__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ( P @ A )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_617_some__eq__trivial,axiom,
! [X4: nat > nat] :
( ( fChoice_nat_nat
@ ^ [Y: nat > nat] : ( Y = X4 ) )
= X4 ) ).
% some_eq_trivial
thf(fact_618_subset__antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_619_subsetI,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_620_subsetI,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ X @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_621_subsetI,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ X @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_622_subsetI,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A2 )
=> ( member4402528950554000163at_nat @ X @ B2 ) )
=> ( ord_le973658574027395234at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_623_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_624_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_625_some__sym__eq__trivial,axiom,
! [X4: nat > nat] :
( ( fChoice_nat_nat
@ ( ^ [Y3: nat > nat,Z: nat > nat] : ( Y3 = Z )
@ X4 ) )
= X4 ) ).
% some_sym_eq_trivial
thf(fact_626_double__diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ( minus_8121590178497047118at_nat @ B2 @ ( minus_8121590178497047118at_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_627_Diff__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_628_Diff__mono,axiom,
! [A2: set_nat_nat,C4: set_nat_nat,D3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C4 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ ( minus_8121590178497047118at_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_629_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_630_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_631_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_nat,Z: set_nat_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_632_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_633_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_634_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_635_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_636_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [T3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T3 @ A4 )
=> ( member_nat_nat_nat2 @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_637_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [T3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T3 @ A4 )
=> ( member_nat_nat_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_638_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat3,B4: set_nat_nat_nat_nat3] :
! [T3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T3 @ A4 )
=> ( member952132173341509300at_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_639_subset__iff,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
! [T3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ T3 @ A4 )
=> ( member4402528950554000163at_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_640_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_641_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [T3: nat > nat] :
( ( member_nat_nat @ T3 @ A4 )
=> ( member_nat_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_642_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_643_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_644_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A4 )
=> ( member_nat_nat_nat2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_645_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A4 )
=> ( member_nat_nat_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_646_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat3,B4: set_nat_nat_nat_nat3] :
! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A4 )
=> ( member952132173341509300at_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_647_subset__eq,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A4 )
=> ( member4402528950554000163at_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_648_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_649_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A4 )
=> ( member_nat_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_650_equalityE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_651_subsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_652_subsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_653_subsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_654_subsetD,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A2 @ B2 )
=> ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_655_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_656_subsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_657_in__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,X4: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_658_in__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,X4: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_659_in__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,X4: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ X4 @ A2 )
=> ( member952132173341509300at_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_660_in__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat,X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A2 @ B2 )
=> ( ( member4402528950554000163at_nat @ X4 @ A2 )
=> ( member4402528950554000163at_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_661_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_662_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X4: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_663_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ? [B5: nat > nat] : ( member_nat_nat @ B5 @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_664_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ? [B5: nat > nat > nat] : ( member_nat_nat_nat2 @ B5 @ ( minus_7721066311745899709at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_665_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ? [B5: ( nat > nat ) > nat] : ( member_nat_nat_nat @ B5 @ ( minus_1221035652888719293at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_666_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ? [B5: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ B5 @ ( minus_4646100876039749548at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_667_psubset__imp__ex__mem,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( ord_le2785809691299232406at_nat @ A2 @ B2 )
=> ? [B5: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ B5 @ ( minus_5225787954611647771at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_668_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_669_psubsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_670_psubsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_671_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_672_psubsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_673_psubsetD,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le2785809691299232406at_nat @ A2 @ B2 )
=> ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_674_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_675_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S3 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_676_someI,axiom,
! [P: ( nat > nat ) > $o,X4: nat > nat] :
( ( P @ X4 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI
thf(fact_677_Eps__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( fChoice_nat_nat @ P )
= ( fChoice_nat_nat @ Q ) ) ) ).
% Eps_cong
thf(fact_678_tfl__some,axiom,
! [P3: ( nat > nat ) > $o,X2: nat > nat] :
( ( P3 @ X2 )
=> ( P3 @ ( fChoice_nat_nat @ P3 ) ) ) ).
% tfl_some
thf(fact_679_some__eq__imp,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat,B: nat > nat] :
( ( ( fChoice_nat_nat @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_680_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A4 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_681_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_682_less__eq__set__def,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat3,B4: set_nat_nat_nat_nat3] :
( ord_le5430825838364970130_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_683_less__eq__set__def,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
( ord_le319988079983864419_nat_o
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_684_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_685_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A4 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_686_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_687_Collect__subset,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_688_Collect__subset,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_689_Collect__subset,axiom,
! [A2: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ord_le973658574027395234at_nat
@ ( collec2410089373097230945at_nat
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_690_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_691_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_692_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A4 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_693_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A4 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_694_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_695_less__set__def,axiom,
( ord_le6177938698872215975at_nat
= ( ^ [A4: set_nat_nat_nat_nat3,B4: set_nat_nat_nat_nat3] :
( ord_le4961065272816086430_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_696_less__set__def,axiom,
( ord_le2785809691299232406at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
( ord_le6599672692516096367_nat_o
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_697_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_698_some1__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ? [X2: nat > nat] :
( ( P @ X2 )
& ! [Y4: nat > nat] :
( ( P @ Y4 )
=> ( Y4 = X2 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_699_some__eq__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
% some_eq_ex
thf(fact_700_someI2__bex,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ? [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) )
=> ( ! [X: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_701_someI2__bex,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ? [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) )
=> ( ! [X: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_702_someI2__bex,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ? [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
& ( P @ X2 ) )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoic52552927678224201at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_703_someI2__bex,axiom,
! [A2: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ? [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A2 )
& ( P @ X2 ) )
=> ( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member4402528950554000163at_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoic2516396905127217208at_nat
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_704_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) )
=> ( ! [X: nat] :
( ( ( member_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_705_someI2__bex,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) )
=> ( ! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_706_someI2__ex,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X_1: nat > nat] : ( P @ X_1 )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_707_someI__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ? [X_1: nat > nat] : ( P @ X_1 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI_ex
thf(fact_708_someI2,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat,Q: ( nat > nat ) > $o] :
( ( P @ A )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2
thf(fact_709_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_710_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_711_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ~ ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_712_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_713_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_714_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_715_psubsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_716_line__is__dim1__subspace__t__ge__1,axiom,
! [N: nat,T: nat,L4: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ T )
=> ( ( hales_is_line @ L4 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L4 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_717_line__is__dim1__subspace__t__1,axiom,
! [N: nat,L4: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( hales_is_line @ L4 @ N @ one_one_nat )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L4 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ one_one_nat ) )
@ one_one_nat
@ N
@ one_one_nat ) ) ) ).
% line_is_dim1_subspace_t_1
thf(fact_718_line__is__dim1__subspace,axiom,
! [N: nat,T: nat,L4: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_line @ L4 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L4 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_719_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_720_classes__subset__cube,axiom,
! [N: nat,T: nat,I4: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I4 ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_721_hj__def,axiom,
( hales_hj
= ( ^ [R2: nat,T3: nat] :
? [N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N6 )
& ! [N7: nat] :
( ( ord_less_eq_nat @ N6 @ N7 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N7 @ T3 )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [L5: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ R2 )
& ( hales_is_line @ L5 @ N7 @ T3 )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ L5 @ ( set_ord_lessThan_nat @ T3 ) ) )
=> ( ( Chi3 @ X3 )
= C2 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_722_image__eqI,axiom,
! [B: nat,F: nat > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_723_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_724_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_725_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_726_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X4: nat > nat > nat,A2: set_nat_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_727_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_728_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_729_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_730_image__eqI,axiom,
! [B: nat > nat > nat,F: ( nat > nat ) > nat > nat > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_3101123049818244468at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_731_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_732_DiffI,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_733_DiffI,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A2 )
=> ( ~ ( member_nat_nat_nat2 @ C @ B2 )
=> ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_734_DiffI,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat_nat @ C @ B2 )
=> ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_735_DiffI,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ A2 )
=> ( ~ ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_736_DiffI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A2 )
=> ( ~ ( member4402528950554000163at_nat @ C @ B2 )
=> ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_737_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_738_Diff__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat @ C @ A2 )
& ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_739_Diff__iff,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat_nat2 @ C @ A2 )
& ~ ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_740_Diff__iff,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat_nat @ C @ A2 )
& ~ ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_741_Diff__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
= ( ( member952132173341509300at_nat @ C @ A2 )
& ~ ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_742_Diff__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
= ( ( member4402528950554000163at_nat @ C @ A2 )
& ~ ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_743_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_744_image__ident,axiom,
! [Y6: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_745_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_746_image__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_747_image__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_748_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_749_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_750_image__restrict__eq,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_1991755285388994676at_nat @ ( restri6011711336257459485at_nat @ F @ A2 ) @ A2 )
= ( image_1991755285388994676at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_751_image__restrict__eq,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_6605983383471867107at_nat @ ( restri1704181820465610764at_nat @ F @ A2 ) @ A2 )
= ( image_6605983383471867107at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_752_image__add__0,axiom,
! [S2: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_753_image__add__0,axiom,
! [S2: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_754_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_755_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_756_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_757_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_758_imageI,axiom,
! [X4: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_759_imageI,axiom,
! [X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_760_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_761_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_762_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( image_3101123049818244468at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_763_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_764_image__iff,axiom,
! [Z2: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ Z2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_765_image__iff,axiom,
! [Z2: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_766_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_767_bex__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X2 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_768_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X2 ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_769_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_770_image__cong,axiom,
! [M6: set_nat_nat,N8: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M6 = N8 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M6 )
= ( image_3205354838064109189at_nat @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_771_image__cong,axiom,
! [M6: set_nat,N8: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M6 = N8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M6 )
= ( image_nat_nat_nat2 @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_772_image__cong,axiom,
! [M6: set_nat,N8: set_nat,F: nat > nat,G: nat > nat] :
( ( M6 = N8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M6 )
= ( image_nat_nat @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_773_ball__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_774_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_775_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_776_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_777_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_778_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_779_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_780_rev__image__eqI,axiom,
! [X4: nat > nat > nat,A2: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_781_rev__image__eqI,axiom,
! [X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_782_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_783_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_784_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: nat > nat > nat,F: ( nat > nat ) > nat > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_3101123049818244468at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_785_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_786_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_787_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_788_Compr__image__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_789_Compr__image__eq,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ ( image_6919068903512877573at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_6919068903512877573at_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_790_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_791_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_913610194320715013at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_913610194320715013at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_792_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_7809927846809980933at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_7809927846809980933at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_793_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_794_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat > nat,A2: set_nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ ( image_6393715451659844596at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_6393715451659844596at_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_795_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat > nat,A2: set_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ ( image_3101123049818244468at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_3101123049818244468at_nat @ F
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_796_image__image,axiom,
! [F: ( nat > nat ) > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_797_image__image,axiom,
! [F: nat > nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_798_image__image,axiom,
! [F: nat > nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_799_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_800_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_3205354838064109189at_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_801_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_802_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_803_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_804_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_805_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_806_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_807_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_808_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat_nat2 @ X @ A2 ) ) ) ).
% imageE
thf(fact_809_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ~ ! [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_810_imageE,axiom,
! [B: nat > nat,F: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat] :
( ( member_nat_nat @ B @ ( image_1545173636400105204at_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat_nat2 @ X @ A2 ) ) ) ).
% imageE
thf(fact_811_imageE,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ A2 ) )
=> ~ ! [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_812_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_813_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ).
% image_mono
thf(fact_814_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_815_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_816_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_817_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_818_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B2: set_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_819_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B2: set_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_820_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_821_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_822_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_823_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_3101123049818244468at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_824_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_825_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_826_subset__imageE,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_827_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B2
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_828_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_829_image__subset__iff,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_830_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_831_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_832_subset__image__iff,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_833_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B2
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_834_DiffE,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_835_DiffE,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_836_DiffE,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_837_DiffE,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_838_DiffE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
=> ~ ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_839_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_840_DiffD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_841_DiffD1,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ( member_nat_nat_nat2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_842_DiffD1,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ( member_nat_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_843_DiffD1,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ( member952132173341509300at_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_844_DiffD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
=> ( member4402528950554000163at_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_845_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_846_DiffD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_847_DiffD2,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat_nat2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_848_DiffD2,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_849_DiffD2,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ~ ( member952132173341509300at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_850_DiffD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
=> ~ ( member4402528950554000163at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_851_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_852_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_853_image__diff__subset,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) @ ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_854_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) @ ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_855_minus__set__def,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ( minus_7240682219522218504_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A4 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_856_minus__set__def,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ( minus_2851842960567056136_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_857_minus__set__def,axiom,
( minus_4646100876039749548at_nat
= ( ^ [A4: set_nat_nat_nat_nat3,B4: set_nat_nat_nat_nat3] :
( collec3567154360959927026at_nat
@ ( minus_7158188067284919257_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_858_minus__set__def,axiom,
( minus_5225787954611647771at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
( collec2410089373097230945at_nat
@ ( minus_6692596912184789802_nat_o
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ A4 )
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_859_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_860_minus__set__def,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ( minus_167519014754328503_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A4 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_861_set__diff__eq,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A4 )
& ~ ( member_nat_nat_nat2 @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_862_set__diff__eq,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A4 )
& ~ ( member_nat_nat_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_863_set__diff__eq,axiom,
( minus_4646100876039749548at_nat
= ( ^ [A4: set_nat_nat_nat_nat3,B4: set_nat_nat_nat_nat3] :
( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A4 )
& ~ ( member952132173341509300at_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_864_set__diff__eq,axiom,
( minus_5225787954611647771at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
( collec2410089373097230945at_nat
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A4 )
& ~ ( member4402528950554000163at_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_865_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ~ ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_866_set__diff__eq,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A4 )
& ~ ( member_nat_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_867_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_868_all__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_869_all__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A2 )
=> ( P @ ( image_3205354838064109189at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_870_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_871_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 )
=> ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_872_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member4402528950554000163at_nat @ Y5
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_873_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( image_5175309785689899713at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member3693257457796161904at_nat @ X
@ ( piE_na4170927303951785078at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat] :
( ( ( member3693257457796161904at_nat @ Y5
@ ( piE_na4170927303951785078at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_874_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_4065942021260649921at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member6416598835793757296at_nat @ X
@ ( piE_na3061559539522535286at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( ( member6416598835793757296at_nat @ Y5
@ ( piE_na3061559539522535286at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_875_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat_nat_nat3] :
( ( ord_le5260717879541182899at_nat @ ( image_6137295791034124976at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat] :
( ( member5171902557168503775at_nat @ X
@ ( piE_na8009433643268206821at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat] :
( ( ( member5171902557168503775at_nat @ Y5
@ ( piE_na8009433643268206821at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_876_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( ord_le973658574027395234at_nat @ ( image_323718453976782111at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat] :
( ( member6105598001968527566at_nat @ X
@ ( piE_na799184809307736020at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member6105598001968527566at_nat @ Y5
@ ( piE_na799184809307736020at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_877_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y5
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_878_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
=> ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_879_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_722231358656203602at_nat @ F @ A2 ) @ B2 )
=> ? [X: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat] :
( ( member1174580258192983937at_nat @ X
@ ( piE_na5629913657871898759at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat] :
( ( ( member1174580258192983937at_nat @ Y5
@ ( piE_na5629913657871898759at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_880_bij__betw__same__card,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( bij_be3563731812766147924at_nat @ F @ A2 @ B2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_881_bij__betw__same__card,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_882_bij__betw__same__card,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,B2: set_nat_nat] :
( ( bij_be4581752835692700517at_nat @ F @ A2 @ B2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_883_bij__betw__same__card,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_884_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_885_bij__betw__same__card,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_886_bij__betw__same__card,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( bij_be5311014265664741861at_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_887_bij__betw__same__card,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_888_bij__betw__same__card,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_889_bij__betw__add,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ ( plus_plus_nat @ A ) @ A2 @ B2 )
= ( ( image_nat_nat @ ( plus_plus_nat @ A ) @ A2 )
= B2 ) ) ).
% bij_betw_add
thf(fact_890_bij__betw__add,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( bij_betw_int_int @ ( plus_plus_int @ A ) @ A2 @ B2 )
= ( ( image_int_int @ ( plus_plus_int @ A ) @ A2 )
= B2 ) ) ).
% bij_betw_add
thf(fact_891_translation__subtract__diff,axiom,
! [A: int,S: set_int,T: set_int] :
( ( image_int_int
@ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
@ ( minus_minus_set_int @ S @ T ) )
= ( minus_minus_set_int
@ ( image_int_int
@ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
@ S )
@ ( image_int_int
@ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_892_bij__betw__byWitness,axiom,
! [A2: set_nat_nat_nat2,F4: nat > ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > nat,A5: set_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ A5 )
=> ( ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F4 @ A5 ) @ A2 )
=> ( bij_be1059735840858801910at_nat @ F @ A2 @ A5 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_893_bij__betw__byWitness,axiom,
! [A2: set_nat,F4: nat > nat,F: nat > nat,A5: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A5 ) @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ A5 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_894_bij__betw__byWitness,axiom,
! [A2: set_nat_nat,F4: nat > nat > nat,F: ( nat > nat ) > nat,A5: set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ A5 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F4 @ A5 ) @ A2 )
=> ( bij_betw_nat_nat_nat @ F @ A2 @ A5 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_895_bij__betw__byWitness,axiom,
! [A2: set_nat,F4: ( nat > nat ) > nat,F: nat > nat > nat,A5: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A5 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F4 @ A5 ) @ A2 )
=> ( bij_betw_nat_nat_nat2 @ F @ A2 @ A5 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_896_bij__betw__byWitness,axiom,
! [A2: set_nat_nat,F4: ( nat > nat ) > nat > nat,F: ( nat > nat ) > nat > nat,A5: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A5 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ A5 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F4 @ A5 ) @ A2 )
=> ( bij_be5678534868967705974at_nat @ F @ A2 @ A5 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_897_bij__betw__subset,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,A5: set_nat,B2: set_nat_nat_nat2,B6: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ A5 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ A2 )
=> ( ( ( image_7809927846809980933at_nat @ F @ B2 )
= B6 )
=> ( bij_be1059735840858801910at_nat @ F @ B2 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_898_bij__betw__subset,axiom,
! [F: nat > nat > nat,A2: set_nat,A5: set_nat_nat,B2: set_nat,B6: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ A5 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat_nat2 @ F @ B2 )
= B6 )
=> ( bij_betw_nat_nat_nat2 @ F @ B2 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_899_bij__betw__subset,axiom,
! [F: nat > nat,A2: set_nat,A5: set_nat,B2: set_nat,B6: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ A5 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat @ F @ B2 )
= B6 )
=> ( bij_betw_nat_nat @ F @ B2 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_900_bij__betw__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,A5: set_nat_nat,B2: set_nat_nat,B6: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ A5 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ( image_3205354838064109189at_nat @ F @ B2 )
= B6 )
=> ( bij_be5678534868967705974at_nat @ F @ B2 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_901_bij__betw__subset,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,A5: set_nat,B2: set_nat_nat,B6: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ A5 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat_nat @ F @ B2 )
= B6 )
=> ( bij_betw_nat_nat_nat @ F @ B2 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_902_translation__diff,axiom,
! [A: int,S: set_int,T: set_int] :
( ( image_int_int @ ( plus_plus_int @ A ) @ ( minus_minus_set_int @ S @ T ) )
= ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A ) @ S ) @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% translation_diff
thf(fact_903_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_904_bij__betwE,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_905_bij__betwE,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_906_bij__betwE,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_907_bij__betwE,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_908_bij__betw__inv,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ? [G2: ( ( nat > nat ) > nat ) > nat] : ( bij_be1059735840858801910at_nat @ G2 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_909_bij__betw__inv,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ? [G2: nat > ( nat > nat ) > nat] : ( bij_be8282881169987224566at_nat @ G2 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_910_bij__betw__inv,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ? [G2: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ G2 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_911_bij__betw__inv,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ? [G2: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ G2 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_912_bij__betw__inv,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ? [G2: nat > nat] : ( bij_betw_nat_nat @ G2 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_913_bij__betw__ball,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat,Phi2: nat > $o] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( Phi2 @ X3 ) ) )
= ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( Phi2 @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_914_bij__betw__ball,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,Phi2: nat > $o] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( Phi2 @ X3 ) ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( Phi2 @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_915_bij__betw__ball,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat,Phi2: ( nat > nat ) > $o] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( Phi2 @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( Phi2 @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_916_bij__betw__ball,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,Phi2: nat > $o] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( Phi2 @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( Phi2 @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_917_bij__betw__cong,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,G: ( ( nat > nat ) > nat ) > nat,A5: set_nat] :
( ! [A6: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_be1059735840858801910at_nat @ F @ A2 @ A5 )
= ( bij_be1059735840858801910at_nat @ G @ A2 @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_918_bij__betw__cong,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat,A5: set_nat] :
( ! [A6: nat > nat] :
( ( member_nat_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_betw_nat_nat_nat @ F @ A2 @ A5 )
= ( bij_betw_nat_nat_nat @ G @ A2 @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_919_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat,A5: set_nat_nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_betw_nat_nat_nat2 @ F @ A2 @ A5 )
= ( bij_betw_nat_nat_nat2 @ G @ A2 @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_920_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat,A5: set_nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
= ( bij_betw_nat_nat @ G @ A2 @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_921_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_922_bij__betw__apply,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,A: nat > nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_923_bij__betw__apply,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat,A: nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_924_bij__betw__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat,A: nat > nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_925_bij__betw__apply,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,B2: set_nat,A: nat > nat > nat] :
( ( bij_be3386790225224311798at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_926_bij__betw__apply,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,B2: set_nat_nat_nat,A: nat] :
( ( bij_be168876897561698550at_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_927_bij__betw__apply,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2,A: nat] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_928_bij__betw__apply,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat,A: ( nat > nat ) > nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_929_bij__betw__apply,axiom,
! [F: ( nat > nat ) > nat > nat > nat,A2: set_nat_nat,B2: set_nat_nat_nat,A: nat > nat] :
( ( bij_be6420382030093991653at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_930_bij__betw__apply,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2,A: nat > nat] :
( ( bij_be5311014265664741861at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_931_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F3: nat > nat,A4: set_nat,B4: set_nat] :
? [G3: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_932_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat_nat
= ( ^ [F3: ( nat > nat ) > nat,A4: set_nat_nat,B4: set_nat] :
? [G3: nat > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A4 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( member_nat_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_933_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat_nat2
= ( ^ [F3: nat > nat > nat,A4: set_nat,B4: set_nat_nat] :
? [G3: ( nat > nat ) > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B4 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_934_bij__betw__iff__bijections,axiom,
( bij_be5678534868967705974at_nat
= ( ^ [F3: ( nat > nat ) > nat > nat,A4: set_nat_nat,B4: set_nat_nat] :
? [G3: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A4 )
=> ( ( member_nat_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B4 )
=> ( ( member_nat_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_935_bij__betw__iff__bijections,axiom,
( bij_be168876897561698550at_nat
= ( ^ [F3: nat > nat > nat > nat,A4: set_nat,B4: set_nat_nat_nat] :
? [G3: ( nat > nat > nat ) > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat_nat_nat2 @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B4 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_936_bij__betw__iff__bijections,axiom,
( bij_be8282881169987224566at_nat
= ( ^ [F3: nat > ( nat > nat ) > nat,A4: set_nat,B4: set_nat_nat_nat2] :
? [G3: ( ( nat > nat ) > nat ) > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat_nat_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B4 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_937_bij__betw__iff__bijections,axiom,
( bij_be3386790225224311798at_nat
= ( ^ [F3: ( nat > nat > nat ) > nat,A4: set_nat_nat_nat,B4: set_nat] :
? [G3: nat > nat > nat > nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A4 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( member_nat_nat_nat2 @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_938_bij__betw__iff__bijections,axiom,
( bij_be1059735840858801910at_nat
= ( ^ [F3: ( ( nat > nat ) > nat ) > nat,A4: set_nat_nat_nat2,B4: set_nat] :
? [G3: nat > ( nat > nat ) > nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A4 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( member_nat_nat_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_939_bij__betw__iff__bijections,axiom,
( bij_be4864432616675852389at_nat
= ( ^ [F3: ( nat > nat > nat ) > nat > nat,A4: set_nat_nat_nat,B4: set_nat_nat] :
? [G3: ( nat > nat ) > nat > nat > nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A4 )
=> ( ( member_nat_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B4 )
=> ( ( member_nat_nat_nat2 @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_940_bij__betw__iff__bijections,axiom,
( bij_be4581752835692700517at_nat
= ( ^ [F3: ( ( nat > nat ) > nat ) > nat > nat,A4: set_nat_nat_nat2,B4: set_nat_nat] :
? [G3: ( nat > nat ) > ( nat > nat ) > nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A4 )
=> ( ( member_nat_nat @ ( F3 @ X3 ) @ B4 )
& ( ( G3 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B4 )
=> ( ( member_nat_nat_nat @ ( G3 @ X3 ) @ A4 )
& ( ( F3 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_941_subspace__elems__embed,axiom,
! [S2: ( nat > nat ) > nat > nat,K2: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S2 @ K2 @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S2 @ ( hales_cube @ K2 @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_942_bij__betw__imp__surj__on,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 )
=> ( ( image_3205354838064109189at_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_943_bij__betw__imp__surj__on,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( image_7809927846809980933at_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_944_bij__betw__imp__surj__on,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( image_nat_nat_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_945_bij__betw__imp__surj__on,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( image_nat_nat_nat2 @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_946_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( image_nat_nat @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_947_Tset__def,axiom,
( tset
= ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [I: nat,S4: nat > nat] :
( ( Uu
= ( hales_join_nat @ ( l_line @ I ) @ S4 @ n2 @ m2 ) )
& ( member_nat @ I @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t2 @ one_one_nat ) ) )
& ( member_nat_nat @ S4 @ ( image_3205354838064109189at_nat @ s @ ( hales_cube @ k @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ) ) ) ).
% Tset_def
thf(fact_948_layered__subspace__def,axiom,
( hales_4783935871306402712at_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N2: nat,T3: nat,R2: nat > nat,Chi3: ( nat > nat ) > nat > nat] :
( ( hales_is_subspace @ S6 @ K4 @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C2: nat > nat] :
( ( ord_less_nat_nat @ C2 @ R2 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K4 @ T3 @ X3 ) )
=> ( ( Chi3 @ ( S6 @ Y ) )
= C2 ) ) ) )
& ( member952132173341509300at_nat @ Chi3
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_or1140352010380016476at_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_949_layered__subspace__def,axiom,
( hales_114318738418697479at_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N2: nat,T3: nat,R2: ( nat > nat ) > nat,Chi3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( hales_is_subspace @ S6 @ K4 @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C2: ( nat > nat ) > nat] :
( ( ord_less_nat_nat_nat @ C2 @ R2 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K4 @ T3 @ X3 ) )
=> ( ( Chi3 @ ( S6 @ Y ) )
= C2 ) ) ) )
& ( member4402528950554000163at_nat @ Chi3
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_or2699333443382148811at_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_950_layered__subspace__def,axiom,
( hales_4259056829518216709ce_int
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N2: nat,T3: nat,R2: int,Chi3: ( nat > nat ) > int] :
( ( hales_is_subspace @ S6 @ K4 @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C2: int] :
( ( ord_less_int @ C2 @ R2 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K4 @ T3 @ X3 ) )
=> ( ( Chi3 @ ( S6 @ Y ) )
= C2 ) ) ) )
& ( member_nat_nat_int @ Chi3
@ ( piE_nat_nat_int @ ( hales_cube @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_int @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_951_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N2: nat,T3: nat,R2: nat,Chi3: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S6 @ K4 @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C2: nat] :
( ( ord_less_nat @ C2 @ R2 )
& ! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_classes @ K4 @ T3 @ X3 ) )
=> ( ( Chi3 @ ( S6 @ Y ) )
= C2 ) ) ) )
& ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N2 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_952_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat,S2: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ R4 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ S2 ) )
= ( ord_le3211623285424100676at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_953_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat2,S2: set_nat_nat_nat2] :
( ( ord_le996020443555834177_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ R4 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ S2 ) )
= ( ord_le5934964663421696068at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_954_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat_nat3,S2: set_nat_nat_nat_nat3] :
( ( ord_le5430825838364970130_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ R4 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ S2 ) )
= ( ord_le5260717879541182899at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_955_pred__subset__eq,axiom,
! [R4: set_na6626867396258451522at_nat,S2: set_na6626867396258451522at_nat] :
( ( ord_le319988079983864419_nat_o
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ R4 )
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X3 @ S2 ) )
= ( ord_le973658574027395234at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_956_pred__subset__eq,axiom,
! [R4: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ S2 ) )
= ( ord_less_eq_set_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_957_pred__subset__eq,axiom,
! [R4: set_nat_nat,S2: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ R4 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ S2 ) )
= ( ord_le9059583361652607317at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_958_some__inv__into__2,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F3: nat > nat] : ( F3 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into_2
thf(fact_959_atMost__eq__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X4 )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X4 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_960_atMost__iff,axiom,
! [I4: nat > nat,K2: nat > nat] :
( ( member_nat_nat @ I4 @ ( set_or9140604705432621368at_nat @ K2 ) )
= ( ord_less_eq_nat_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_961_atMost__iff,axiom,
! [I4: nat > nat > nat,K2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ ( set_or6142498856979658663at_nat @ K2 ) )
= ( ord_le3127000006974329230at_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_962_atMost__iff,axiom,
! [I4: ( nat > nat ) > nat,K2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ ( set_or5033131092550408871at_nat @ K2 ) )
= ( ord_le2017632242545079438at_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_963_atMost__iff,axiom,
! [I4: ( nat > nat ) > nat > nat,K2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ ( set_or3591701359631937174at_nat @ K2 ) )
= ( ord_le747776305331315197at_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_964_atMost__iff,axiom,
! [I4: ( nat > nat ) > ( nat > nat ) > nat,K2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ ( set_or7738058496500099077at_nat @ K2 ) )
= ( ord_le3015115239550301420at_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_965_atMost__iff,axiom,
! [I4: set_nat_nat,K2: set_nat_nat] :
( ( member_set_nat_nat @ I4 @ ( set_or250740698829186286at_nat @ K2 ) )
= ( ord_le9059583361652607317at_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_966_atMost__iff,axiom,
! [I4: int,K2: int] :
( ( member_int @ I4 @ ( set_ord_atMost_int @ K2 ) )
= ( ord_less_eq_int @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_967_atMost__iff,axiom,
! [I4: nat,K2: nat] :
( ( member_nat @ I4 @ ( set_ord_atMost_nat @ K2 ) )
= ( ord_less_eq_nat @ I4 @ K2 ) ) ).
% atMost_iff
thf(fact_968_atMost__subset__iff,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X4 ) @ ( set_or250740698829186286at_nat @ Y2 ) )
= ( ord_le9059583361652607317at_nat @ X4 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_969_atMost__subset__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X4 ) @ ( set_ord_atMost_int @ Y2 ) )
= ( ord_less_eq_int @ X4 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_970_atMost__subset__iff,axiom,
! [X4: nat > nat,Y2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X4 ) @ ( set_or9140604705432621368at_nat @ Y2 ) )
= ( ord_less_eq_nat_nat @ X4 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_971_atMost__subset__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X4 ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_972_image__add__atMost,axiom,
! [C: int,A: int] :
( ( image_int_int @ ( plus_plus_int @ C ) @ ( set_ord_atMost_int @ A ) )
= ( set_ord_atMost_int @ ( plus_plus_int @ C @ A ) ) ) ).
% image_add_atMost
thf(fact_973_exE__some,axiom,
! [P: ( nat > nat ) > $o,C: nat > nat] :
( ? [X_1: nat > nat] : ( P @ X_1 )
=> ( ( C
= ( fChoice_nat_nat @ P ) )
=> ( P @ C ) ) ) ).
% exE_some
thf(fact_974_setcompr__eq__image,axiom,
! [F: nat > nat,P: nat > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_975_setcompr__eq__image,axiom,
! [F: ( nat > nat ) > nat,P: ( nat > nat ) > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_nat @ F @ ( collect_nat_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_976_setcompr__eq__image,axiom,
! [F: nat > nat > nat,P: nat > $o] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_977_setcompr__eq__image,axiom,
! [F: ( nat > nat ) > nat > nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X3: nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( P @ X3 ) ) )
= ( image_3205354838064109189at_nat @ F @ ( collect_nat_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_978_Setcompr__eq__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat @ X3 @ A2 ) ) )
= ( image_nat_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_979_Setcompr__eq__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat_nat @ X3 @ A2 ) ) )
= ( image_nat_nat_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_980_Setcompr__eq__image,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X3: nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat @ X3 @ A2 ) ) )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_981_Setcompr__eq__image,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat > nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat_nat_nat2 @ X3 @ A2 ) ) )
= ( image_913610194320715013at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_982_Setcompr__eq__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: ( nat > nat ) > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat_nat_nat @ X3 @ A2 ) ) )
= ( image_7809927846809980933at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_983_Setcompr__eq__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X3: nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat_nat @ X3 @ A2 ) ) )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_984_Setcompr__eq__image,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > nat,A2: set_nat_nat_nat_nat3] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: ( nat > nat ) > nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member952132173341509300at_nat @ X3 @ A2 ) ) )
= ( image_8194121248528334964at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_985_Setcompr__eq__image,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X3: nat > nat > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat_nat_nat2 @ X3 @ A2 ) ) )
= ( image_1545173636400105204at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_986_Setcompr__eq__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2] :
( ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [X3: ( nat > nat ) > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member_nat_nat_nat @ X3 @ A2 ) ) )
= ( image_1262493855416953332at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_987_Setcompr__eq__image,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,A2: set_na6626867396258451522at_nat] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( Uu
= ( F @ X3 ) )
& ( member4402528950554000163at_nat @ X3 @ A2 ) ) )
= ( image_3521005150465447523at_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_988_atMost__def,axiom,
( set_or9140604705432621368at_nat
= ( ^ [U: nat > nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] : ( ord_less_eq_nat_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_989_atMost__def,axiom,
( set_or250740698829186286at_nat
= ( ^ [U: set_nat_nat] :
( collect_set_nat_nat
@ ^ [X3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_990_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_991_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_992_bij__unique__inv,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,X4: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ X4 @ B2 )
=> ( ( member_nat @ ( the_inv_into_nat_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_inv_into_nat_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_inv_into_nat_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_993_bij__unique__inv,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat,X4: nat > nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ X4 @ B2 )
=> ( ( member_nat @ ( the_in3844390324871770692at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_in3844390324871770692at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in3844390324871770692at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_994_bij__unique__inv,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,X4: nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ X4 @ B2 )
=> ( ( member_nat_nat @ ( the_in5300466440149791684at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5 @ A2 )
& ( ( the_in5300466440149791684at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in5300466440149791684at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_995_bij__unique__inv,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat,X4: nat > nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ X4 @ B2 )
=> ( ( member_nat_nat @ ( the_in2963963264082133811at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5 @ A2 )
& ( ( the_in2963963264082133811at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in2963963264082133811at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_996_bij__unique__inv,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,B2: set_nat_nat_nat,X4: nat > nat > nat] :
( ( bij_be168876897561698550at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ X4 @ B2 )
=> ( ( member_nat @ ( the_in6677677329530902195at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_in6677677329530902195at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in6677677329530902195at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_997_bij__unique__inv,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2,X4: ( nat > nat ) > nat] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ X4 @ B2 )
=> ( ( member_nat @ ( the_in5568309565101652403at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_in5568309565101652403at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in5568309565101652403at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_998_bij__unique__inv,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,B2: set_nat,X4: nat] :
( ( bij_be3386790225224311798at_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ X4 @ B2 )
=> ( ( member_nat_nat_nat2 @ ( the_in672218620338739635at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5 @ A2 )
& ( ( the_in672218620338739635at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in672218620338739635at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_999_bij__unique__inv,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat,X4: nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ X4 @ B2 )
=> ( ( member_nat_nat_nat @ ( the_in7568536272828005555at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5 @ A2 )
& ( ( the_in7568536272828005555at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in7568536272828005555at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_1000_bij__unique__inv,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat,X4: nat > nat] :
( ( bij_be4864432616675852389at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ X4 @ B2 )
=> ( ( member_nat_nat_nat2 @ ( the_in6738486182373217954at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5 @ A2 )
& ( ( the_in6738486182373217954at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in6738486182373217954at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_1001_bij__unique__inv,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,B2: set_nat_nat,X4: nat > nat] :
( ( bij_be4581752835692700517at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ X4 @ B2 )
=> ( ( member_nat_nat_nat @ ( the_in6455806401390066082at_nat @ A2 @ F @ X4 ) @ A2 )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5 @ A2 )
& ( ( the_in6455806401390066082at_nat @ A2 @ F @ X4 )
= Y5 ) )
=> ( Y5
= ( the_in6455806401390066082at_nat @ A2 @ F @ X4 ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_1002_bij__betw__the__inv__into,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ( bij_be1059735840858801910at_nat @ ( the_in5568309565101652403at_nat @ A2 @ F ) @ B2 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_1003_bij__betw__the__inv__into,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( bij_be8282881169987224566at_nat @ ( the_in7568536272828005555at_nat @ A2 @ F ) @ B2 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_1004_bij__betw__the__inv__into,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( bij_betw_nat_nat_nat @ ( the_in3844390324871770692at_nat @ A2 @ F ) @ B2 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_1005_bij__betw__the__inv__into,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( bij_betw_nat_nat @ ( the_inv_into_nat_nat @ A2 @ F ) @ B2 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_1006_bij__betw__the__inv__into,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( bij_betw_nat_nat_nat2 @ ( the_in5300466440149791684at_nat @ A2 @ F ) @ B2 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_1007_f__the__inv__into__f__bij__betw,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat,X4: nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ( F @ ( the_in7568536272828005555at_nat @ A2 @ F @ X4 ) )
= X4 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1008_f__the__inv__into__f__bij__betw,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat,X4: nat > nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( member_nat_nat @ X4 @ B2 ) )
=> ( ( F @ ( the_in3844390324871770692at_nat @ A2 @ F @ X4 ) )
= X4 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1009_f__the__inv__into__f__bij__betw,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,X4: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A2 @ F @ X4 ) )
= X4 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1010_f__the__inv__into__f__bij__betw,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,X4: nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ( F @ ( the_in5300466440149791684at_nat @ A2 @ F @ X4 ) )
= X4 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1011_layered__eq__classes,axiom,
! [S2: ( nat > nat ) > nat > nat,K2: nat,N: nat,T: nat,R: nat,Chi2: ( nat > nat ) > nat] :
( ( hales_4261547300027266985ce_nat @ S2 @ K2 @ N @ T @ R @ Chi2 )
=> ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K2 ) )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ ( hales_classes @ K2 @ T @ X2 ) )
=> ! [Xb: nat > nat] :
( ( member_nat_nat @ Xb @ ( hales_classes @ K2 @ T @ X2 ) )
=> ( ( Chi2 @ ( S2 @ Xa ) )
= ( Chi2 @ ( S2 @ Xb ) ) ) ) ) ) ) ).
% layered_eq_classes
thf(fact_1012_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_1013_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_1014_inv__into__cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F3: nat > nat] : ( F3 @ zero_zero_nat )
@ S
@ zero_zero_nat )
= S ) ) ).
% inv_into_cube_props(2)
thf(fact_1015_inv__into__cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F3: nat > nat] : ( F3 @ zero_zero_nat )
@ S )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% inv_into_cube_props(1)
thf(fact_1016_some__inv__into,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F3: nat > nat] : ( F3 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into
thf(fact_1017_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1018_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1019_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_1020_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_1021_order__refl,axiom,
! [X4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_1022_order__refl,axiom,
! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).
% order_refl
thf(fact_1023_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B2: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1024_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ ( collect_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1025_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1026_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1027_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1028_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( member_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( collect_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1029_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat3] :
( ! [X: nat] :
( ( P @ X )
=> ( member952132173341509300at_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ ( image_6393715451659844596at_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1030_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_3101123049818244468at_nat @ F @ ( collect_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1031_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( member_nat_nat_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ ( collect_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1032_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > ( nat > nat ) > ( nat > nat ) > nat,B2: set_na6626867396258451522at_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member4402528950554000163at_nat @ ( F @ X ) @ B2 ) )
=> ( ord_le973658574027395234at_nat @ ( image_3941236537129881699at_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1033_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1034_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1035_le__cases3,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1036_le__cases3,axiom,
! [X4: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X4 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X4 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1037_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1038_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat_nat,Z: set_nat_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
& ( ord_le9059583361652607317at_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1039_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
& ( ord_less_eq_int @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1040_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1041_ord__eq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1042_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1043_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1044_ord__le__eq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1045_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1046_order__antisym,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_1047_order__antisym,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_1048_order__antisym,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_1049_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_1050_order_Otrans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_1051_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_1052_order__trans,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_1053_order__trans,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_1054_order__trans,axiom,
! [X4: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_1055_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: nat,B5: nat] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1056_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A6: int,B5: int] :
( ( ord_less_eq_int @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: int,B5: int] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1057_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1058_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat_nat,Z: set_nat_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1059_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1060_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1061_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1062_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1063_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1064_dual__order_Otrans,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1065_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1066_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1067_antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1068_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1069_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1070_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat_nat,Z: set_nat_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1071_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1072_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1073_order__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1074_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1075_order__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1076_order__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1077_order__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1078_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1079_order__subst1,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1080_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1081_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1082_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1083_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1084_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1085_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1086_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1087_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1088_order__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1089_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1090_order__eq__refl,axiom,
! [X4: nat,Y2: nat] :
( ( X4 = Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_1091_order__eq__refl,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( X4 = Y2 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_1092_order__eq__refl,axiom,
! [X4: int,Y2: int] :
( ( X4 = Y2 )
=> ( ord_less_eq_int @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_1093_linorder__linear,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_1094_linorder__linear,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_1095_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1096_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1097_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1098_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1099_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1100_ord__eq__le__subst,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1101_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1102_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1103_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1104_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1105_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1106_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1107_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1108_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1109_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1110_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1111_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1112_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1113_linorder__le__cases,axiom,
! [X4: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_1114_linorder__le__cases,axiom,
! [X4: int,Y2: int] :
( ~ ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_1115_order__antisym__conv,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_1116_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X4 )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_1117_order__antisym__conv,axiom,
! [Y2: int,X4: int] :
( ( ord_less_eq_int @ Y2 @ X4 )
=> ( ( ord_less_eq_int @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_1118_lt__ex,axiom,
! [X4: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X4 ) ).
% lt_ex
thf(fact_1119_gt__ex,axiom,
! [X4: nat] :
? [X_12: nat] : ( ord_less_nat @ X4 @ X_12 ) ).
% gt_ex
thf(fact_1120_gt__ex,axiom,
! [X4: int] :
? [X_12: int] : ( ord_less_int @ X4 @ X_12 ) ).
% gt_ex
thf(fact_1121_less__imp__neq,axiom,
! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( X4 != Y2 ) ) ).
% less_imp_neq
thf(fact_1122_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1123_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B5: nat] :
( ( P @ A6 @ B5 )
= ( P @ B5 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B5: nat] :
( ( P @ A6 @ B5 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1124_cube1__alt__def,axiom,
! [N: nat] :
( ( hales_cube @ N @ one_one_nat )
= ( insert_nat_nat
@ ( restrict_nat_nat
@ ^ [X3: nat] : zero_zero_nat
@ ( set_ord_lessThan_nat @ N ) )
@ bot_bot_set_nat_nat ) ) ).
% cube1_alt_def
thf(fact_1125_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_1126_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_1127_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1128_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1129_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1130_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1131_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1132_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_1133_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_1134_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1135_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1136_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1137_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1138_zdiff__int__split,axiom,
! [P: int > $o,X4: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X4 @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1139_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1140_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1141_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1142_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1143_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M5: nat,N4: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1144_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1145_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1146_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1147_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1148_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1149_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1150_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1151_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1152_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1153_bij__betw__Suc,axiom,
! [M6: set_nat,N8: set_nat] :
( ( bij_betw_nat_nat @ suc @ M6 @ N8 )
= ( ( image_nat_nat @ suc @ M6 )
= N8 ) ) ).
% bij_betw_Suc
thf(fact_1154_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1155_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1156_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1157_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1158_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1159_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1160_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1161_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1162_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1163_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1164_nat__power__eq__Suc__0__iff,axiom,
! [X4: nat,M: nat] :
( ( ( power_power_nat @ X4 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X4
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1165_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1166_card__atMost,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U2 ) )
= ( suc @ U2 ) ) ).
% card_atMost
thf(fact_1167_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1168_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1169_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I4 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I4 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1170_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1171_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1172_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost_nat @ ( suc @ K2 ) )
= ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_1173_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_1174_Suc__inject,axiom,
! [X4: nat,Y2: nat] :
( ( ( suc @ X4 )
= ( suc @ Y2 ) )
=> ( X4 = Y2 ) ) ).
% Suc_inject
thf(fact_1175_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1176_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( set_ord_atMost_nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_1177_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1178_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1179_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1180_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1181_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1182_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1183_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1184_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1185_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1186_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K4: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1187_less__add__Suc2,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).
% less_add_Suc2
thf(fact_1188_less__add__Suc1,axiom,
! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1189_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1190_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1191_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1192_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1193_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1194_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1195_inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I4 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I4 ) ) ) ) ).
% inc_induct
thf(fact_1196_dec__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P @ I4 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I4 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1197_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1198_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1199_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1200_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1201_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1202_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1203_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1204_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1205_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1206_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1207_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1208_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1209_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1210_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1211_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1212_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X: nat,Y4: nat] :
( ( P @ X @ Y4 )
=> ( P @ ( suc @ X ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1213_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1214_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1215_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1216_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1217_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1218_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1219_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1220_strict__inc__induct,axiom,
! [I4: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_1221_less__Suc__induct,axiom,
! [I4: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I4 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1222_less__trans__Suc,axiom,
! [I4: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I4 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1223_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1224_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1225_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1226_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1227_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1228_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1229_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1230_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1231_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1232_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1233_Suc__lessE,axiom,
! [I4: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1234_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1235_Nat_OlessE,axiom,
! [I4: nat,K2: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( ( K2
!= ( suc @ I4 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1236_nat__arith_Osuc1,axiom,
! [A2: nat,K2: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1237_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1238_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1239_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I4: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I4 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1240_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1241_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1242_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1243_Suc__le__D,axiom,
! [N: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
=> ? [M5: nat] :
( M8
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1244_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1245_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1246_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1247_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1248_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1249_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R4 @ X @ X )
=> ( ! [X: nat,Y4: nat,Z4: nat] :
( ( R4 @ X @ Y4 )
=> ( ( R4 @ Y4 @ Z4 )
=> ( R4 @ X @ Z4 ) ) )
=> ( ! [N4: nat] : ( R4 @ N4 @ ( suc @ N4 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1250_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1251_diff__Suc__less,axiom,
! [N: nat,I4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1252_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1253_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_1254_nat__one__le__power,axiom,
! [I4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I4 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I4 @ N ) ) ) ).
% nat_one_le_power
thf(fact_1255_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1256_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1257_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_1258_card__less,axiom,
! [M6: set_nat,I4: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M6 )
& ( ord_less_nat @ K4 @ ( suc @ I4 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1259_card__less__Suc,axiom,
! [M6: set_nat,I4: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M6 )
& ( ord_less_nat @ K4 @ I4 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M6 )
& ( ord_less_nat @ K4 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1260_card__less__Suc2,axiom,
! [M6: set_nat,I4: nat] :
( ~ ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M6 )
& ( ord_less_nat @ K4 @ I4 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M6 )
& ( ord_less_nat @ K4 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1261_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1262_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1263_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1264_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
% Helper facts (9)
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,
! [X4: nat,Y2: nat] :
( ( if_nat @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y2: nat] :
( ( if_nat @ $true @ X4 @ Y2 )
= X4 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_T,axiom,
! [P: ( ( nat > nat ) > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat @ P ) )
= ( ? [X6: ( nat > nat ) > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( nat > nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat2 @ P ) )
= ( ? [X6: nat > nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( P @ ( fChoic52552927678224201at_nat @ P ) )
= ( ? [X6: ( nat > nat ) > nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_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_T,axiom,
! [P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( P @ ( fChoic2516396905127217208at_nat @ P ) )
= ( ? [X6: ( nat > nat ) > ( nat > nat ) > nat] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member952132173341509300at_nat @ ( t @ x )
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ k @ ( plus_plus_nat @ t2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( hales_cube @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t2 @ one_one_nat ) ) ) ) ).
%------------------------------------------------------------------------------