TPTP Problem File: SLH0211^1.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_01251_054635__5861406_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1476 ( 537 unt; 202 typ; 0 def)
% Number of atoms : 3675 (1116 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11333 ( 255 ~; 51 |; 255 &;9167 @)
% ( 0 <=>;1605 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 2179 (2179 >; 0 *; 0 +; 0 <<)
% Number of symbols : 187 ( 184 usr; 18 con; 0-6 aty)
% Number of variables : 3787 ( 424 ^;3268 !; 95 ?;3787 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:23.274
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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_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_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 (184)
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_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_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_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_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6840239867990089257at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat ) > set_na8843485148432118594at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat_nat5 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na7122919648973241129at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat_nat ) > set_na8778986904112484418at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat2: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat ) > set_nat_nat_nat_nat4 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_na7569501297962130601at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat2 ) > set_na6626867396258451522at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na8678869062391380393at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat ) > set_na3764207730537033026at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat3: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
piE_nat_nat_int: set_nat_nat > ( ( nat > nat ) > set_int ) > set_nat_nat_int ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat4: set_nat > ( nat > set_nat_nat_nat2 ) > set_nat_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_nat_nat_nat_nat5: set_nat > ( nat > set_nat_nat_nat ) > set_nat_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_Orestrict_001_062_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_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_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_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_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_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_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_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_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_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_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le996020443555834177_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5384859702510996545_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_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_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_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_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_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_Mt__Nat__Onat_J,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1262493855416953332at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_7809927846809980933at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1545173636400105204at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_913610194320715013at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_1991755285388994676at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3101123049818244468at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_nat_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_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_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_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_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_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 ).
thf(sy_c_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
fChoice_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > ( nat > nat ) > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fChoice_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > nat > nat > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member4489290058226556451at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_na8843485148432118594at_nat > $o ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member2991261302380110260at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat_nat5 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
member5318315686745620148at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat_nat4 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member4402528950554000163at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member1679187572556404771at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_na3764207730537033026at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member952132173341509300at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
member_nat_nat_int: ( ( nat > nat ) > int ) > set_nat_nat_int > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member2740455936716430260at_nat: ( nat > ( nat > nat ) > nat ) > set_nat_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member17114558718834868at_nat: ( nat > nat > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_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__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,
s: nat ).
thf(sy_v_t,type,
t: nat ).
thf(sy_v_x____,type,
x: nat > nat ).
% Relevant facts (1265)
thf(fact_0_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_1_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_2_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_3_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_4_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_5_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_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_14_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_15_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_16_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_17_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_18_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_19_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_20_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_21_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_22_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_23_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_24_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_25_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_26_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_27_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_28_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_29_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_30_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_31_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_43_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_44_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_45_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_46_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_47_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_48_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_49_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_50_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_51_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_52_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_53_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_54_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_55_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_56_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_57_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_58_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( collec2410089373097230945at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_65_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_66_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_67_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_68_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_69_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_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_79_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_80_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_81_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_82_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_83_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_84_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_85_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_86_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_87_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_88_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_89_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_90_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_91_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_92_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_93_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_94_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_95_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_96_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_97_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_98_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_99_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_100_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_101_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_102_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_103_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_104_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_105_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_106_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_107_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_108_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_109_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_110_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_111_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_112_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_113_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_114_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_115_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_116_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_117_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_118_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_119_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_120_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_121_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_122_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_123_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_124_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_125_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_126_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_127_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_128_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_129_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_130_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_131_cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X3 @ zero_zero_nat )
= S ) ) ) ).
% cube_props(1)
thf(fact_132_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_133_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_134_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_135_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_136_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_137_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_138_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_139_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_140_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_141_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: int,Chi: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S2 @ one_one_nat @ N @ T @ R @ Chi )
=> ! [S3: nat] :
( ( ord_less_nat @ S3 @ T )
=> ! [L2: nat] :
( ( ord_less_nat @ L2 @ T )
=> ( ( ( Chi
@ ( 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 )
= S3 ) ) ) ) )
= ( Chi
@ ( 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 )
= L2 ) ) ) ) ) )
& ( ord_less_int
@ ( Chi
@ ( 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 )
= S3 ) ) ) ) )
@ R ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_142_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: nat,Chi: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S2 @ one_one_nat @ N @ T @ R @ Chi )
=> ! [S3: nat] :
( ( ord_less_nat @ S3 @ T )
=> ! [L2: nat] :
( ( ord_less_nat @ L2 @ T )
=> ( ( ( Chi
@ ( 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 )
= S3 ) ) ) ) )
= ( Chi
@ ( 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 )
= L2 ) ) ) ) ) )
& ( ord_less_nat
@ ( Chi
@ ( 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 )
= S3 ) ) ) ) )
@ R ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_143_dim1__layered__subspace__as__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: int,Chi: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S2 @ one_one_nat @ N @ T @ R @ Chi )
=> ? [C1: int,C22: int] :
( ( ord_less_int @ C1 @ R )
& ( ord_less_int @ C22 @ R )
& ! [S3: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( ( Chi
@ ( 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 )
= S3 ) ) ) ) )
= C1 ) )
& ( ( Chi
@ ( 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_144_dim1__layered__subspace__as__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R: nat,Chi: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S2 @ one_one_nat @ N @ T @ R @ Chi )
=> ? [C1: nat,C22: nat] :
( ( ord_less_nat @ C1 @ R )
& ( ord_less_nat @ C22 @ R )
& ! [S3: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( ( Chi
@ ( 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 )
= S3 ) ) ) ) )
= C1 ) )
& ( ( Chi
@ ( 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_145_calculation_I2_J,axiom,
( ( l_line @ zero_zero_nat )
= ( l
@ ( 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 )
= zero_zero_nat ) ) ) ) ) ).
% calculation(2)
thf(fact_146_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_147_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_148_some__sym__eq__trivial,axiom,
! [X: nat > nat] :
( ( fChoice_nat_nat
@ ( ^ [Y2: nat > nat,Z: nat > nat] : ( Y2 = Z )
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_149_some__eq__trivial,axiom,
! [X: nat > nat] :
( ( fChoice_nat_nat
@ ^ [Y3: nat > nat] : ( Y3 = X ) )
= X ) ).
% some_eq_trivial
thf(fact_150_some__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ( P @ A )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( X3 = A ) )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_151_a,axiom,
member_nat_nat @ x @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ).
% a
thf(fact_152_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_153_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_154_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_155_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_156_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_157_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_158_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_159_someI,axiom,
! [P: ( nat > nat ) > $o,X: nat > nat] :
( ( P @ X )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI
thf(fact_160_Eps__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice_nat_nat @ P )
= ( fChoice_nat_nat @ Q ) ) ) ).
% Eps_cong
thf(fact_161_tfl__some,axiom,
! [P3: ( nat > nat ) > $o,X4: nat > nat] :
( ( P3 @ X4 )
=> ( P3 @ ( fChoice_nat_nat @ P3 ) ) ) ).
% tfl_some
thf(fact_162_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_163_someI2,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat,Q: ( nat > nat ) > $o] :
( ( P @ A )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2
thf(fact_164_someI__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ? [X_1: nat > nat] : ( P @ X_1 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI_ex
thf(fact_165_someI2__ex,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X_1: nat > nat] : ( P @ X_1 )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_166_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat] :
( ( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_167_someI2__bex,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_168_someI2__bex,axiom,
! [A2: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ? [X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member4402528950554000163at_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoic2516396905127217208at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_169_someI2__bex,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_170_someI2__bex,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X3: nat > nat] :
( ( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_171_some__eq__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X5: nat > nat] : ( P @ X5 ) ) ) ).
% some_eq_ex
thf(fact_172_some1__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ? [X4: nat > nat] :
( ( P @ X4 )
& ! [Y4: nat > nat] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_173_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_174_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_175_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_176_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_177_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_178_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 @ t @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% L_line_def
thf(fact_179_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_180_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_181_s__def,axiom,
( s
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t @ one_one_nat ) @ m2 ) ) ) ).
% s_def
thf(fact_182_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 )
@ ^ [F: nat > nat] : ( F @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into_2
thf(fact_183_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 )
@ ^ [F: nat > nat] : ( F @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into
thf(fact_184__092_060open_0620_A_060_As_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ s ).
% \<open>0 < s\<close>
thf(fact_185_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_186_join__cubes,axiom,
! [F2: nat > nat,N: nat,T: nat,G: nat > nat,M: nat] :
( ( member_nat_nat @ F2 @ ( 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 @ F2 @ G @ N @ M ) @ ( hales_cube @ ( plus_plus_nat @ N @ M ) @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_187_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_188_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_189_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_190_lessThan__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_191_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_192_lessThan__iff,axiom,
! [I: ( nat > nat ) > ( nat > nat ) > nat,K: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I @ ( set_or6177432841829679145at_nat @ K ) )
= ( ord_le7877100967975825120at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_193_lessThan__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I @ K ) ) ).
% lessThan_iff
thf(fact_194_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_195_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_196_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_197_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_198_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_199_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_200_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_201_cube__props_I3_J,axiom,
! [S: nat,T: nat,S2: ( nat > nat ) > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat
@ ^ [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
@ ^ [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_202_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_203_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 )
@ ^ [F: nat > nat] : ( F @ zero_zero_nat )
@ S )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% inv_into_cube_props(1)
thf(fact_204_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 )
@ ^ [F: nat > nat] : ( F @ zero_zero_nat )
@ S
@ zero_zero_nat )
= S ) ) ).
% inv_into_cube_props(2)
thf(fact_205_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_206_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_207_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_208_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_209_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_210_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_211_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_212_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_213_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_214_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_215_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_216_L__line__base__prop,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) )
=> ( member_nat_nat @ ( l_line @ X4 ) @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% L_line_base_prop
thf(fact_217__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 @ t @ one_one_nat ) ) ) )
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t @ one_one_nat ) @ m2 ) ) ) ).
% \<open>r ^ card (cube m (t + 1)) = r ^ (t + 1) ^ m\<close>
thf(fact_218_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_219_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_220_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_221_L__prop,axiom,
hales_4261547300027266985ce_nat @ l @ one_one_nat @ n2 @ t @ s @ chi_L_s ).
% L_prop
thf(fact_222_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_223__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_224_cube__restrict,axiom,
! [J: nat,N: nat,Y: nat > nat,T: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( member_nat_nat @ Y @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T ) ) ) ) ).
% cube_restrict
thf(fact_225_split__cube_I2_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y3: nat] : ( X @ ( plus_plus_nat @ Y3 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_226_split__cube_I1_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_227_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_228_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_229_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_230_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_231_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_232_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_233_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_234_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_235_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_236_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_237_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_238_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_239_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_240_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_241_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_242_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_243_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_244_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_245_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_246__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,
~ ! [L3: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ L3 @ one_one_nat @ n2 @ t @ s @ 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_247_A,axiom,
! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) ) )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) )
=> ( member_nat @ ( chi @ ( hales_join_nat @ X4 @ Xa @ n2 @ m2 ) ) @ ( set_ord_lessThan_nat @ r ) ) ) ) ).
% A
thf(fact_248__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 @ t @ one_one_nat ) ) ) )
= ( power_power_nat @ r @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ) ).
% \<open>card {..<r} ^ card (cube m (t + 1)) = r ^ card (cube m (t + 1))\<close>
thf(fact_249_calculation_I1_J,axiom,
( ( chi_S @ x )
= ( chi @ ( hales_join_nat @ ( l_line @ zero_zero_nat ) @ x @ n2 @ m2 ) ) ) ).
% calculation(1)
thf(fact_250__092_060chi_062S__def,axiom,
( chi_S
= ( restrict_nat_nat_nat
@ ^ [Y3: nat > nat] : ( chi @ ( hales_join_nat @ ( l_line @ zero_zero_nat ) @ Y3 @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% \<chi>S_def
thf(fact_251_line__subspace__s,axiom,
! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] :
( ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ n2 @ t @ s @ Chi )
& ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S5
@ ( 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 )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) )
@ n2
@ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% line_subspace_s
thf(fact_252__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 @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s ) ) ) ).
% \<open>\<chi>L_s \<in> cube n (t + 1) \<rightarrow>\<^sub>E {..<s}\<close>
thf(fact_253_m__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ m2 )
& ! [M3: nat] :
( ( ord_less_eq_nat @ m2 @ M3 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ M3 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ k @ M3 @ t @ r @ Chi2 ) ) ) ) ).
% m_props
thf(fact_254_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_255_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_256_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_257__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_258__092_060chi_062__prop,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ m @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<chi>_prop
thf(fact_259_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_260_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_261_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_262_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_263_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_264_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_265_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_266_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_267_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_268_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_269_n_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [N4: nat] :
( ( ord_less_eq_nat @ n @ N4 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ N4 @ t @ s @ Chi2 ) ) ) ) ).
% n'_props
thf(fact_270_card__lessThan,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U2 ) )
= U2 ) ).
% card_lessThan
thf(fact_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_assms_I4_J,axiom,
! [K3: nat,R: nat] :
( ( ord_less_eq_nat @ K3 @ k )
=> ( hales_lhj @ R @ t @ K3 ) ) ).
% assms(4)
thf(fact_282__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 )
& ! [N4: nat] :
( ( ord_less_eq_nat @ N5 @ N4 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ N4 @ t @ s @ Chi2 ) ) ) ) ).
% \<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_283__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,
~ ! [M4: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ! [M3: nat] :
( ( ord_less_eq_nat @ M4 @ M3 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ M3 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ k @ M3 @ t @ r @ Chi2 ) ) ) ) ).
% \<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_284_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_285_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_286_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_287_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_288_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_289_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_290__092_060chi_062L__s__def,axiom,
( chi_L_s
= ( restrict_nat_nat_nat
@ ^ [X2: nat > nat] : ( phi @ ( chi_L @ X2 ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% \<chi>L_s_def
thf(fact_291_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_292_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_293_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B3: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_294_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B3: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_295_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B3: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_296_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B3: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_297_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B3 )
=> ? [X3: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X3
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_298_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B3: set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B3 )
=> ? [X3: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X3
@ ( piE_nat_nat_nat_nat5 @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_299_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B3: set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X3
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I2: ( nat > nat ) > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_300_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B3: set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X3: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X3
@ ( piE_nat_nat_nat_nat2 @ A2
@ ^ [I2: nat > nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_301_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B3: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B3 )
=> ? [X3: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X3
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_302_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat > nat,B3: set_nat_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member4489290058226556451at_nat @ X3
@ ( piE_na6840239867990089257at_nat @ A2
@ ^ [I2: ( nat > nat ) > nat] : B3 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_303_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_304_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_305_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_306_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_307_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_308_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_309_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_310_power__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_311_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_312_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_313_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_314_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_315_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_316_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_317_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_318_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_319_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_320_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_321_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_322_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_323_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_324_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_325_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_326_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_327_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_328_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_329_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_330_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_331_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_332_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_333_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_334_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_335_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_336_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_337_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_338_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N6: nat] :
( ( ord_less_eq_nat @ M6 @ N6 )
& ( M6 != N6 ) ) ) ) ).
% nat_less_le
thf(fact_339_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_340_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N6: nat] :
( ( ord_less_nat @ M6 @ N6 )
| ( M6 = N6 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_341_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_342_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_343_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F2 @ I3 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_344_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N6: nat] :
? [K4: nat] :
( N6
= ( plus_plus_nat @ M6 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_345_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_346_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_347_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_348_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_349_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_350_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_351_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_352_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_353_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_354_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_355_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_356_power__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_357_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_358_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_359_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_360_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_361_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_362_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_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_374_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_375_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_376_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_377_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_378_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_379_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_380_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_381_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_382_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_383_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_384_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_396_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F2 @ M4 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S5 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_dim0__layered__subspace__ex,axiom,
! [Chi: ( nat > nat ) > nat,N: nat,T: nat,R: nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ zero_zero_nat @ N @ T @ R @ Chi ) ) ).
% dim0_layered_subspace_ex
thf(fact_423_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_424_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_425__092_060chi_062L__def,axiom,
( chi_L
= ( restri6011711336257459485at_nat
@ ^ [X2: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y3: nat > nat] : ( chi @ ( hales_join_nat @ X2 @ Y3 @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% \<chi>L_def
thf(fact_426_lhj__def,axiom,
( hales_lhj
= ( ^ [R2: nat,T2: nat,K4: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S6: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S6 @ K4 @ N8 @ T2 @ R2 @ Chi3 ) ) ) ) ) ) ).
% lhj_def
thf(fact_427__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 @ t @ one_one_nat ) )
@ ^ [I2: 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 @ t @ one_one_nat ) ) ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = card {..<r} ^ card (cube m (t + 1))\<close>
thf(fact_428__092_060phi_062__prop,axiom,
( bij_be1059735840858801910at_nat @ phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s ) ) ).
% \<phi>_prop
thf(fact_429_s__coloured,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= s ) ).
% s_coloured
thf(fact_430_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_431__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 @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t @ one_one_nat ) @ m2 ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = r ^ (t + 1) ^ m\<close>
thf(fact_432__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 @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s ) ) ).
% \<open>\<And>thesis. (\<And>\<phi>. bij_betw \<phi> (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) {..<s} \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_433_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_434_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_435_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_436_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_437_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_438_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_439_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_440_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_441_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_451_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_452_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_453_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_454_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_455_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_456_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_457_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_458_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_459_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_460_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_461_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_462_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_463_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_464_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_465_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_466_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_467_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_468__092_060chi_062L__prop,axiom,
( member4402528950554000163at_nat @ chi_L
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] :
( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [J3: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ) ).
% \<chi>L_prop
thf(fact_469_bij__betw__same__card,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F2 @ A2 @ B3 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_470_bij__betw__same__card,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat] :
( ( bij_betw_nat_nat_nat @ F2 @ A2 @ B3 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_471_bij__betw__same__card,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat_nat_nat2] :
( ( bij_be5311014265664741861at_nat @ F2 @ A2 @ B3 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite1794908990118856198at_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_472_bij__betw__same__card,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B3 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_473_bij__betw__same__card,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ B3 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_474_bij__betw__same__card,axiom,
! [F2: nat > ( nat > nat ) > nat,A2: set_nat,B3: set_nat_nat_nat2] :
( ( bij_be8282881169987224566at_nat @ F2 @ A2 @ B3 )
=> ( ( finite_card_nat @ A2 )
= ( finite1794908990118856198at_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_475_bij__betw__same__card,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,B3: set_nat_nat] :
( ( bij_be4581752835692700517at_nat @ F2 @ A2 @ B3 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_476_bij__betw__same__card,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat] :
( ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B3 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_477_bij__betw__same__card,axiom,
! [F2: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( bij_be3563731812766147924at_nat @ F2 @ A2 @ B3 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite1794908990118856198at_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_478_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_479_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_480_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_481_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_482_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_483_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_484_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_485_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_486_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_487_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_488_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_489_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_490_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_491_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_492_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_493_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_494_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_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_503_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_504_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_505_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_506_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_507_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_508_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_509_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_510_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_511_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_512_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_513_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_514_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_515_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_516_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_517_bij__unique__inv,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat,X: nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat @ X @ B3 )
=> ( ( member_nat @ ( the_inv_into_nat_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_inv_into_nat_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_inv_into_nat_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_518_bij__unique__inv,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat,X: nat > nat] :
( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B3 )
=> ( ( member_nat_nat @ X @ B3 )
=> ( ( member_nat @ ( the_in3844390324871770692at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_in3844390324871770692at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in3844390324871770692at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_519_bij__unique__inv,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat,X: nat] :
( ( bij_betw_nat_nat_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat @ X @ B3 )
=> ( ( member_nat_nat @ ( the_in5300466440149791684at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5 @ A2 )
& ( ( the_in5300466440149791684at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in5300466440149791684at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_520_bij__unique__inv,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat,X: nat > nat] :
( ( bij_be5678534868967705974at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat_nat @ X @ B3 )
=> ( ( member_nat_nat @ ( the_in2963963264082133811at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5 @ A2 )
& ( ( the_in2963963264082133811at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in2963963264082133811at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_521_bij__unique__inv,axiom,
! [F2: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,B3: set_nat,X: nat] :
( ( bij_be3386790225224311798at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat @ X @ B3 )
=> ( ( member_nat_nat_nat2 @ ( the_in672218620338739635at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5 @ A2 )
& ( ( the_in672218620338739635at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in672218620338739635at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_522_bij__unique__inv,axiom,
! [F2: nat > ( nat > nat ) > nat,A2: set_nat,B3: set_nat_nat_nat2,X: ( nat > nat ) > nat] :
( ( bij_be8282881169987224566at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ X @ B3 )
=> ( ( member_nat @ ( the_in5568309565101652403at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_in5568309565101652403at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in5568309565101652403at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_523_bij__unique__inv,axiom,
! [F2: nat > nat > nat > nat,A2: set_nat,B3: set_nat_nat_nat,X: nat > nat > nat] :
( ( bij_be168876897561698550at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ X @ B3 )
=> ( ( member_nat @ ( the_in6677677329530902195at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat] :
( ( ( member_nat @ Y5 @ A2 )
& ( ( the_in6677677329530902195at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in6677677329530902195at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_524_bij__unique__inv,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat,X: nat] :
( ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat @ X @ B3 )
=> ( ( member_nat_nat_nat @ ( the_in7568536272828005555at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5 @ A2 )
& ( ( the_in7568536272828005555at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in7568536272828005555at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_525_bij__unique__inv,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,B3: set_nat_nat,X: nat > nat] :
( ( bij_be4581752835692700517at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat_nat @ X @ B3 )
=> ( ( member_nat_nat_nat @ ( the_in6455806401390066082at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5 @ A2 )
& ( ( the_in6455806401390066082at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in6455806401390066082at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_526_bij__unique__inv,axiom,
! [F2: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat,X: nat > nat] :
( ( bij_be4864432616675852389at_nat @ F2 @ A2 @ B3 )
=> ( ( member_nat_nat @ X @ B3 )
=> ( ( member_nat_nat_nat2 @ ( the_in6738486182373217954at_nat @ A2 @ F2 @ X ) @ A2 )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5 @ A2 )
& ( ( the_in6738486182373217954at_nat @ A2 @ F2 @ X )
= Y5 ) )
=> ( Y5
= ( the_in6738486182373217954at_nat @ A2 @ F2 @ X ) ) ) ) ) ) ).
% bij_unique_inv
thf(fact_527_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_528_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_542_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_543_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_544_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_545_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_546_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_547_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_548_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_549_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_550_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_551_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_552_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_553_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_554_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_555_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_556_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_557_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_558_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_559_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_560_cube__def,axiom,
( hales_cube
= ( ^ [N6: nat,T2: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N6 )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) ) ) ).
% cube_def
thf(fact_561_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_562_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_563_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_564_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_565_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 )
=> ? [F3: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ F3 @ A2 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% ex_bij_betw_nat_finite_2
thf(fact_566_ex__bij__betw__nat__finite__2,axiom,
! [A2: set_nat,N: nat] :
( ( ( finite_card_nat @ A2 )
= N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [F3: nat > nat] : ( bij_betw_nat_nat @ F3 @ A2 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% ex_bij_betw_nat_finite_2
thf(fact_567_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 )
=> ? [F3: ( ( nat > nat ) > nat ) > nat] : ( bij_be1059735840858801910at_nat @ F3 @ A2 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% ex_bij_betw_nat_finite_2
thf(fact_568_cube__card,axiom,
! [N: nat,T: nat] :
( ( finite_card_nat_nat
@ ( piE_nat_nat @ ( set_ord_lessThan_nat @ N )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
= ( power_power_nat @ T @ N ) ) ).
% cube_card
thf(fact_569_is__line__def,axiom,
( hales_is_line
= ( ^ [L5: nat > nat > nat,N6: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L5
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I2: nat] : ( hales_cube @ N6 @ T2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ N6 )
=> ( ! [X2: nat] :
( ( ord_less_nat @ X2 @ T2 )
=> ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ T2 )
=> ( ( L5 @ X2 @ J3 )
= ( L5 @ Y3 @ J3 ) ) ) )
| ! [S4: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( ( L5 @ S4 @ J3 )
= S4 ) ) ) )
& ? [J3: nat] :
( ( ord_less_nat @ J3 @ N6 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( ( L5 @ S4 @ J3 )
= S4 ) ) ) ) ) ) ).
% is_line_def
thf(fact_570_PiE__restrict,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F2 @ ( piE_nat_nat_nat2 @ A2 @ B3 ) )
=> ( ( restrict_nat_nat_nat2 @ F2 @ A2 )
= F2 ) ) ).
% PiE_restrict
thf(fact_571_PiE__restrict,axiom,
! [F2: nat > nat,A2: set_nat,B3: nat > set_nat] :
( ( member_nat_nat @ F2 @ ( piE_nat_nat @ A2 @ B3 ) )
=> ( ( restrict_nat_nat @ F2 @ A2 )
= F2 ) ) ).
% PiE_restrict
thf(fact_572_PiE__restrict,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F2 @ ( piE_nat_nat_nat @ A2 @ B3 ) )
=> ( ( restrict_nat_nat_nat @ F2 @ A2 )
= F2 ) ) ).
% PiE_restrict
thf(fact_573_PiE__restrict,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat_nat2] :
( ( member4402528950554000163at_nat @ F2 @ ( piE_na7569501297962130601at_nat @ A2 @ B3 ) )
=> ( ( restri6011711336257459485at_nat @ F2 @ A2 )
= F2 ) ) ).
% PiE_restrict
thf(fact_574_PiE__restrict,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F2 @ ( piE_nat_nat_nat_nat3 @ A2 @ B3 ) )
=> ( ( restri4446420529079022766at_nat @ F2 @ A2 )
= F2 ) ) ).
% PiE_restrict
thf(fact_575_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_576_bij__betw__restrict__eq,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat] :
( ( bij_be1059735840858801910at_nat @ ( restri9050993537824894510at_nat @ F2 @ A2 ) @ A2 @ B3 )
= ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B3 ) ) ).
% bij_betw_restrict_eq
thf(fact_577_bij__betw__restrict__eq,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F2 @ A2 ) @ A2 @ B3 )
= ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B3 ) ) ).
% bij_betw_restrict_eq
thf(fact_578_bij__betw__restrict__eq,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ ( restrict_nat_nat @ F2 @ A2 ) @ A2 @ B3 )
= ( bij_betw_nat_nat @ F2 @ A2 @ B3 ) ) ).
% bij_betw_restrict_eq
thf(fact_579_bij__betw__restrict__eq,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat] :
( ( bij_betw_nat_nat_nat @ ( restrict_nat_nat_nat @ F2 @ A2 ) @ A2 @ B3 )
= ( bij_betw_nat_nat_nat @ F2 @ A2 @ B3 ) ) ).
% bij_betw_restrict_eq
thf(fact_580_bij__betw__restrict__eq,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat_nat_nat2] :
( ( bij_be5311014265664741861at_nat @ ( restri6011711336257459485at_nat @ F2 @ A2 ) @ A2 @ B3 )
= ( bij_be5311014265664741861at_nat @ F2 @ A2 @ B3 ) ) ).
% bij_betw_restrict_eq
thf(fact_581_bij__betw__restrict__eq,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ ( restri4446420529079022766at_nat @ F2 @ A2 ) @ A2 @ B3 )
= ( bij_be5678534868967705974at_nat @ F2 @ A2 @ B3 ) ) ).
% bij_betw_restrict_eq
thf(fact_582_psubsetI,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_583_subset__antisym,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_584_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_585_subsetI,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ X3 @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_586_subsetI,axiom,
! [A2: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
=> ( member4402528950554000163at_nat @ X3 @ B3 ) )
=> ( ord_le973658574027395234at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_587_subsetI,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ X3 @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_588_subsetI,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ X3 @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_589_Diff__mono,axiom,
! [A2: set_nat_nat,C4: set_nat_nat,D3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C4 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B3 ) @ ( minus_8121590178497047118at_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_590_Diff__subset,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_591_double__diff,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ( minus_8121590178497047118at_nat @ B3 @ ( minus_8121590178497047118at_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_592_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ? [B4: nat > nat] : ( member_nat_nat @ B4 @ ( minus_8121590178497047118at_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_593_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_594_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( ord_le371403230139555384at_nat @ A2 @ B3 )
=> ? [B4: ( nat > nat ) > nat] : ( member_nat_nat_nat @ B4 @ ( minus_1221035652888719293at_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_595_psubset__imp__ex__mem,axiom,
! [A2: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( ord_le2785809691299232406at_nat @ A2 @ B3 )
=> ? [B4: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ B4 @ ( minus_5225787954611647771at_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_596_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B3 )
=> ? [B4: nat > nat > nat] : ( member_nat_nat_nat2 @ B4 @ ( minus_7721066311745899709at_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_597_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_598_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_599_set__eq__subset,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B5 )
& ( ord_le9059583361652607317at_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_600_subset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_601_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_602_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_603_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_604_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_605_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B5: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A5 )
=> ( member_nat_nat_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_606_subset__iff,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A5: set_na6626867396258451522at_nat,B5: set_na6626867396258451522at_nat] :
! [T2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ T2 @ A5 )
=> ( member4402528950554000163at_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_607_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B5: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A5 )
=> ( member_nat_nat_nat2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_608_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A5 )
=> ( member_nat_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_609_equalityD2,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_610_equalityD1,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_611_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_612_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B5: set_nat_nat_nat2] :
! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A5 )
=> ( member_nat_nat_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_613_subset__eq,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A5: set_na6626867396258451522at_nat,B5: set_na6626867396258451522at_nat] :
! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A5 )
=> ( member4402528950554000163at_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_614_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B5: set_nat_nat_nat] :
! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A5 )
=> ( member_nat_nat_nat2 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_615_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A5 )
=> ( member_nat_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_616_equalityE,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ~ ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_617_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_618_subsetD,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_619_subsetD,axiom,
! [A2: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A2 @ B3 )
=> ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_620_subsetD,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_621_subsetD,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_622_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_623_in__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,X: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_624_in__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A2 @ B3 )
=> ( ( member4402528950554000163at_nat @ X @ A2 )
=> ( member4402528950554000163at_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_625_in__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,X: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B3 ) ) ) ).
% in_mono
thf(fact_626_in__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_627_psubsetD,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_628_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_629_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_630_psubsetD,axiom,
! [A2: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le2785809691299232406at_nat @ A2 @ B3 )
=> ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_631_psubsetD,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_632_PiE__ext,axiom,
! [X: ( nat > nat ) > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat,Y: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ( member_nat_nat_nat @ Y @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_633_PiE__ext,axiom,
! [X: ( nat > nat ) > ( nat > nat ) > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat_nat_nat2,Y: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ ( piE_na7569501297962130601at_nat @ K @ S ) )
=> ( ( member4402528950554000163at_nat @ Y @ ( piE_na7569501297962130601at_nat @ K @ S ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_634_PiE__ext,axiom,
! [X: nat > nat,K: set_nat,S: nat > set_nat,Y: nat > nat] :
( ( member_nat_nat @ X @ ( piE_nat_nat @ K @ S ) )
=> ( ( member_nat_nat @ Y @ ( piE_nat_nat @ K @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_635_PiE__ext,axiom,
! [X: nat > nat > nat,K: set_nat,S: nat > set_nat_nat,Y: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ( member_nat_nat_nat2 @ Y @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_636_PiE__mem,axiom,
! [F2: nat > nat,S2: set_nat,T3: nat > set_nat,X: nat] :
( ( member_nat_nat @ F2 @ ( piE_nat_nat @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_637_PiE__mem,axiom,
! [F2: ( nat > nat ) > nat,S2: set_nat_nat,T3: ( nat > nat ) > set_nat,X: nat > nat] :
( ( member_nat_nat_nat @ F2 @ ( piE_nat_nat_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat @ X @ S2 )
=> ( member_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_638_PiE__mem,axiom,
! [F2: nat > nat > nat,S2: set_nat,T3: nat > set_nat_nat,X: nat] :
( ( member_nat_nat_nat2 @ F2 @ ( piE_nat_nat_nat2 @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_639_PiE__mem,axiom,
! [F2: ( nat > nat ) > nat > nat,S2: set_nat_nat,T3: ( nat > nat ) > set_nat_nat,X: nat > nat] :
( ( member952132173341509300at_nat @ F2 @ ( piE_nat_nat_nat_nat3 @ S2 @ T3 ) )
=> ( ( member_nat_nat @ X @ S2 )
=> ( member_nat_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_640_PiE__mem,axiom,
! [F2: nat > ( nat > nat ) > nat,S2: set_nat,T3: nat > set_nat_nat_nat2,X: nat] :
( ( member2740455936716430260at_nat @ F2 @ ( piE_nat_nat_nat_nat4 @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat_nat_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_641_PiE__mem,axiom,
! [F2: nat > nat > nat > nat,S2: set_nat,T3: nat > set_nat_nat_nat,X: nat] :
( ( member17114558718834868at_nat @ F2 @ ( piE_nat_nat_nat_nat5 @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat_nat_nat2 @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_642_PiE__mem,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,S2: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat,X: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F2 @ ( piE_nat_nat_nat_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat_nat @ X @ S2 )
=> ( member_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_643_PiE__mem,axiom,
! [F2: ( nat > nat > nat ) > nat,S2: set_nat_nat_nat,T3: ( nat > nat > nat ) > set_nat,X: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F2 @ ( piE_nat_nat_nat_nat2 @ S2 @ T3 ) )
=> ( ( member_nat_nat_nat2 @ X @ S2 )
=> ( member_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_644_PiE__mem,axiom,
! [F2: ( nat > nat ) > nat > nat > nat,S2: set_nat_nat,T3: ( nat > nat ) > set_nat_nat_nat,X: nat > nat] :
( ( member1679187572556404771at_nat @ F2 @ ( piE_na8678869062391380393at_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat @ X @ S2 )
=> ( member_nat_nat_nat2 @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_645_PiE__mem,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat > nat,S2: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat_nat,X: ( nat > nat ) > nat] :
( ( member4489290058226556451at_nat @ F2 @ ( piE_na6840239867990089257at_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat_nat @ X @ S2 )
=> ( member_nat_nat @ ( F2 @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_646_PiE__cong,axiom,
! [I5: set_nat_nat,A2: ( nat > nat ) > set_nat,B3: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat_nat @ I5 @ A2 )
= ( piE_nat_nat_nat @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_647_PiE__cong,axiom,
! [I5: set_nat_nat,A2: ( nat > nat ) > set_nat_nat_nat2,B3: ( nat > nat ) > set_nat_nat_nat2] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_na7569501297962130601at_nat @ I5 @ A2 )
= ( piE_na7569501297962130601at_nat @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_648_PiE__cong,axiom,
! [I5: set_nat,A2: nat > set_nat,B3: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat @ I5 @ A2 )
= ( piE_nat_nat @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_649_PiE__cong,axiom,
! [I5: set_nat,A2: nat > set_nat_nat,B3: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat_nat2 @ I5 @ A2 )
= ( piE_nat_nat_nat2 @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_650_one__dim__cube__eq__nat__set,axiom,
! [K: nat] :
( bij_betw_nat_nat_nat
@ ^ [F: nat > nat] : ( F @ zero_zero_nat )
@ ( hales_cube @ one_one_nat @ K )
@ ( set_ord_lessThan_nat @ K ) ) ).
% one_dim_cube_eq_nat_set
thf(fact_651_nat__set__eq__one__dim__cube,axiom,
! [K: nat] :
( bij_betw_nat_nat_nat2
@ ^ [X2: nat] :
( restrict_nat_nat
@ ^ [Y3: nat] : X2
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K )
@ ( hales_cube @ one_one_nat @ K ) ) ).
% nat_set_eq_one_dim_cube
thf(fact_652_Collect__subset,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_653_Collect__subset,axiom,
! [A2: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ord_le973658574027395234at_nat
@ ( collec2410089373097230945at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_654_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_655_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_656_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_657_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_658_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B5: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_659_less__eq__set__def,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A5: set_na6626867396258451522at_nat,B5: set_na6626867396258451522at_nat] :
( ord_le319988079983864419_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_660_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B5: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A5 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_661_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_662_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_663_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_664_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A5: set_nat_nat_nat2,B5: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_665_less__set__def,axiom,
( ord_le2785809691299232406at_nat
= ( ^ [A5: set_na6626867396258451522at_nat,B5: set_na6626867396258451522at_nat] :
( ord_le6599672692516096367_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_666_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A5: set_nat_nat_nat,B5: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A5 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_667_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F2: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F2 @ A2 @ X )
= ( F2 @ X ) ) ) ).
% restrict_apply'
thf(fact_668_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F2: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat @ F2 @ A2 @ X )
= ( F2 @ X ) ) ) ).
% restrict_apply'
thf(fact_669_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F2: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat @ F2 @ A2 @ X )
= ( F2 @ X ) ) ) ).
% restrict_apply'
thf(fact_670_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restri6011711336257459485at_nat @ F2 @ A2 @ X )
= ( F2 @ X ) ) ) ).
% restrict_apply'
thf(fact_671_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F2: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restri4446420529079022766at_nat @ F2 @ A2 @ X )
= ( F2 @ X ) ) ) ).
% restrict_apply'
thf(fact_672_restrict__ext,axiom,
! [A2: set_nat,F2: nat > nat > nat,G: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat2 @ F2 @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_673_restrict__ext,axiom,
! [A2: set_nat,F2: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat @ F2 @ A2 )
= ( restrict_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_674_restrict__ext,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat @ F2 @ A2 )
= ( restrict_nat_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_675_restrict__ext,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > ( nat > nat ) > nat,G: ( nat > nat ) > ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri6011711336257459485at_nat @ F2 @ A2 )
= ( restri6011711336257459485at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_676_restrict__ext,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri4446420529079022766at_nat @ F2 @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_677_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_less_set_nat_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_678_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat_nat @ B3 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_679_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B5 )
& ~ ( ord_le9059583361652607317at_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_680_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_681_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_682_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_683_psubsetE,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_684_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
@ ^ [Y3: nat > nat] : ( L4 @ ( Y3 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_685_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
@ ^ [Y3: nat > nat] : ( L4 @ ( Y3 @ 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_686_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
@ ^ [Y3: nat > nat] : ( L4 @ ( Y3 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_687_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B3 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_688_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat_nat2,C4: ( nat > nat ) > set_nat_nat_nat2] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le5934964663421696068at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le973658574027395234at_nat @ ( piE_na7569501297962130601at_nat @ A2 @ B3 ) @ ( piE_na7569501297962130601at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_689_PiE__mono,axiom,
! [A2: set_nat,B3: nat > set_nat,C4: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B3 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_690_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B3 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_691_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B3 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_692_PiE__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B3: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le4724818764771537408at_nat @ ( piE_na5629913657871898759at_nat @ A2 @ B3 ) @ ( piE_na5629913657871898759at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_693_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat_nat,C4: ( nat > nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B3 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_694_PiE__mono,axiom,
! [A2: set_nat,B3: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B3 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_695_restrict__PiE__iff,axiom,
! [F2: nat > nat > nat,I5: set_nat,X6: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F2 @ I5 ) @ ( piE_nat_nat_nat2 @ I5 @ X6 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( member_nat_nat @ ( F2 @ X2 ) @ ( X6 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_696_restrict__PiE__iff,axiom,
! [F2: nat > nat,I5: set_nat,X6: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F2 @ I5 ) @ ( piE_nat_nat @ I5 @ X6 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( member_nat @ ( F2 @ X2 ) @ ( X6 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_697_restrict__PiE__iff,axiom,
! [F2: ( nat > nat ) > nat,I5: set_nat_nat,X6: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F2 @ I5 ) @ ( piE_nat_nat_nat @ I5 @ X6 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat @ ( F2 @ X2 ) @ ( X6 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_698_restrict__PiE__iff,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat,I5: set_nat_nat,X6: ( nat > nat ) > set_nat_nat_nat2] :
( ( member4402528950554000163at_nat @ ( restri6011711336257459485at_nat @ F2 @ I5 ) @ ( piE_na7569501297962130601at_nat @ I5 @ X6 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat_nat_nat @ ( F2 @ X2 ) @ ( X6 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_699_restrict__PiE__iff,axiom,
! [F2: ( nat > nat ) > nat > nat,I5: set_nat_nat,X6: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F2 @ I5 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ X6 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat_nat @ ( F2 @ X2 ) @ ( X6 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_700_classes__subset__cube,axiom,
! [N: nat,T: nat,I: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_701_hj__def,axiom,
( hales_hj
= ( ^ [R2: nat,T2: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ T2 )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [L5: nat > nat > nat,C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ( hales_is_line @ L5 @ N8 @ T2 )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ L5 @ ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( Chi3 @ X2 )
= C3 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_702_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_703_order__refl,axiom,
! [X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ X ) ).
% order_refl
thf(fact_704_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_705_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_706_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_707_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_708_bij__betw__the__inv__into,axiom,
! [F2: nat > ( nat > nat ) > nat,A2: set_nat,B3: set_nat_nat_nat2] :
( ( bij_be8282881169987224566at_nat @ F2 @ A2 @ B3 )
=> ( bij_be1059735840858801910at_nat @ ( the_in5568309565101652403at_nat @ A2 @ F2 ) @ B3 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_709_bij__betw__the__inv__into,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat] :
( ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B3 )
=> ( bij_be8282881169987224566at_nat @ ( the_in7568536272828005555at_nat @ A2 @ F2 ) @ B3 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_710_bij__betw__the__inv__into,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B3 )
=> ( bij_betw_nat_nat_nat @ ( the_in3844390324871770692at_nat @ A2 @ F2 ) @ B3 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_711_bij__betw__the__inv__into,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ B3 )
=> ( bij_betw_nat_nat @ ( the_inv_into_nat_nat @ A2 @ F2 ) @ B3 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_712_bij__betw__the__inv__into,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat] :
( ( bij_betw_nat_nat_nat @ F2 @ A2 @ B3 )
=> ( bij_betw_nat_nat_nat2 @ ( the_in5300466440149791684at_nat @ A2 @ F2 ) @ B3 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_713_image__ident,axiom,
! [Y6: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_714_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_715_image__restrict__eq,axiom,
! [F2: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F2 @ A2 ) @ A2 )
= ( image_nat_nat_nat2 @ F2 @ A2 ) ) ).
% image_restrict_eq
thf(fact_716_image__restrict__eq,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F2 @ A2 ) @ A2 )
= ( image_nat_nat @ F2 @ A2 ) ) ).
% image_restrict_eq
thf(fact_717_image__restrict__eq,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat @ ( restrict_nat_nat_nat @ F2 @ A2 ) @ A2 )
= ( image_nat_nat_nat @ F2 @ A2 ) ) ).
% image_restrict_eq
thf(fact_718_image__restrict__eq,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_1991755285388994676at_nat @ ( restri6011711336257459485at_nat @ F2 @ A2 ) @ A2 )
= ( image_1991755285388994676at_nat @ F2 @ A2 ) ) ).
% image_restrict_eq
thf(fact_719_image__restrict__eq,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F2 @ A2 ) @ A2 )
= ( image_3205354838064109189at_nat @ F2 @ A2 ) ) ).
% image_restrict_eq
thf(fact_720_image__add__0,axiom,
! [S2: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_721_image__add__0,axiom,
! [S2: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_722_bij__betw__add,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ ( plus_plus_nat @ A ) @ A2 @ B3 )
= ( ( image_nat_nat @ ( plus_plus_nat @ A ) @ A2 )
= B3 ) ) ).
% bij_betw_add
thf(fact_723_bij__betw__add,axiom,
! [A: int,A2: set_int,B3: set_int] :
( ( bij_betw_int_int @ ( plus_plus_int @ A ) @ A2 @ B3 )
= ( ( image_int_int @ ( plus_plus_int @ A ) @ A2 )
= B3 ) ) ).
% bij_betw_add
thf(fact_724_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_725_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ ( image_nat_nat_nat2 @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_726_image__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F2: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ ( image_3205354838064109189at_nat @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_727_image__subsetI,axiom,
! [A2: set_nat,F2: nat > nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_728_image__subsetI,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat,B3: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_729_image__subsetI,axiom,
! [A2: set_nat,F2: nat > nat > nat,B3: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_730_image__subsetI,axiom,
! [A2: set_nat,F2: nat > ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_731_image__subsetI,axiom,
! [A2: set_nat,F2: nat > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_732_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > nat,B3: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_733_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F2: ( nat > nat > nat ) > nat,B3: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_734_image__subsetI,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_735_image__subsetI,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_736_image__subsetI,axiom,
! [A2: set_nat_nat,F2: ( nat > nat ) > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_3101123049818244468at_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_737_subset__imageE,axiom,
! [B3: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B3
!= ( image_nat_nat @ F2 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_738_subset__imageE,axiom,
! [B3: set_nat_nat,F2: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B3
!= ( image_nat_nat_nat2 @ F2 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_739_subset__imageE,axiom,
! [B3: set_nat_nat,F2: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F2 @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B3
!= ( image_3205354838064109189at_nat @ F2 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_740_image__subset__iff,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_741_image__subset__iff,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F2 @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_742_image__subset__iff,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ B3 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F2 @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_743_subset__image__iff,axiom,
! [B3: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_744_subset__image__iff,axiom,
! [B3: set_nat_nat,F2: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat_nat2 @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_745_subset__image__iff,axiom,
! [B3: set_nat_nat,F2: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B3
= ( image_3205354838064109189at_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_746_image__diff__subset,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B3 ) ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_747_image__diff__subset,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ ( image_nat_nat_nat2 @ F2 @ B3 ) ) @ ( image_nat_nat_nat2 @ F2 @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_748_image__diff__subset,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ ( image_3205354838064109189at_nat @ F2 @ B3 ) ) @ ( image_3205354838064109189at_nat @ F2 @ ( minus_8121590178497047118at_nat @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_749_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_750_translation__subtract__diff,axiom,
! [A: int,S: set_int,T: set_int] :
( ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A )
@ ( minus_minus_set_int @ S @ T ) )
= ( minus_minus_set_int
@ ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A )
@ S )
@ ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_751_set__diff__eq,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A5 )
& ~ ( member_nat_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_752_set__diff__eq,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A5: set_nat_nat_nat2,B5: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A5 )
& ~ ( member_nat_nat_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_753_set__diff__eq,axiom,
( minus_5225787954611647771at_nat
= ( ^ [A5: set_na6626867396258451522at_nat,B5: set_na6626867396258451522at_nat] :
( collec2410089373097230945at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A5 )
& ~ ( member4402528950554000163at_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_754_set__diff__eq,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A5: set_nat_nat_nat,B5: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A5 )
& ~ ( member_nat_nat_nat2 @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_755_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ~ ( member_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_756_minus__set__def,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( collect_nat_nat
@ ( minus_167519014754328503_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_757_minus__set__def,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A5: set_nat_nat_nat2,B5: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ( minus_2851842960567056136_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_758_minus__set__def,axiom,
( minus_5225787954611647771at_nat
= ( ^ [A5: set_na6626867396258451522at_nat,B5: set_na6626867396258451522at_nat] :
( collec2410089373097230945at_nat
@ ( minus_6692596912184789802_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_759_minus__set__def,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A5: set_nat_nat_nat,B5: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ( minus_7240682219522218504_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A5 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_760_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_761_Compr__image__eq,axiom,
! [F2: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_762_Compr__image__eq,axiom,
! [F2: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat2 @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_763_Compr__image__eq,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat @ F2
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_764_Compr__image__eq,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_3205354838064109189at_nat @ F2
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_765_Compr__image__eq,axiom,
! [F2: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( image_5809701139083627781at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5809701139083627781at_nat @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_766_Compr__image__eq,axiom,
! [F2: nat > nat > nat > nat,A2: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( image_6919068903512877573at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6919068903512877573at_nat @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_767_Compr__image__eq,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_7809927846809980933at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_7809927846809980933at_nat @ F2
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_768_Compr__image__eq,axiom,
! [F2: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_913610194320715013at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_913610194320715013at_nat @ F2
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_769_Compr__image__eq,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_1262493855416953332at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1262493855416953332at_nat @ F2
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_770_Compr__image__eq,axiom,
! [F2: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_1545173636400105204at_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1545173636400105204at_nat @ F2
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_771_image__image,axiom,
! [F2: ( nat > nat ) > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat @ F2 @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_772_image__image,axiom,
! [F2: nat > nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat2 @ F2 @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_773_image__image,axiom,
! [F2: nat > nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F2 @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_774_image__image,axiom,
! [F2: ( nat > nat ) > nat > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_3205354838064109189at_nat @ F2 @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_775_image__image,axiom,
! [F2: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F2 @ ( image_3205354838064109189at_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_776_image__image,axiom,
! [F2: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F2 @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_777_imageE,axiom,
! [B: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_778_imageE,axiom,
! [B: nat > nat,F2: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_779_imageE,axiom,
! [B: nat,F2: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_780_imageE,axiom,
! [B: nat > nat,F2: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_781_imageE,axiom,
! [B: nat,F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F2 @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_782_imageE,axiom,
! [B: nat,F2: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_783_imageE,axiom,
! [B: ( nat > nat ) > nat,F2: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_784_imageE,axiom,
! [B: nat > nat > nat,F2: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_785_imageE,axiom,
! [B: nat > nat,F2: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F2 @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_786_imageE,axiom,
! [B: nat > nat,F2: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat] :
( ( member_nat_nat @ B @ ( image_1545173636400105204at_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_787_all__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_nat @ F2 @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_788_all__subset__image,axiom,
! [F2: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ ( image_nat_nat_nat2 @ F2 @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F2 @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_789_all__subset__image,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ ( image_3205354838064109189at_nat @ F2 @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ A2 )
=> ( P @ ( image_3205354838064109189at_nat @ F2 @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_790_bij__betw__subset,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,A6: set_nat,B3: set_nat_nat_nat2,B6: set_nat] :
( ( bij_be1059735840858801910at_nat @ F2 @ A2 @ A6 )
=> ( ( ord_le5934964663421696068at_nat @ B3 @ A2 )
=> ( ( ( image_7809927846809980933at_nat @ F2 @ B3 )
= B6 )
=> ( bij_be1059735840858801910at_nat @ F2 @ B3 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_791_bij__betw__subset,axiom,
! [F2: nat > nat > nat,A2: set_nat,A6: set_nat_nat,B3: set_nat,B6: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ A6 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ( image_nat_nat_nat2 @ F2 @ B3 )
= B6 )
=> ( bij_betw_nat_nat_nat2 @ F2 @ B3 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_792_bij__betw__subset,axiom,
! [F2: nat > nat,A2: set_nat,A6: set_nat,B3: set_nat,B6: set_nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ A6 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ( image_nat_nat @ F2 @ B3 )
= B6 )
=> ( bij_betw_nat_nat @ F2 @ B3 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_793_bij__betw__subset,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,A6: set_nat_nat,B3: set_nat_nat,B6: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F2 @ A2 @ A6 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( ( ( image_3205354838064109189at_nat @ F2 @ B3 )
= B6 )
=> ( bij_be5678534868967705974at_nat @ F2 @ B3 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_794_bij__betw__subset,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,A6: set_nat,B3: set_nat_nat,B6: set_nat] :
( ( bij_betw_nat_nat_nat @ F2 @ A2 @ A6 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( ( ( image_nat_nat_nat @ F2 @ B3 )
= B6 )
=> ( bij_betw_nat_nat_nat @ F2 @ B3 @ B6 ) ) ) ) ).
% bij_betw_subset
thf(fact_795_bij__betw__byWitness,axiom,
! [A2: set_nat_nat_nat2,F4: nat > ( nat > nat ) > nat,F2: ( ( nat > nat ) > nat ) > nat,A6: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ( F4 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( F2 @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F2 @ A2 ) @ A6 )
=> ( ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F4 @ A6 ) @ A2 )
=> ( bij_be1059735840858801910at_nat @ F2 @ A2 @ A6 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_796_bij__betw__byWitness,axiom,
! [A2: set_nat,F4: nat > nat,F2: nat > nat,A6: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F4 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( F2 @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ A6 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A6 ) @ A2 )
=> ( bij_betw_nat_nat @ F2 @ A2 @ A6 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_797_bij__betw__byWitness,axiom,
! [A2: set_nat_nat,F4: nat > nat > nat,F2: ( nat > nat ) > nat,A6: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F4 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( F2 @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ A6 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F4 @ A6 ) @ A2 )
=> ( bij_betw_nat_nat_nat @ F2 @ A2 @ A6 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_798_bij__betw__byWitness,axiom,
! [A2: set_nat,F4: ( nat > nat ) > nat,F2: nat > nat > nat,A6: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F4 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A6 )
=> ( ( F2 @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ A6 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F4 @ A6 ) @ A2 )
=> ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ A6 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_799_bij__betw__byWitness,axiom,
! [A2: set_nat_nat,F4: ( nat > nat ) > nat > nat,F2: ( nat > nat ) > nat > nat,A6: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F4 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A6 )
=> ( ( F2 @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ A6 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F4 @ A6 ) @ A2 )
=> ( bij_be5678534868967705974at_nat @ F2 @ A2 @ A6 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_800_PiE__uniqueness,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F2 @ A2 ) @ B3 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F2 @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F2 @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_801_PiE__uniqueness,axiom,
! [F2: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F2 @ A2 ) @ B3 )
=> ? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F2 @ Xa ) ) )
& ! [Y5: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member4402528950554000163at_nat @ Y5
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F2 @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_802_PiE__uniqueness,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F2 @ Xa ) ) )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F2 @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_803_PiE__uniqueness,axiom,
! [F2: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ A2 ) @ B3 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F2 @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y5
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F2 @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_804_PiE__uniqueness,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ A2 ) @ B3 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F2 @ Xa ) ) )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F2 @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_805_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_806_order__antisym__conv,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_807_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_808_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_809_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_810_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_811_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_812_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_813_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_814_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_815_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_816_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_817_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_818_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_819_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_820_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F2: nat > set_nat_nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_821_ord__eq__le__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_822_ord__eq__le__subst,axiom,
! [A: nat,F2: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_823_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F2: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_824_ord__eq__le__subst,axiom,
! [A: int,F2: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_825_ord__eq__le__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_826_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F2: int > set_nat_nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_827_ord__eq__le__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_828_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_829_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_830_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_831_order__eq__refl,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( X = Y )
=> ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_832_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_833_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_834_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_835_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_836_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_837_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_838_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_839_order__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_840_order__subst2,axiom,
! [A: int,B: int,F2: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_841_order__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_842_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_843_order__subst1,axiom,
! [A: nat,F2: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_844_order__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_845_order__subst1,axiom,
! [A: set_nat_nat,F2: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_846_order__subst1,axiom,
! [A: set_nat_nat,F2: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_847_order__subst1,axiom,
! [A: set_nat_nat,F2: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_848_order__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_849_order__subst1,axiom,
! [A: int,F2: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_850_order__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_851_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_852_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_853_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_854_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_855_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_856_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_857_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_858_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_859_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_860_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_861_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_862_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_863_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_864_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_865_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_866_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_867_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_868_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_869_order__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_870_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_871_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_872_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_873_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_874_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_875_order__antisym,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_876_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_877_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_878_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_879_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_880_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_881_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_882_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_883_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_884_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat_nat,Z: set_nat_nat] : ( Y2 = Z ) )
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
& ( ord_le9059583361652607317at_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_885_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_886_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_887_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_888_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_889_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_890_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_891_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_892_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_893_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_894_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_895_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_896_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_897_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_898_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_899_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_900_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_901_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_902_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_903_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_904_order__less__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_905_order__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_906_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_907_order__less__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_908_order__less__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_909_order__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_910_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_911_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_912_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_913_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_914_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_915_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_916_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_917_ord__eq__less__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_918_ord__eq__less__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_919_ord__eq__less__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_920_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_921_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_922_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_923_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_924_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_925_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_926_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_927_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_928_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_929_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_930_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_931_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_932_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_933_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_934_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_935_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_936_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_937_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_938_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_939_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_940_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_941_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_942_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N6: nat] :
( ( P5 @ N6 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N6 )
=> ~ ( P5 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_943_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_944_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_945_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_946_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_947_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_948_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_949_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_950_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_951_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_952_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_953_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_954_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_955_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_956_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_957_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_958_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_959_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_960_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_961_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_962_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_963_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_964_order__le__imp__less__or__eq,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_less_set_nat_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_965_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_966_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_967_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_968_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_969_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_970_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_971_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_972_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_973_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_974_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_975_order__less__le__subst1,axiom,
! [A: set_nat_nat,F2: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_less_set_nat_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_976_order__less__le__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_977_order__less__le__subst1,axiom,
! [A: nat,F2: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_978_order__less__le__subst1,axiom,
! [A: set_nat_nat,F2: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_979_order__less__le__subst1,axiom,
! [A: int,F2: set_nat_nat > int,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_980_order__less__le__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_981_order__less__le__subst1,axiom,
! [A: set_nat_nat,F2: int > set_nat_nat,B: int,C: int] :
( ( ord_less_set_nat_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_982_order__less__le__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_983_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_984_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_985_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_986_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_987_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_988_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F2: set_nat_nat > int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_989_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_990_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_991_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_992_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_993_order__le__less__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_994_order__le__less__subst1,axiom,
! [A: set_nat_nat,F2: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_995_order__le__less__subst1,axiom,
! [A: set_nat_nat,F2: int > set_nat_nat,B: int,C: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_set_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_996_order__le__less__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_997_order__le__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_998_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_999_order__less__le__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z2 )
=> ( ord_less_set_nat_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1000_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1001_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1002_order__le__less__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_less_set_nat_nat @ Y @ Z2 )
=> ( ord_less_set_nat_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1003_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1004_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1005_order__neq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A != B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1006_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1007_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1008_order__le__neq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1009_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1010_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1011_order__less__imp__le,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y )
=> ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1012_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1013_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1014_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1015_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1016_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1017_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1018_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1019_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1020_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1021_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1022_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1023_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1024_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1025_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1026_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1027_order_Ostrict__implies__order,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1028_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1029_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1030_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ~ ( ord_le9059583361652607317at_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1031_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1032_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1033_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1034_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1035_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1036_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_less_set_nat_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1037_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1038_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1039_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1040_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1041_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1042_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1043_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1044_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1045_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ~ ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1046_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1047_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1048_order_Ostrict__trans2,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1049_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1050_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1051_order_Ostrict__trans1,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1052_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1053_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1054_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1055_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1056_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1057_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1058_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1059_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1060_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1061_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1062_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
& ~ ( ord_le9059583361652607317at_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1063_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1064_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1065_antisym__conv2,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1066_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1067_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1068_antisym__conv1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1069_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1070_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1071_nless__le,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A @ B ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1072_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1073_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1074_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_1075_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1076_leD,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ~ ( ord_less_set_nat_nat @ X @ Y ) ) ).
% leD
thf(fact_1077_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_1078_f__the__inv__into__f__bij__betw,axiom,
! [F2: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat,X: nat] :
( ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B3 )
=> ( ( ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B3 )
=> ( member_nat @ X @ B3 ) )
=> ( ( F2 @ ( the_in7568536272828005555at_nat @ A2 @ F2 @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1079_f__the__inv__into__f__bij__betw,axiom,
! [F2: nat > nat > nat,A2: set_nat,B3: set_nat_nat,X: nat > nat] :
( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B3 )
=> ( ( ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B3 )
=> ( member_nat_nat @ X @ B3 ) )
=> ( ( F2 @ ( the_in3844390324871770692at_nat @ A2 @ F2 @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1080_f__the__inv__into__f__bij__betw,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat,X: nat] :
( ( bij_betw_nat_nat @ F2 @ A2 @ B3 )
=> ( ( ( bij_betw_nat_nat @ F2 @ A2 @ B3 )
=> ( member_nat @ X @ B3 ) )
=> ( ( F2 @ ( the_inv_into_nat_nat @ A2 @ F2 @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1081_f__the__inv__into__f__bij__betw,axiom,
! [F2: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat,X: nat] :
( ( bij_betw_nat_nat_nat @ F2 @ A2 @ B3 )
=> ( ( ( bij_betw_nat_nat_nat @ F2 @ A2 @ B3 )
=> ( member_nat @ X @ B3 ) )
=> ( ( F2 @ ( the_in5300466440149791684at_nat @ A2 @ F2 @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1082_layered__subspace__def,axiom,
( hales_114318738418697479at_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N6: nat,T2: nat,R2: ( nat > nat ) > nat,Chi3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( hales_is_subspace @ S6 @ K4 @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C3: ( nat > nat ) > nat] :
( ( ord_less_nat_nat_nat @ C3 @ R2 )
& ! [Y3: nat > nat] :
( ( member_nat_nat @ Y3 @ ( hales_classes @ K4 @ T2 @ X2 ) )
=> ( ( Chi3 @ ( S6 @ Y3 ) )
= C3 ) ) ) )
& ( member4402528950554000163at_nat @ Chi3
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_or2699333443382148811at_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1083_layered__subspace__def,axiom,
( hales_4259056829518216709ce_int
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N6: nat,T2: nat,R2: int,Chi3: ( nat > nat ) > int] :
( ( hales_is_subspace @ S6 @ K4 @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C3: int] :
( ( ord_less_int @ C3 @ R2 )
& ! [Y3: nat > nat] :
( ( member_nat_nat @ Y3 @ ( hales_classes @ K4 @ T2 @ X2 ) )
=> ( ( Chi3 @ ( S6 @ Y3 ) )
= C3 ) ) ) )
& ( member_nat_nat_int @ Chi3
@ ( piE_nat_nat_int @ ( hales_cube @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_int @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1084_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K4: nat,N6: nat,T2: nat,R2: nat,Chi3: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S6 @ K4 @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K4 ) )
=> ? [C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ! [Y3: nat > nat] :
( ( member_nat_nat @ Y3 @ ( hales_classes @ K4 @ T2 @ X2 ) )
=> ( ( Chi3 @ ( S6 @ Y3 ) )
= C3 ) ) ) )
& ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1085_pred__subset__eq,axiom,
! [R4: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S2 ) )
= ( ord_less_eq_set_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_1086_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat2,S2: set_nat_nat_nat2] :
( ( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ R4 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ S2 ) )
= ( ord_le5934964663421696068at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_1087_pred__subset__eq,axiom,
! [R4: set_na6626867396258451522at_nat,S2: set_na6626867396258451522at_nat] :
( ( ord_le319988079983864419_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ R4 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ S2 ) )
= ( ord_le973658574027395234at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_1088_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat,S2: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ R4 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ S2 ) )
= ( ord_le3211623285424100676at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_1089_pred__subset__eq,axiom,
! [R4: set_nat_nat,S2: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ R4 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ S2 ) )
= ( ord_le9059583361652607317at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_1090_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > nat,B3: set_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1091_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1092_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > ( nat > nat ) > ( nat > nat ) > nat,B3: set_na6626867396258451522at_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member4402528950554000163at_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le973658574027395234at_nat @ ( image_3941236537129881699at_nat @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1093_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat_nat_nat2 @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1094_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F2: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( member_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F2 @ ( collect_nat_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1095_image__Collect__subsetI,axiom,
! [P: nat > $o,F2: nat > nat > nat,B3: set_nat_nat] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( member_nat_nat @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F2 @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1096_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1097_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_1098_subspace__elems__embed,axiom,
! [S2: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S2 @ K @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S2 @ ( hales_cube @ K @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_1099_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_1100_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_1101_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1102_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_1103_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_1104_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1105_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1106_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1107_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1108_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1109_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1110_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1111_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1112_cube1__alt__def,axiom,
! [N: nat] :
( ( hales_cube @ N @ one_one_nat )
= ( insert_nat_nat
@ ( restrict_nat_nat
@ ^ [X2: nat] : zero_zero_nat
@ ( set_ord_lessThan_nat @ N ) )
@ bot_bot_set_nat_nat ) ) ).
% cube1_alt_def
thf(fact_1113_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1114_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1115_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1116_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1117_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1118_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1119_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N6: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N6 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1120_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1121_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1122_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1123_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P6: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_1124_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P6: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1125_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1126_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1127_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1128_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1129_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1130_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1131_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1132_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1133_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1134_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1135_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1136_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1137_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_1138_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1139_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1140_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1141_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1142_bij__betw__Suc,axiom,
! [M5: set_nat,N3: set_nat] :
( ( bij_betw_nat_nat @ suc @ M5 @ N3 )
= ( ( image_nat_nat @ suc @ M5 )
= N3 ) ) ).
% bij_betw_Suc
thf(fact_1143_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1144_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1145_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_1146_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_1147_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_1148_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1149_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_1150_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1151_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1152_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1153_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1154_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1155_card__atMost,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U2 ) )
= ( suc @ U2 ) ) ).
% card_atMost
thf(fact_1156_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1157_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_1158_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1159_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1160_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_1161_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1162_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1163_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1164_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1165_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1166_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_1167_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_1168_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_1169_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N6: nat] : ( ord_less_eq_nat @ ( suc @ N6 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1170_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_1171_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_1172_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_1173_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1174_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1175_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_1176_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1177_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1178_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N6: nat] :
? [K4: nat] :
( N6
= ( suc @ ( plus_plus_nat @ M6 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1179_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1180_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1181_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_1182_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
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_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_1185_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_1186_Suc__eq__plus1,axiom,
( suc
= ( ^ [N6: nat] : ( plus_plus_nat @ N6 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1187_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1188_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1189_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_1190_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1191_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1192_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1193_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1194_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1195_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_1196_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1197_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1198_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1199_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_1200_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1201_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1202_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_1203_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1204_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_1205_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1206_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1207_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1208_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_1209_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1210_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_1211_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1212_Suc__le__D,axiom,
! [N: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
=> ? [M4: nat] :
( M8
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1213_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_1214_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1215_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_1216_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1217_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1218_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R4 @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( R4 @ X3 @ Y4 )
=> ( ( R4 @ Y4 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R4 @ N2 @ ( suc @ N2 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1219_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1220_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1221_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1222_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1223_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1224_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1225_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1226_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1227_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1228_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1229_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1230_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1231_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1232_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_1233_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1234_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1235_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1236_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1237_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_1238_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1239_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1240_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1241_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1242_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1243_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1244_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z3: int] :
? [N6: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N6 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1245_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_1246_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_1247_card__less,axiom,
! [M5: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M5 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M5 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1248_card__less__Suc,axiom,
! [M5: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M5 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M5 )
& ( ord_less_nat @ K4 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M5 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1249_card__less__Suc2,axiom,
! [M5: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M5 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M5 )
& ( ord_less_nat @ K4 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M5 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1250_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_1251_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_1252_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N6 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N6 ) ) ) ) ) ).
% add_eq_if
thf(fact_1253_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1254_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1255_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1256_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1257_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1258_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1259_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1260_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1261_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1262_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1263_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1264_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
% Helper facts (8)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X5: nat] : ( P @ X5 ) ) ) ).
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 ) )
= ( ? [X5: nat > nat] : ( P @ X5 ) ) ) ).
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 ) )
= ( ? [X5: ( nat > nat ) > nat] : ( P @ X5 ) ) ) ).
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 ) )
= ( ? [X5: nat > nat > nat] : ( P @ X5 ) ) ) ).
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 ) )
= ( ? [X5: ( nat > nat ) > ( nat > nat ) > nat] : ( P @ X5 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_nat_nat
@ ( 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 )
= zero_zero_nat ) ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ).
%------------------------------------------------------------------------------