TPTP Problem File: SLH0717^1.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_01231_053375__5857430_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1550 ( 472 unt; 279 typ; 0 def)
% Number of atoms : 3948 (1099 equ; 0 cnn)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 12262 ( 240 ~; 18 |; 456 &;9827 @)
% ( 0 <=>;1721 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 3686 (3686 >; 0 *; 0 +; 0 <<)
% Number of symbols : 248 ( 247 usr; 16 con; 0-6 aty)
% Number of variables : 3908 ( 442 ^;3245 !; 221 ?;3908 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:22.005
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
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% Explicit typings (247)
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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__Int__Oint,type,
piE_nat_nat_nat_int: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_int ) > set_nat_nat_nat_int ).
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__Int__Oint_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_int_nat_nat_nat: set_int > ( int > set_nat_nat_nat2 ) > set_int_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_int_nat_nat: set_int > ( int > set_nat_nat ) > set_int_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001t__Int__Oint,type,
piE_int_int: set_int > ( int > set_int ) > set_int_int ).
thf(sy_c_FuncSet_OPiE_001t__Int__Oint_001t__Nat__Onat,type,
piE_int_nat: set_int > ( int > set_nat ) > set_int_nat ).
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__Int__Oint,type,
piE_nat_int: set_nat > ( nat > set_int ) > set_nat_int ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
restri5928693893068016200at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
restri3376681761679556074at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri4486049526108805866at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restri3045778531447280891at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
restri7842575624313966988at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat ) > set_na6626867396258451522at_nat > ( ( nat > nat ) > ( nat > nat ) > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restri9050993537824894510at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
restri6011711336257459485at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restri4446420529079022766at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restrict_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restrict_nat_nat_nat2: ( nat > nat > nat ) > set_nat > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: ( nat > nat ) > set_nat > nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_6692596912184789802_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_7158188067284919257_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
minus_2851842960567056136_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_7240682219522218504_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
minus_167519014754328503_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
minus_5225787954611647771at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_4646100876039749548at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
minus_1221035652888719293at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_7721066311745899709at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hales__Jewett_Oclasses,type,
hales_classes: nat > nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ohj,type,
hales_hj: nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Hales__Jewett_Olayered__subspace_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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le6599672692516096367_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le4961065272816086430_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le8812218136411540557_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le3977685358511927117_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le7877100967975825120at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le4629963735342356977at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_less_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_less_nat_nat_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_nat_nat_nat2: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le2785809691299232406at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6177938698872215975at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le371403230139555384at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6871433888996735800at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le319988079983864419_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5430825838364970130_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le996020443555834177_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5384859702510996545_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le3015115239550301420at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le747776305331315197at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le2017632242545079438at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3127000006974329230at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4724818764771537408at_nat: set_na6273678875609698720at_nat > set_na6273678875609698720at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
ord_le8898325182481281041at_nat: set_na8032705079450369969at_nat > set_na8032705079450369969at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le9041126503034175505at_nat: set_na8175506400003264433at_nat > set_na8175506400003264433at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3190276326201062306at_nat: set_na8843485148432118594at_nat > set_na8843485148432118594at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3125778081881428130at_nat: set_na8778986904112484418at_nat > set_na8778986904112484418at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le973658574027395234at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5260717879541182899at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le5934964663421696068at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3211623285424100676at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
collec2410089373097230945at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na6626867396258451522at_nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec3567154360959927026at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
image_323718453976782111at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_4065942021260649921at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_5175309785689899713at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ) > set_na6626867396258451522at_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_722231358656203602at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat ) > set_na6626867396258451522at_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_3521005150465447523at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat ) > set_na6626867396258451522at_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_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_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__Int__Oint,type,
image_nat_nat_int: ( ( nat > nat ) > int ) > set_nat_nat > set_int ).
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_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_5036371617510403937at_nat: ( int > ( nat > nat ) > nat ) > set_int > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_int_nat_nat: ( int > nat > nat ) > set_int > set_nat_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__Int__Oint_001t__Nat__Onat,type,
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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__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
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,
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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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or7562748684798938298at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or2699333443382148811at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
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set_or3808701207811398603at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
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set_or1140352010380016476at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
fChoic2516396905127217208at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > ( nat > nat ) > ( nat > nat ) > nat ).
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fChoic52552927678224201at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( nat > nat ) > nat > nat ).
thf(sy_c_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_fChoice_001t__Nat__Onat,type,
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member6416598835793757296at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_na5550323840042141967at_nat > $o ).
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member3693257457796161904at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat ) > set_na2687664174320723471at_nat > $o ).
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member1174580258192983937at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat ) > set_na6273678875609698720at_nat > $o ).
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member4107745174335074322at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat ) > set_na8032705079450369969at_nat > $o ).
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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__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member2740455936716430260at_nat: ( nat > ( nat > nat ) > nat ) > set_nat_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member17114558718834868at_nat: ( nat > nat > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_v_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_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 (1264)
thf(fact_0_a,axiom,
member_nat_nat @ x @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ).
% a
thf(fact_1__092_060chi_062__prop,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ m @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<chi>_prop
thf(fact_2_A,axiom,
! [X: nat > nat] :
( ( member_nat_nat @ X @ ( 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 @ X @ Xa @ n2 @ m2 ) ) @ ( set_ord_lessThan_nat @ r ) ) ) ) ).
% A
thf(fact_3_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_4__092_060chi_062L__def,axiom,
( chi_L
= ( restri6011711336257459485at_nat
@ ^ [X2: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y: nat > nat] : ( chi @ ( hales_join_nat @ X2 @ Y @ 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_5__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_6_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_7_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_8_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_9_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B2: set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X3
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I: ( nat > nat ) > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_10_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_11_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B2 )
=> ? [X3: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X3
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_12_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X3: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X3
@ ( piE_nat_nat_nat_nat5 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_13_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X3
@ ( piE_nat_nat_nat_nat2 @ A2
@ ^ [I: nat > nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_14_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member4489290058226556451at_nat @ X3
@ ( piE_na6840239867990089257at_nat @ A2
@ ^ [I: ( nat > nat ) > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_15_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X3
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_16_join__cubes,axiom,
! [F: nat > nat,N: nat,T: nat,G: nat > nat,M: nat] :
( ( member_nat_nat @ F @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( ( member_nat_nat @ G @ ( hales_cube @ M @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( member_nat_nat @ ( hales_join_nat @ F @ G @ N @ M ) @ ( hales_cube @ ( plus_plus_nat @ N @ M ) @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_17__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_18_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat] :
( ( member4107745174335074322at_nat @ F @ ( piE_na2138371880555796248at_nat @ A2 @ B2 ) )
=> ( ( restri7842575624313966988at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_19_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ( member1174580258192983937at_nat @ F @ ( piE_na5629913657871898759at_nat @ A2 @ B2 ) )
=> ( ( restri3045778531447280891at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_20_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ( member3693257457796161904at_nat @ F @ ( piE_na4170927303951785078at_nat @ A2 @ B2 ) )
=> ( ( restri4486049526108805866at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_21_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2] :
( ( member6416598835793757296at_nat @ F @ ( piE_na3061559539522535286at_nat @ A2 @ B2 ) )
=> ( ( restri3376681761679556074at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_22_PiE__restrict,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat] :
( ( member6105598001968527566at_nat @ F @ ( piE_na799184809307736020at_nat @ A2 @ B2 ) )
=> ( ( restri5928693893068016200at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_23_PiE__restrict,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat_nat2] :
( ( member4402528950554000163at_nat @ F @ ( piE_na7569501297962130601at_nat @ A2 @ B2 ) )
=> ( ( restri6011711336257459485at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_24_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_25_PiE__restrict,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_26_PiE__restrict,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_27_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_28__092_060chi_062L__prop,axiom,
( member4402528950554000163at_nat @ chi_L
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] :
( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [J: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ) ).
% \<chi>L_prop
thf(fact_29_s__def,axiom,
( s
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t @ one_one_nat ) @ m2 ) ) ) ).
% s_def
thf(fact_30_lessThan__eq__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X4 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X4 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_31_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_32_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_33_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_34_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_35_s__coloured,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= s ) ).
% s_coloured
thf(fact_36_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,I2: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat] :
( ( member4107745174335074322at_nat @ ( restri7842575624313966988at_nat @ F @ I2 ) @ ( piE_na2138371880555796248at_nat @ I2 @ X5 ) )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ I2 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_37_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,I2: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ( member1174580258192983937at_nat @ ( restri3045778531447280891at_nat @ F @ I2 ) @ ( piE_na5629913657871898759at_nat @ I2 @ X5 ) )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ I2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_38_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,I2: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ( member3693257457796161904at_nat @ ( restri4486049526108805866at_nat @ F @ I2 ) @ ( piE_na4170927303951785078at_nat @ I2 @ X5 ) )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ I2 )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_39_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,I2: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2] :
( ( member6416598835793757296at_nat @ ( restri3376681761679556074at_nat @ F @ I2 ) @ ( piE_na3061559539522535286at_nat @ I2 @ X5 ) )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ I2 )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_40_restrict__PiE__iff,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat,I2: set_na6626867396258451522at_nat,X5: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat] :
( ( member6105598001968527566at_nat @ ( restri5928693893068016200at_nat @ F @ I2 ) @ ( piE_na799184809307736020at_nat @ I2 @ X5 ) )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ I2 )
=> ( member4402528950554000163at_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_41_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,I2: set_nat_nat,X5: ( nat > nat ) > set_nat_nat_nat2] :
( ( member4402528950554000163at_nat @ ( restri6011711336257459485at_nat @ F @ I2 ) @ ( piE_na7569501297962130601at_nat @ I2 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I2 )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_42_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I2: set_nat_nat,X5: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I2 ) @ ( piE_nat_nat_nat @ I2 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I2 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_43_restrict__PiE__iff,axiom,
! [F: nat > nat,I2: set_nat,X5: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I2 ) @ ( piE_nat_nat @ I2 @ X5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I2 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_44_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I2: set_nat,X5: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I2 ) @ ( piE_nat_nat_nat2 @ I2 @ X5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_45_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I2: set_nat_nat,X5: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I2 ) @ ( piE_nat_nat_nat_nat3 @ I2 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_46__092_060phi_062__prop,axiom,
( bij_be1059735840858801910at_nat @ phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s ) ) ).
% \<phi>_prop
thf(fact_47__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 ) )
@ ^ [I: 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_48_bij__betw__restrict__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ ( restri9050993537824894510at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_49_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( bij_be5311014265664741861at_nat @ ( restri6011711336257459485at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be5311014265664741861at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_50_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_betw_nat_nat_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_51_bij__betw__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_betw_nat_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_52_bij__betw__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 @ B2 )
= ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_53_bij__betw__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 @ B2 )
= ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 ) ) ).
% bij_betw_restrict_eq
thf(fact_54_calculation,axiom,
( ( chi_L_s @ x )
= ( phi @ ( chi_L @ x ) ) ) ).
% calculation
thf(fact_55__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 ) )
@ ^ [I: 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_56_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_57_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_58_cube__def,axiom,
( hales_cube
= ( ^ [N2: nat,T2: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N2 )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T2 ) ) ) ) ).
% cube_def
thf(fact_59_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_60_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_61_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_62_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_63_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_64_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_65_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_66_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_67_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_68_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_69_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_70_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_71_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_72_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_73_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_74_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_75_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_76_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( I3 = J2 )
& ( K = L ) )
=> ( ( plus_plus_int @ I3 @ K )
= ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_77_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_78_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_79_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_80_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_81_PiE__cong,axiom,
! [I2: set_nat_nat,A2: ( nat > nat ) > set_nat,B2: ( nat > nat ) > set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat_nat @ I2 @ A2 )
= ( piE_nat_nat_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_82_PiE__cong,axiom,
! [I2: set_nat_nat,A2: ( nat > nat ) > set_nat_nat_nat2,B2: ( nat > nat ) > set_nat_nat_nat2] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_na7569501297962130601at_nat @ I2 @ A2 )
= ( piE_na7569501297962130601at_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_83_PiE__cong,axiom,
! [I2: set_nat,A2: nat > set_nat,B2: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat @ I2 @ A2 )
= ( piE_nat_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_84_PiE__cong,axiom,
! [I2: set_nat,A2: nat > set_nat_nat,B2: nat > set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat_nat2 @ I2 @ A2 )
= ( piE_nat_nat_nat2 @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_85_PiE__cong,axiom,
! [I2: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat] :
( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_na2138371880555796248at_nat @ I2 @ A2 )
= ( piE_na2138371880555796248at_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_86_PiE__cong,axiom,
! [I2: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_na5629913657871898759at_nat @ I2 @ A2 )
= ( piE_na5629913657871898759at_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_87_PiE__cong,axiom,
! [I2: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat] :
( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_na4170927303951785078at_nat @ I2 @ A2 )
= ( piE_na4170927303951785078at_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_88_PiE__cong,axiom,
! [I2: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2] :
( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_na3061559539522535286at_nat @ I2 @ A2 )
= ( piE_na3061559539522535286at_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_89_PiE__cong,axiom,
! [I2: set_na6626867396258451522at_nat,A2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat] :
( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ I2 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_na799184809307736020at_nat @ I2 @ A2 )
= ( piE_na799184809307736020at_nat @ I2 @ B2 ) ) ) ).
% PiE_cong
thf(fact_90_PiE__mem,axiom,
! [F: nat > nat,S: set_nat,T3: nat > set_nat,X4: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S @ T3 ) )
=> ( ( member_nat @ X4 @ S )
=> ( member_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_91_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat,X4: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S @ T3 ) )
=> ( ( member_nat_nat @ X4 @ S )
=> ( member_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_92_PiE__mem,axiom,
! [F: nat > nat > nat,S: set_nat,T3: nat > set_nat_nat,X4: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ S @ T3 ) )
=> ( ( member_nat @ X4 @ S )
=> ( member_nat_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_93_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,S: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat,X4: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F @ ( piE_nat_nat_nat_nat @ S @ T3 ) )
=> ( ( member_nat_nat_nat @ X4 @ S )
=> ( member_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_94_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat,S: set_nat,T3: nat > set_nat_nat_nat2,X4: nat] :
( ( member2740455936716430260at_nat @ F @ ( piE_nat_nat_nat_nat4 @ S @ T3 ) )
=> ( ( member_nat @ X4 @ S )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_95_PiE__mem,axiom,
! [F: nat > nat > nat > nat,S: set_nat,T3: nat > set_nat_nat_nat,X4: nat] :
( ( member17114558718834868at_nat @ F @ ( piE_nat_nat_nat_nat5 @ S @ T3 ) )
=> ( ( member_nat @ X4 @ S )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_96_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat,S: set_nat_nat_nat,T3: ( nat > nat > nat ) > set_nat,X4: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F @ ( piE_nat_nat_nat_nat2 @ S @ T3 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ S )
=> ( member_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_97_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat_nat,X4: nat > nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ S @ T3 ) )
=> ( ( member_nat_nat @ X4 @ S )
=> ( member_nat_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_98_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,S: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat_nat,X4: ( nat > nat ) > nat] :
( ( member4489290058226556451at_nat @ F @ ( piE_na6840239867990089257at_nat @ S @ T3 ) )
=> ( ( member_nat_nat_nat @ X4 @ S )
=> ( member_nat_nat @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_99_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat > nat,S: set_nat_nat,T3: ( nat > nat ) > set_nat_nat_nat,X4: nat > nat] :
( ( member1679187572556404771at_nat @ F @ ( piE_na8678869062391380393at_nat @ S @ T3 ) )
=> ( ( member_nat_nat @ X4 @ S )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( T3 @ X4 ) ) ) ) ).
% PiE_mem
thf(fact_100_PiE__ext,axiom,
! [X4: ( nat > nat ) > nat > nat,K: set_nat_nat,S2: ( nat > nat ) > set_nat_nat,Y2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ ( piE_nat_nat_nat_nat3 @ K @ S2 ) )
=> ( ( member952132173341509300at_nat @ Y2 @ ( piE_nat_nat_nat_nat3 @ K @ S2 ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_101_PiE__ext,axiom,
! [X4: ( nat > nat ) > nat,K: set_nat_nat,S2: ( nat > nat ) > set_nat,Y2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ ( piE_nat_nat_nat @ K @ S2 ) )
=> ( ( member_nat_nat_nat @ Y2 @ ( piE_nat_nat_nat @ K @ S2 ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_102_PiE__ext,axiom,
! [X4: ( nat > nat ) > ( nat > nat ) > nat,K: set_nat_nat,S2: ( nat > nat ) > set_nat_nat_nat2,Y2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X4 @ ( piE_na7569501297962130601at_nat @ K @ S2 ) )
=> ( ( member4402528950554000163at_nat @ Y2 @ ( piE_na7569501297962130601at_nat @ K @ S2 ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_103_PiE__ext,axiom,
! [X4: nat > nat,K: set_nat,S2: nat > set_nat,Y2: nat > nat] :
( ( member_nat_nat @ X4 @ ( piE_nat_nat @ K @ S2 ) )
=> ( ( member_nat_nat @ Y2 @ ( piE_nat_nat @ K @ S2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_104_PiE__ext,axiom,
! [X4: nat > nat > nat,K: set_nat,S2: nat > set_nat_nat,Y2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ ( piE_nat_nat_nat2 @ K @ S2 ) )
=> ( ( member_nat_nat_nat2 @ Y2 @ ( piE_nat_nat_nat2 @ K @ S2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_105_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,K: set_na6626867396258451522at_nat,S2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat] :
( ( member4107745174335074322at_nat @ X4 @ ( piE_na2138371880555796248at_nat @ K @ S2 ) )
=> ( ( member4107745174335074322at_nat @ Y2 @ ( piE_na2138371880555796248at_nat @ K @ S2 ) )
=> ( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_106_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,K: set_na6626867396258451522at_nat,S2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat] :
( ( member1174580258192983937at_nat @ X4 @ ( piE_na5629913657871898759at_nat @ K @ S2 ) )
=> ( ( member1174580258192983937at_nat @ Y2 @ ( piE_na5629913657871898759at_nat @ K @ S2 ) )
=> ( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_107_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,K: set_na6626867396258451522at_nat,S2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member3693257457796161904at_nat @ X4 @ ( piE_na4170927303951785078at_nat @ K @ S2 ) )
=> ( ( member3693257457796161904at_nat @ Y2 @ ( piE_na4170927303951785078at_nat @ K @ S2 ) )
=> ( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_108_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,K: set_na6626867396258451522at_nat,S2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member6416598835793757296at_nat @ X4 @ ( piE_na3061559539522535286at_nat @ K @ S2 ) )
=> ( ( member6416598835793757296at_nat @ Y2 @ ( piE_na3061559539522535286at_nat @ K @ S2 ) )
=> ( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_109_PiE__ext,axiom,
! [X4: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat,K: set_na6626867396258451522at_nat,S2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_nat,Y2: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat] :
( ( member6105598001968527566at_nat @ X4 @ ( piE_na799184809307736020at_nat @ K @ S2 ) )
=> ( ( member6105598001968527566at_nat @ Y2 @ ( piE_na799184809307736020at_nat @ K @ S2 ) )
=> ( ! [I4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I4 @ K )
=> ( ( X4 @ I4 )
= ( Y2 @ I4 ) ) )
=> ( X4 = Y2 ) ) ) ) ).
% PiE_ext
thf(fact_110_split__cube_I1_J,axiom,
! [X4: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X4 @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_111_split__cube_I2_J,axiom,
! [X4: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y: nat] : ( X4 @ ( plus_plus_nat @ Y @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_112_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restri6011711336257459485at_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_113_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restrict_nat_nat_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_114_restrict__apply_H,axiom,
! [X4: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( restrict_nat_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_115_restrict__apply_H,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_116_restrict__apply_H,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( restri4446420529079022766at_nat @ F @ A2 @ X4 )
= ( F @ X4 ) ) ) ).
% restrict_apply'
thf(fact_117_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat,G: ( nat > nat ) > ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri6011711336257459485at_nat @ F @ A2 )
= ( restri6011711336257459485at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_118_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= ( restrict_nat_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_119_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= ( restrict_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_120_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_121_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_122_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_123_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_124_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_125_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_126_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ A @ ( collec3567154360959927026at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_129_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_130_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_131_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_132_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_133_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_136_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_137_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_138__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_139_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_140_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_141_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_142__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_143__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 ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= ( power_power_nat @ ( finite_card_nat @ ( set_ord_lessThan_nat @ r ) ) @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = card {..<r} ^ card (cube m (t + 1))\<close>
thf(fact_144_bij__betw__same__card,axiom,
! [F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( bij_be3563731812766147924at_nat @ F @ A2 @ B2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_145_bij__betw__same__card,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,B2: set_nat_nat] :
( ( bij_be4581752835692700517at_nat @ F @ A2 @ B2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_146_bij__betw__same__card,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_147_bij__betw__same__card,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( bij_be5311014265664741861at_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_148_bij__betw__same__card,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_149_bij__betw__same__card,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_150_bij__betw__same__card,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_151_bij__betw__same__card,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_152_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_153__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_154_card__funcsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat_nat
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
= ( power_power_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_155_card__funcsetE,axiom,
! [A2: set_int,B2: set_nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_card_int_nat
@ ( piE_int_nat @ A2
@ ^ [I: int] : B2 ) )
= ( power_power_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_int @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_156_card__funcsetE,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
= ( power_power_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_157_card__funcsetE,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2904276754548105990at_nat
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
= ( power_power_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_158_card__funcsetE,axiom,
! [A2: set_int,B2: set_nat_nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite5031643484550168418at_nat
@ ( piE_int_nat_nat @ A2
@ ^ [I: int] : B2 ) )
= ( power_power_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_int @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_159_card__funcsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite6187392128852221813at_nat
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
= ( power_power_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_160_card__funcsetE,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite8226521257890822773at_nat
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I: ( nat > nat ) > nat] : B2 ) )
= ( power_power_nat @ ( finite_card_nat @ B2 ) @ ( finite1794908990118856198at_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_161_card__funcsetE,axiom,
! [A2: set_nat,B2: set_nat_nat_nat2] :
( ( finite_finite_nat @ A2 )
=> ( ( finite7975715892227142773at_nat
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I: nat] : B2 ) )
= ( power_power_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_162_card__funcsetE,axiom,
! [A2: set_int,B2: set_nat_nat_nat2] :
( ( finite_finite_int @ A2 )
=> ( ( finite7202386370653918929at_nat
@ ( piE_int_nat_nat_nat @ A2
@ ^ [I: int] : B2 ) )
= ( power_power_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite_card_int @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_163_card__funcsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite4448215331531677284at_nat
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
= ( power_power_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% card_funcsetE
thf(fact_164_n_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [N3: nat] :
( ( ord_less_eq_nat @ n @ N3 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ N3 @ t @ s @ Chi ) ) ) ) ).
% n'_props
thf(fact_165_line__subspace__s,axiom,
! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] :
( ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ n2 @ t @ s @ Chi2 )
& ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ 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_166_m__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ m2 )
& ! [M2: nat] :
( ( ord_less_eq_nat @ m2 @ M2 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ M2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ k @ M2 @ t @ r @ Chi ) ) ) ) ).
% m_props
thf(fact_167_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_168_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_169_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_170__092_060open_0620_A_060_As_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ s ).
% \<open>0 < s\<close>
thf(fact_171_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_172_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_173_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_174_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_175_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_176_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_177_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_178_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_179_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_180_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_181_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_182_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_183_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_184_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_185_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_186_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_187_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_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y2: nat] :
( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_196_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y2 ) )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_197_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_198_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_199_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_200_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_201_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_202_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_203_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_204_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_205_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_206_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_207_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_208_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_209_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_210_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_211_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_lessThan__subset__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X4 ) @ ( set_ord_lessThan_int @ Y2 ) )
= ( ord_less_eq_int @ X4 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_221_lessThan__subset__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_222_lessThan__iff,axiom,
! [I3: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_223_lessThan__iff,axiom,
! [I3: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I3 @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_224_lessThan__iff,axiom,
! [I3: ( nat > nat ) > ( nat > nat ) > nat,K: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I3 @ ( set_or6177432841829679145at_nat @ K ) )
= ( ord_le7877100967975825120at_nat @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_225_lessThan__iff,axiom,
! [I3: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_226_lessThan__iff,axiom,
! [I3: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ ( set_or7562748684798938298at_nat @ K ) )
= ( ord_le4629963735342356977at_nat @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_227_lessThan__iff,axiom,
! [I3: int,K: int] :
( ( member_int @ I3 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_228_lessThan__iff,axiom,
! [I3: nat,K: nat] :
( ( member_nat @ I3 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_229_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_230_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_card_Oinfinite,axiom,
! [A2: set_nat_nat_nat2] :
( ~ ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite1794908990118856198at_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_257_card_Oinfinite,axiom,
! [A2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_card_nat_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_258_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_259_card_Oinfinite,axiom,
! [A2: set_int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_card_int @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_260_assms_I4_J,axiom,
! [K2: nat,R: nat] :
( ( ord_less_eq_nat @ K2 @ k )
=> ( hales_lhj @ R @ t @ K2 ) ) ).
% assms(4)
thf(fact_261_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_262__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,
~ ! [N4: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
& ! [N3: nat] :
( ( ord_less_eq_nat @ N4 @ N3 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ s ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ one_one_nat @ N3 @ t @ s @ Chi ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n'. 0 < n' \<and> (\<forall>N\<ge>n'. \<forall>\<chi>. \<chi> \<in> cube N (t + 1) \<rightarrow>\<^sub>E {..<s} \<longrightarrow> (\<exists>S. layered_subspace S 1 N t s \<chi>)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_263_power__strict__increasing__iff,axiom,
! [B: nat,X4: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X4 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_264_power__strict__increasing__iff,axiom,
! [B: int,X4: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X4 @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_265_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_266_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_267__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,
~ ! [M3: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ! [M2: nat] :
( ( ord_less_eq_nat @ M3 @ M2 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ M2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ k @ M2 @ t @ r @ Chi ) ) ) ) ).
% \<open>\<And>thesis. (\<And>m. 0 < m \<and> (\<forall>M'\<ge>m. \<forall>\<chi>. \<chi> \<in> cube M' (t + 1) \<rightarrow>\<^sub>E {..<r} \<longrightarrow> (\<exists>S. layered_subspace S k M' t r \<chi>)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_268_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_269_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_270_power__increasing__iff,axiom,
! [B: nat,X4: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_271_power__increasing__iff,axiom,
! [B: int,X4: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_293_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_308_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_309_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_310_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_311_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_312_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_313_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_314_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_315_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_316_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_317_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_318_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_319_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_320_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_321_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_322_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_323_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_324_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_325_card__ge__0__finite,axiom,
! [A2: set_nat_nat_nat2] :
( ( ord_less_nat @ zero_zero_nat @ ( finite1794908990118856198at_nat @ A2 ) )
=> ( finite3753911285555252421at_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_326_card__ge__0__finite,axiom,
! [A2: set_nat_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat_nat @ A2 ) )
=> ( finite2115694454571419734at_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_327_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_328_card__ge__0__finite,axiom,
! [A2: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( finite_finite_int @ A2 ) ) ).
% card_ge_0_finite
thf(fact_329_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_330_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M4: nat] :
( ( P @ X4 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_331_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_332_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_333_nat__power__less__imp__less,axiom,
! [I3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M ) @ ( power_power_nat @ I3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_334_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_335_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_int,R2: nat > int > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_336_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_nat,R2: int > nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_337_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_int,R2: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_338_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat,B2: set_nat,R2: ( nat > nat ) > nat > $o] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_339_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat,B2: set_int,R2: ( nat > nat ) > int > $o] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_340_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat,R2: ( ( nat > nat ) > nat ) > nat > $o] :
( ~ ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [A3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_341_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat_nat,B2: set_nat,R2: ( nat > nat > nat ) > nat > $o] :
( ~ ( finite4863279049984502213at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [A3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_342_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat_nat2,B2: set_int,R2: ( ( nat > nat ) > nat ) > int > $o] :
( ~ ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [A3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_343_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_nat_nat,B2: set_int,R2: ( nat > nat > nat ) > int > $o] :
( ~ ( finite4863279049984502213at_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [A3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_344_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I3 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_345_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_346_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_347_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_348_finite__has__minimal2,axiom,
! [A2: set_nat_nat_nat2,A: ( nat > nat ) > nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( member_nat_nat_nat @ A @ A2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( ord_le2017632242545079438at_nat @ X3 @ A )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A2 )
=> ( ( ord_le2017632242545079438at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_349_finite__has__minimal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( ord_less_eq_nat_nat @ X3 @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_350_finite__has__minimal2,axiom,
! [A2: set_na6626867396258451522at_nat,A: ( nat > nat ) > ( nat > nat ) > nat] :
( ( finite9050609800256966691at_nat @ A2 )
=> ( ( member4402528950554000163at_nat @ A @ A2 )
=> ? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
& ( ord_le3015115239550301420at_nat @ X3 @ A )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( ord_le3015115239550301420at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_351_finite__has__minimal2,axiom,
! [A2: set_nat_nat_nat,A: nat > nat > nat] :
( ( finite4863279049984502213at_nat @ A2 )
=> ( ( member_nat_nat_nat2 @ A @ A2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( ord_le3127000006974329230at_nat @ X3 @ A )
& ! [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ A2 )
=> ( ( ord_le3127000006974329230at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_352_finite__has__minimal2,axiom,
! [A2: set_nat_nat_nat_nat3,A: ( nat > nat ) > nat > nat] :
( ( finite7276464013330389940at_nat @ A2 )
=> ( ( member952132173341509300at_nat @ A @ A2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( ord_le747776305331315197at_nat @ X3 @ A )
& ! [Xa: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ Xa @ A2 )
=> ( ( ord_le747776305331315197at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_353_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_354_finite__has__minimal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A2 )
& ( ord_le9059583361652607317at_nat @ X3 @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_355_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_356_finite__has__maximal2,axiom,
! [A2: set_nat_nat_nat2,A: ( nat > nat ) > nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( member_nat_nat_nat @ A @ A2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( ord_le2017632242545079438at_nat @ A @ X3 )
& ! [Xa: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Xa @ A2 )
=> ( ( ord_le2017632242545079438at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_357_finite__has__maximal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( ord_less_eq_nat_nat @ A @ X3 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_358_finite__has__maximal2,axiom,
! [A2: set_na6626867396258451522at_nat,A: ( nat > nat ) > ( nat > nat ) > nat] :
( ( finite9050609800256966691at_nat @ A2 )
=> ( ( member4402528950554000163at_nat @ A @ A2 )
=> ? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
& ( ord_le3015115239550301420at_nat @ A @ X3 )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( ord_le3015115239550301420at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_359_finite__has__maximal2,axiom,
! [A2: set_nat_nat_nat,A: nat > nat > nat] :
( ( finite4863279049984502213at_nat @ A2 )
=> ( ( member_nat_nat_nat2 @ A @ A2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( ord_le3127000006974329230at_nat @ A @ X3 )
& ! [Xa: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Xa @ A2 )
=> ( ( ord_le3127000006974329230at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_360_finite__has__maximal2,axiom,
! [A2: set_nat_nat_nat_nat3,A: ( nat > nat ) > nat > nat] :
( ( finite7276464013330389940at_nat @ A2 )
=> ( ( member952132173341509300at_nat @ A @ A2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( ord_le747776305331315197at_nat @ A @ X3 )
& ! [Xa: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ Xa @ A2 )
=> ( ( ord_le747776305331315197at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_361_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_362_finite__has__maximal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A2 )
& ( ord_le9059583361652607317at_nat @ A @ X3 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_363_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_364_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_365_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_366_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_367_power__strict__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_368_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_369_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_370_power__decreasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_371_power__decreasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_393_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_394_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_395_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_404_add__nonpos__eq__0__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X4 @ Y2 )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_405_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_406_add__nonneg__eq__0__iff,axiom,
! [X4: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X4 @ Y2 )
= zero_zero_int )
= ( ( X4 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_420_card__le__if__inj__on__rel,axiom,
! [B2: set_int,A2: set_nat,R: nat > int > $o] :
( ( finite_finite_int @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B4: int] :
( ( member_int @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: int] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_int @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_421_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_nat,A2: set_nat,R: nat > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B4: nat > nat] :
( ( member_nat_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat > nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_422_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_nat,R: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat > nat] :
( ( member_nat_nat @ A4 @ A2 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat > nat,A22: nat > nat,B5: nat] :
( ( member_nat_nat @ A1 @ A2 )
=> ( ( member_nat_nat @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_423_card__le__if__inj__on__rel,axiom,
! [B2: set_int,A2: set_nat_nat,R: ( nat > nat ) > int > $o] :
( ( finite_finite_int @ B2 )
=> ( ! [A4: nat > nat] :
( ( member_nat_nat @ A4 @ A2 )
=> ? [B4: int] :
( ( member_int @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat > nat,A22: nat > nat,B5: int] :
( ( member_nat_nat @ A1 @ A2 )
=> ( ( member_nat_nat @ A22 @ A2 )
=> ( ( member_int @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_424_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,R: ( nat > nat ) > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ! [A4: nat > nat] :
( ( member_nat_nat @ A4 @ A2 )
=> ? [B4: nat > nat] :
( ( member_nat_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat > nat,A22: nat > nat,B5: nat > nat] :
( ( member_nat_nat @ A1 @ A2 )
=> ( ( member_nat_nat @ A22 @ A2 )
=> ( ( member_nat_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_425_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_nat_nat,A2: set_nat,R: nat > ( nat > nat > nat ) > $o] :
( ( finite4863279049984502213at_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat > nat > nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_nat_nat2 @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite2904276754548105990at_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_426_card__le__if__inj__on__rel,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat,R: nat > ( ( nat > nat ) > nat ) > $o] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat,A22: nat,B5: ( nat > nat ) > nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat_nat_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_427_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_nat_nat,R: ( nat > nat > nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ A4 @ A2 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: nat > nat > nat,A22: nat > nat > nat,B5: nat] :
( ( member_nat_nat_nat2 @ A1 @ A2 )
=> ( ( member_nat_nat_nat2 @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite2904276754548105990at_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_428_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat_nat_nat2,R: ( ( nat > nat ) > nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ A4 @ A2 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B2 )
& ( R @ A4 @ B4 ) ) )
=> ( ! [A1: ( nat > nat ) > nat,A22: ( nat > nat ) > nat,B5: nat] :
( ( member_nat_nat_nat @ A1 @ A2 )
=> ( ( member_nat_nat_nat @ A22 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_429_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_430_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_431_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_432_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_433_bij__betw__finite,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( finite3753911285555252421at_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_434_bij__betw__finite,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( finite2115694454571419734at_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_435_bij__betw__finite,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_436_bij__betw__finite,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_437_bij__betw__finite,axiom,
! [F: nat > int,A2: set_nat,B2: set_int] :
( ( bij_betw_nat_int @ F @ A2 @ B2 )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_int @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_438_bij__betw__finite,axiom,
! [F: int > nat,A2: set_int,B2: set_nat] :
( ( bij_betw_int_nat @ F @ A2 @ B2 )
=> ( ( finite_finite_int @ A2 )
= ( finite_finite_nat @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_439_bij__betw__finite,axiom,
! [F: int > int,A2: set_int,B2: set_int] :
( ( bij_betw_int_int @ F @ A2 @ B2 )
=> ( ( finite_finite_int @ A2 )
= ( finite_finite_int @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_440_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_441_power__strict__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_nat @ N @ N6 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_strict_increasing
thf(fact_442_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_443_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_444_dim1__layered__subspace__as__line,axiom,
! [T: nat,S: ( nat > nat ) > nat > nat,N: nat,R: int,Chi2: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S @ one_one_nat @ N @ T @ R @ Chi2 )
=> ? [C1: int,C22: int] :
( ( ord_less_int @ C1 @ R )
& ( ord_less_int @ C22 @ R )
& ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ( ( Chi2
@ ( S
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= C1 ) )
& ( ( Chi2
@ ( S
@ ( 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_445_dim1__layered__subspace__as__line,axiom,
! [T: nat,S: ( nat > nat ) > nat > nat,N: nat,R: nat,Chi2: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S @ one_one_nat @ N @ T @ R @ Chi2 )
=> ? [C1: nat,C22: nat] :
( ( ord_less_nat @ C1 @ R )
& ( ord_less_nat @ C22 @ R )
& ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ( ( Chi2
@ ( S
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= C1 ) )
& ( ( Chi2
@ ( S
@ ( 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_446_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S: ( nat > nat ) > nat > nat,N: nat,R: int,Chi2: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S @ one_one_nat @ N @ T @ R @ Chi2 )
=> ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ! [L2: nat] :
( ( ord_less_nat @ L2 @ T )
=> ( ( ( Chi2
@ ( S
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= ( Chi2
@ ( S
@ ( 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
@ ( Chi2
@ ( S
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
@ R ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_447_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S: ( nat > nat ) > nat > nat,N: nat,R: nat,Chi2: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S @ one_one_nat @ N @ T @ R @ Chi2 )
=> ! [S5: nat] :
( ( ord_less_nat @ S5 @ T )
=> ! [L2: nat] :
( ( ord_less_nat @ L2 @ T )
=> ( ( ( Chi2
@ ( S
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
= ( Chi2
@ ( S
@ ( 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
@ ( Chi2
@ ( S
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P2 @ zero_zero_nat )
= S5 ) ) ) ) )
@ R ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_448_power__increasing,axiom,
! [N: nat,N6: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_449_power__increasing,axiom,
! [N: nat,N6: nat,A: int] :
( ( ord_less_eq_nat @ N @ N6 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).
% power_increasing
thf(fact_450_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_451_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_452_finite__PiE,axiom,
! [S: set_nat,T3: nat > set_nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( finite_finite_nat @ ( T3 @ I4 ) ) )
=> ( finite2115694454571419734at_nat @ ( piE_nat_nat @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_453_finite__PiE,axiom,
! [S: set_nat,T3: nat > set_int] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( finite_finite_int @ ( T3 @ I4 ) ) )
=> ( finite7161215471916998834at_int @ ( piE_nat_int @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_454_finite__PiE,axiom,
! [S: set_int,T3: int > set_nat] :
( ( finite_finite_int @ S )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S )
=> ( finite_finite_nat @ ( T3 @ I4 ) ) )
=> ( finite3115048166472474290nt_nat @ ( piE_int_nat @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_455_finite__PiE,axiom,
! [S: set_int,T3: int > set_int] :
( ( finite_finite_int @ S )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S )
=> ( finite_finite_int @ ( T3 @ I4 ) ) )
=> ( finite8160569183818053390nt_int @ ( piE_int_int @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_456_finite__PiE,axiom,
! [S: set_nat_nat,T3: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ S )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ S )
=> ( finite_finite_nat @ ( T3 @ I4 ) ) )
=> ( finite3753911285555252421at_nat @ ( piE_nat_nat_nat @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_457_finite__PiE,axiom,
! [S: set_nat_nat,T3: ( nat > nat ) > set_int] :
( ( finite2115694454571419734at_nat @ S )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ S )
=> ( finite_finite_int @ ( T3 @ I4 ) ) )
=> ( finite8799432302900831521at_int @ ( piE_nat_nat_int @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_458_finite__PiE,axiom,
! [S: set_nat,T3: nat > set_nat_nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( finite2115694454571419734at_nat @ ( T3 @ I4 ) ) )
=> ( finite4863279049984502213at_nat @ ( piE_nat_nat_nat2 @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_459_finite__PiE,axiom,
! [S: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat] :
( ( finite3753911285555252421at_nat @ S )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ S )
=> ( finite_finite_nat @ ( T3 @ I4 ) ) )
=> ( finite92221105514215092at_nat @ ( piE_nat_nat_nat_nat @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_460_finite__PiE,axiom,
! [S: set_nat_nat_nat,T3: ( nat > nat > nat ) > set_nat] :
( ( finite4863279049984502213at_nat @ S )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ S )
=> ( finite_finite_nat @ ( T3 @ I4 ) ) )
=> ( finite2419275489879724980at_nat @ ( piE_nat_nat_nat_nat2 @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_461_finite__PiE,axiom,
! [S: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_int] :
( ( finite3753911285555252421at_nat @ S )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ S )
=> ( finite_finite_int @ ( T3 @ I4 ) ) )
=> ( finite5137742122859794192at_int @ ( piE_nat_nat_nat_int @ S @ T3 ) ) ) ) ).
% finite_PiE
thf(fact_462_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_463_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_464_line__points__in__cube__unfolded,axiom,
! [L3: nat > nat > nat,N: nat,T: nat,S2: nat,J2: nat] :
( ( hales_is_line @ L3 @ N @ T )
=> ( ( ord_less_nat @ S2 @ T )
=> ( ( ord_less_nat @ J2 @ N )
=> ( member_nat @ ( L3 @ S2 @ J2 ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_465_line__points__in__cube,axiom,
! [L3: nat > nat > nat,N: nat,T: nat,S2: nat] :
( ( hales_is_line @ L3 @ N @ T )
=> ( ( ord_less_nat @ S2 @ T )
=> ( member_nat_nat @ ( L3 @ S2 ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_466_cube__props_I4_J,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S2 ) ) )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% cube_props(4)
thf(fact_467_cube__props_I2_J,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S2 ) )
@ zero_zero_nat )
= S2 ) ) ).
% cube_props(2)
thf(fact_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_cube__props_I1_J,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X3 @ zero_zero_nat )
= S2 ) ) ) ).
% cube_props(1)
thf(fact_480_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_481_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_482_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_483_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( I3 = J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_484_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_485_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_486_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_487_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_488_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_489_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_490_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_491_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_492_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_493_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_494_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_495_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_496_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_497_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_498_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_499_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( I3 = J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_500_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_501_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_510_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_511_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_512_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_513_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_514_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_515_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_516_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_517_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_518_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_519_infinite__Iio,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% infinite_Iio
thf(fact_520_cube__props_I3_J,axiom,
! [S2: nat,T: nat,S: ( nat > nat ) > nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( restrict_nat_nat
@ ^ [S4: nat] :
( S
@ ( 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 )
@ S2 )
= ( restrict_nat_nat
@ ^ [S4: nat] :
( S
@ ( 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 )
= S2 ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_521_cube__props_I3_J,axiom,
! [S2: nat,T: nat,S: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S
@ ( 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 )
@ S2 )
= ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S
@ ( 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 )
= S2 ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_522_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_523_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_524_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_525_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_526_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_527_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_528_is__line__def,axiom,
( hales_is_line
= ( ^ [L4: nat > nat > nat,N2: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L4
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I: nat] : ( hales_cube @ N2 @ T2 ) ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ N2 )
=> ( ! [X2: nat] :
( ( ord_less_nat @ X2 @ T2 )
=> ! [Y: nat] :
( ( ord_less_nat @ Y @ T2 )
=> ( ( L4 @ X2 @ J )
= ( L4 @ Y @ J ) ) ) )
| ! [S4: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( ( L4 @ S4 @ J )
= S4 ) ) ) )
& ? [J: nat] :
( ( ord_less_nat @ J @ N2 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( ( L4 @ S4 @ J )
= S4 ) ) ) ) ) ) ).
% is_line_def
thf(fact_529_one__dim__cube__eq__nat__set,axiom,
! [K: nat] :
( bij_betw_nat_nat_nat
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ ( hales_cube @ one_one_nat @ K )
@ ( set_ord_lessThan_nat @ K ) ) ).
% one_dim_cube_eq_nat_set
thf(fact_530_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_531_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_532_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_533_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_534_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_int @ B2 ) )
=> ? [H: nat > int] : ( bij_betw_nat_int @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_535_finite__same__card__bij,axiom,
! [A2: set_int,B2: set_nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_int @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: int > nat] : ( bij_betw_int_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_536_finite__same__card__bij,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( finite_card_int @ A2 )
= ( finite_card_int @ B2 ) )
=> ? [H: int > int] : ( bij_betw_int_int @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_537_finite__same__card__bij,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_538_finite__same__card__bij,axiom,
! [A2: set_nat_nat,B2: set_int] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_int @ B2 ) )
=> ? [H: ( nat > nat ) > int] : ( bij_betw_nat_nat_int @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_539_finite__same__card__bij,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) )
=> ? [H: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_540_finite__same__card__bij,axiom,
! [A2: set_int,B2: set_nat_nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ( finite_card_int @ A2 )
= ( finite_card_nat_nat @ B2 ) )
=> ? [H: int > nat > nat] : ( bij_betw_int_nat_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_541_finite__same__card__bij,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) )
=> ? [H: ( nat > nat ) > nat > nat] : ( bij_be5678534868967705974at_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_542_finite__same__card__bij,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ? [H: ( ( nat > nat ) > nat ) > nat] : ( bij_be1059735840858801910at_nat @ H @ A2 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_543_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: nat > nat] : ( bij_betw_nat_nat @ F2 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_544_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ? [F2: nat > int] : ( bij_betw_nat_int @ F2 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_int @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_545_bij__betw__iff__card,axiom,
! [A2: set_int,B2: set_nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: int > nat] : ( bij_betw_int_nat @ F2 @ A2 @ B2 ) )
= ( ( finite_card_int @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_546_bij__betw__iff__card,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ? [F2: int > int] : ( bij_betw_int_int @ F2 @ A2 @ B2 ) )
= ( ( finite_card_int @ A2 )
= ( finite_card_int @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_547_bij__betw__iff__card,axiom,
! [A2: set_nat_nat,B2: set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: ( nat > nat ) > nat] : ( bij_betw_nat_nat_nat @ F2 @ A2 @ B2 ) )
= ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_548_bij__betw__iff__card,axiom,
! [A2: set_nat_nat,B2: set_int] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ? [F2: ( nat > nat ) > int] : ( bij_betw_nat_nat_int @ F2 @ A2 @ B2 ) )
= ( ( finite_card_nat_nat @ A2 )
= ( finite_card_int @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_549_bij__betw__iff__card,axiom,
! [A2: set_nat,B2: set_nat_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ? [F2: nat > nat > nat] : ( bij_betw_nat_nat_nat2 @ F2 @ A2 @ B2 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_550_bij__betw__iff__card,axiom,
! [A2: set_int,B2: set_nat_nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ? [F2: int > nat > nat] : ( bij_betw_int_nat_nat @ F2 @ A2 @ B2 ) )
= ( ( finite_card_int @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_551_bij__betw__iff__card,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ? [F2: ( nat > nat ) > nat > nat] : ( bij_be5678534868967705974at_nat @ F2 @ A2 @ B2 ) )
= ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_552_bij__betw__iff__card,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: ( ( nat > nat ) > nat ) > nat] : ( bij_be1059735840858801910at_nat @ F2 @ A2 @ B2 ) )
= ( ( finite1794908990118856198at_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_553_cube__card,axiom,
! [N: nat,T: nat] :
( ( finite_card_nat_nat
@ ( piE_nat_nat @ ( set_ord_lessThan_nat @ N )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T ) ) )
= ( power_power_nat @ T @ N ) ) ).
% cube_card
thf(fact_554_cube__restrict,axiom,
! [J2: nat,N: nat,Y2: nat > nat,T: nat] :
( ( ord_less_nat @ J2 @ N )
=> ( ( member_nat_nat @ Y2 @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y2 @ ( set_ord_lessThan_nat @ J2 ) ) @ ( hales_cube @ J2 @ T ) ) ) ) ).
% cube_restrict
thf(fact_555_nat__set__eq__one__dim__cube,axiom,
! [K: nat] :
( bij_betw_nat_nat_nat2
@ ^ [X2: nat] :
( restrict_nat_nat
@ ^ [Y: 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_556_dim0__layered__subspace__ex,axiom,
! [Chi2: ( nat > nat ) > nat,N: nat,T: nat,R: nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ zero_zero_nat @ N @ T @ R @ Chi2 ) ) ).
% dim0_layered_subspace_ex
thf(fact_557_lhj__def,axiom,
( hales_lhj
= ( ^ [R3: nat,T2: nat,K3: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [S6: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S6 @ K3 @ N8 @ T2 @ R3 @ Chi3 ) ) ) ) ) ) ).
% lhj_def
thf(fact_558_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_559_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_560_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_561_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_562_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_563_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_564_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_565_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_566_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_567_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_568_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_569_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_570_finite__Diff,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_571_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_572_finite__Diff2,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
= ( finite_finite_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_573_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_574_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_575_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_576_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_577_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_578_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_579_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_580_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_581_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_582_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_583_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_584_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_585_diff__diff__left,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_586_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_587_finite__Collect__subsets,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_588_finite__Collect__subsets,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( finite3586981331298542604at_nat
@ ( collect_set_nat_nat
@ ^ [B6: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B6 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_589_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_590_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_591_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat,C4: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_592_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_593_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B2 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_594_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_595_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat_nat2,C4: ( nat > nat ) > set_nat_nat_nat2] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le5934964663421696068at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le973658574027395234at_nat @ ( piE_na7569501297962130601at_nat @ A2 @ B2 ) @ ( piE_na7569501297962130601at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_596_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( 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 @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B2 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_597_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B2: ( 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 @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B2 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_598_PiE__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat,C4: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le8898325182481281041at_nat @ ( piE_na2138371880555796248at_nat @ A2 @ B2 ) @ ( piE_na2138371880555796248at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_599_PiE__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le9041126503034175505at_nat @ ( piE_na6564615839001774232at_nat @ A2 @ B2 ) @ ( piE_na6564615839001774232at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_600_PiE__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le4724818764771537408at_nat @ ( piE_na5629913657871898759at_nat @ A2 @ B2 ) @ ( piE_na5629913657871898759at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_601_Diff__infinite__finite,axiom,
! [T3: set_nat,S: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_602_Diff__infinite__finite,axiom,
! [T3: set_int,S: set_int] :
( ( finite_finite_int @ T3 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_603_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A5 )
=> ( P @ B7 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_604_finite__psubset__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A5: set_int] :
( ( finite_finite_int @ A5 )
=> ( ! [B7: set_int] :
( ( ord_less_set_int @ B7 @ A5 )
=> ( P @ B7 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_605_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_606_finite__subset,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( finite_finite_int @ B2 )
=> ( finite_finite_int @ A2 ) ) ) ).
% finite_subset
thf(fact_607_finite__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_608_infinite__super,axiom,
! [S: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_609_infinite__super,axiom,
! [S: set_int,T3: set_int] :
( ( ord_less_eq_set_int @ S @ T3 )
=> ( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ T3 ) ) ) ).
% infinite_super
thf(fact_610_infinite__super,axiom,
! [S: set_nat_nat,T3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ T3 )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_611_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_612_rev__finite__subset,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( finite_finite_int @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_613_rev__finite__subset,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_614_card__Diff__subset,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ A2 )
=> ( ( finite1794908990118856198at_nat @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_615_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_616_card__Diff__subset,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_617_card__Diff__subset,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_618_card__psubset,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) )
=> ( ord_le371403230139555384at_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_619_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_620_card__psubset,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
=> ( ord_less_set_int @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_621_card__psubset,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_622_infinite__arbitrarily__large,axiom,
! [A2: set_nat_nat_nat2,N: nat] :
( ~ ( finite3753911285555252421at_nat @ A2 )
=> ? [B8: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B8 )
& ( ( finite1794908990118856198at_nat @ B8 )
= N )
& ( ord_le5934964663421696068at_nat @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_623_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N )
& ( ord_less_eq_set_nat @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_624_infinite__arbitrarily__large,axiom,
! [A2: set_int,N: nat] :
( ~ ( finite_finite_int @ A2 )
=> ? [B8: set_int] :
( ( finite_finite_int @ B8 )
& ( ( finite_card_int @ B8 )
= N )
& ( ord_less_eq_set_int @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_625_infinite__arbitrarily__large,axiom,
! [A2: set_nat_nat,N: nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ? [B8: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B8 )
& ( ( finite_card_nat_nat @ B8 )
= N )
& ( ord_le9059583361652607317at_nat @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_626_card__subset__eq,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( ( finite1794908990118856198at_nat @ A2 )
= ( finite1794908990118856198at_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_627_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_628_card__subset__eq,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ( finite_card_int @ A2 )
= ( finite_card_int @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_629_card__subset__eq,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_630_card__less__sym__Diff,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_less_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) )
=> ( ord_less_nat @ ( finite1794908990118856198at_nat @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) ) @ ( finite1794908990118856198at_nat @ ( minus_1221035652888719293at_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_631_card__less__sym__Diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_632_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_633_card__less__sym__Diff,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_634_card__le__sym__Diff,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) ) @ ( finite1794908990118856198at_nat @ ( minus_1221035652888719293at_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_635_card__le__sym__Diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_636_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_637_card__le__sym__Diff,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_638_psubset__card__mono,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_639_psubset__card__mono,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_640_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_641_psubset__card__mono,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_set_int @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_642_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_643_linorder__neqE__linordered__idom,axiom,
! [X4: int,Y2: int] :
( ( X4 != Y2 )
=> ( ~ ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ Y2 @ X4 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_644_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat_nat_nat2,C4: nat] :
( ! [G2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ G2 @ F4 )
=> ( ( finite3753911285555252421at_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ G2 ) @ C4 ) ) )
=> ( ( finite3753911285555252421at_nat @ F4 )
& ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_645_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C4: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F4 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C4 ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_646_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_int,C4: nat] :
( ! [G2: set_int] :
( ( ord_less_eq_set_int @ G2 @ F4 )
=> ( ( finite_finite_int @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_int @ G2 ) @ C4 ) ) )
=> ( ( finite_finite_int @ F4 )
& ( ord_less_eq_nat @ ( finite_card_int @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_647_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat_nat,C4: nat] :
( ! [G2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ G2 @ F4 )
=> ( ( finite2115694454571419734at_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ G2 ) @ C4 ) ) )
=> ( ( finite2115694454571419734at_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ F4 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_648_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat_nat_nat2] :
( ( ord_less_eq_nat @ N @ ( finite1794908990118856198at_nat @ S ) )
=> ~ ! [T4: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ T4 @ S )
=> ( ( ( finite1794908990118856198at_nat @ T4 )
= N )
=> ~ ( finite3753911285555252421at_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_649_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_650_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
=> ~ ! [T4: set_int] :
( ( ord_less_eq_set_int @ T4 @ S )
=> ( ( ( finite_card_int @ T4 )
= N )
=> ~ ( finite_finite_int @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_651_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ S ) )
=> ~ ! [T4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ T4 @ S )
=> ( ( ( finite_card_nat_nat @ T4 )
= N )
=> ~ ( finite2115694454571419734at_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_652_exists__subset__between,axiom,
! [A2: set_nat_nat_nat2,N: nat,C4: set_nat_nat_nat2] :
( ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite1794908990118856198at_nat @ C4 ) )
=> ( ( ord_le5934964663421696068at_nat @ A2 @ C4 )
=> ( ( finite3753911285555252421at_nat @ C4 )
=> ? [B8: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ A2 @ B8 )
& ( ord_le5934964663421696068at_nat @ B8 @ C4 )
& ( ( finite1794908990118856198at_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_653_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C4: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C4 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( finite_finite_nat @ C4 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C4 )
& ( ( finite_card_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_654_exists__subset__between,axiom,
! [A2: set_int,N: nat,C4: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_int @ C4 ) )
=> ( ( ord_less_eq_set_int @ A2 @ C4 )
=> ( ( finite_finite_int @ C4 )
=> ? [B8: set_int] :
( ( ord_less_eq_set_int @ A2 @ B8 )
& ( ord_less_eq_set_int @ B8 @ C4 )
& ( ( finite_card_int @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_655_exists__subset__between,axiom,
! [A2: set_nat_nat,N: nat,C4: set_nat_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ C4 ) )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C4 )
=> ( ( finite2115694454571419734at_nat @ C4 )
=> ? [B8: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B8 )
& ( ord_le9059583361652607317at_nat @ B8 @ C4 )
& ( ( finite_card_nat_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_656_card__seteq,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite1794908990118856198at_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_657_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_658_card__seteq,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_659_card__seteq,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_660_card__mono,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_661_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_662_card__mono,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ).
% card_mono
thf(fact_663_card__mono,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_664_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_665_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_666_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_667_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_668_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_669_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_670_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
=> ( P @ M6 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_671_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_672_linorder__neqE__nat,axiom,
! [X4: nat,Y2: nat] :
( ( X4 != Y2 )
=> ( ~ ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ Y2 @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_673_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_674_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_675_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_676_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_677_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_678_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_679_diff__commute,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_680_diff__card__le__card__Diff,axiom,
! [B2: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( finite3753911285555252421at_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite1794908990118856198at_nat @ A2 ) @ ( finite1794908990118856198at_nat @ B2 ) ) @ ( finite1794908990118856198at_nat @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_681_diff__card__le__card__Diff,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B2 ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_682_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_683_diff__card__le__card__Diff,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_684_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_685_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_686_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_687_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_688_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_689_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_690_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_691_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_692_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_693_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_694_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_695_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_696_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_697_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_698_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_699_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_700_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_701_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_702_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_703_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_704_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_705_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_706_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_707_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( M5 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_708_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_709_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
| ( M5 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_710_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_711_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_712_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_713_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_714_trans__less__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_715_trans__less__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_716_add__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_717_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_718_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_719_add__less__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_720_add__lessD1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_721_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_722_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_723_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_724_trans__le__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_725_trans__le__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_726_add__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_727_add__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_728_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_729_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_730_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_731_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_732_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_733_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_734_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_735_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_736_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_737_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_738_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_739_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_740_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_741_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_742_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_743_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_744_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_745_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_746_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_747_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_748_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_749_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_750_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_751_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_752_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_753_add__less__zeroD,axiom,
! [X4: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X4 @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X4 @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_754_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_755_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_756_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_757_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_758_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_759_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_760_add__le__add__imp__diff__le,axiom,
! [I3: nat,K: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_761_add__le__add__imp__diff__le,axiom,
! [I3: int,K: int,N: int,J2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_762_add__le__imp__le__diff,axiom,
! [I3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_763_add__le__imp__le__diff,axiom,
! [I3: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
=> ( ord_less_eq_int @ I3 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_764_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_765_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_766_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_767_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_768_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_769_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_770_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_771_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I3 @ K4 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_772_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_773_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_774_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_775_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_776_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_777_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_778_less__diff__conv,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_779_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_780_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_781_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_782_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_783_le__diff__conv,axiom,
! [J2: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_784_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_785_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_786_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_787_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_788_less__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_789_hj__imp__lhj__base,axiom,
! [T: nat,R: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ! [R4: nat] : ( hales_hj @ R4 @ T )
=> ( hales_lhj @ R @ T @ one_one_nat ) ) ) ).
% hj_imp_lhj_base
thf(fact_790_psubsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_791_dim1__subspace__is__line,axiom,
! [T: nat,S: ( nat > nat ) > nat > nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_subspace @ S @ one_one_nat @ N @ T )
=> ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S
@ ( 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_792_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_793_some__sym__eq__trivial,axiom,
! [X4: nat > nat] :
( ( fChoice_nat_nat
@ ( ^ [Y3: nat > nat,Z: nat > nat] : ( Y3 = Z )
@ X4 ) )
= X4 ) ).
% some_sym_eq_trivial
thf(fact_794_subsetI,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ X3 @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_795_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_796_subsetI,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A2 )
=> ( member4402528950554000163at_nat @ X3 @ B2 ) )
=> ( ord_le973658574027395234at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_797_subsetI,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ X3 @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_798_subsetI,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( member952132173341509300at_nat @ X3 @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_799_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ X3 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_800_subset__antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_801_Diff__iff,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat_nat @ C @ A2 )
& ~ ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_802_Diff__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat @ C @ A2 )
& ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_803_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_804_Diff__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
= ( ( member4402528950554000163at_nat @ C @ A2 )
& ~ ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_805_Diff__iff,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat_nat2 @ C @ A2 )
& ~ ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_806_Diff__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
= ( ( member952132173341509300at_nat @ C @ A2 )
& ~ ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_807_DiffI,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat_nat @ C @ B2 )
=> ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_808_DiffI,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_809_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_810_DiffI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A2 )
=> ( ~ ( member4402528950554000163at_nat @ C @ B2 )
=> ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_811_DiffI,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A2 )
=> ( ~ ( member_nat_nat_nat2 @ C @ B2 )
=> ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_812_DiffI,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ A2 )
=> ( ~ ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_813_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_814_some__eq__trivial,axiom,
! [X4: nat > nat] :
( ( fChoice_nat_nat
@ ^ [Y: nat > nat] : ( Y = X4 ) )
= X4 ) ).
% some_eq_trivial
thf(fact_815_set__diff__eq,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A6 )
& ~ ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_816_set__diff__eq,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A6 )
& ~ ( member_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_817_set__diff__eq,axiom,
( minus_5225787954611647771at_nat
= ( ^ [A6: set_na6626867396258451522at_nat,B6: set_na6626867396258451522at_nat] :
( collec2410089373097230945at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A6 )
& ~ ( member4402528950554000163at_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_818_set__diff__eq,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A6 )
& ~ ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_819_set__diff__eq,axiom,
( minus_4646100876039749548at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A6 )
& ~ ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_820_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_821_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A6 )
& ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_822_minus__set__def,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ( minus_2851842960567056136_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_823_minus__set__def,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( collect_nat_nat
@ ( minus_167519014754328503_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A6 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_824_minus__set__def,axiom,
( minus_5225787954611647771at_nat
= ( ^ [A6: set_na6626867396258451522at_nat,B6: set_na6626867396258451522at_nat] :
( collec2410089373097230945at_nat
@ ( minus_6692596912184789802_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_825_minus__set__def,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ( minus_7240682219522218504_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A6 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_826_minus__set__def,axiom,
( minus_4646100876039749548at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( collec3567154360959927026at_nat
@ ( minus_7158188067284919257_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_827_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_828_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_829_DiffD2,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_830_DiffD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_831_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_832_DiffD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
=> ~ ( member4402528950554000163at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_833_DiffD2,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat_nat2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_834_DiffD2,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ~ ( member952132173341509300at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_835_DiffD1,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ( member_nat_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_836_DiffD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_837_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_838_DiffD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
=> ( member4402528950554000163at_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_839_DiffD1,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ( member_nat_nat_nat2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_840_DiffD1,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ( member952132173341509300at_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_841_DiffE,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_842_DiffE,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_843_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_844_DiffE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A2 @ B2 ) )
=> ~ ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_845_DiffE,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_846_DiffE,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_847_in__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,X4: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_848_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_849_in__mono,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat,X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A2 @ B2 )
=> ( ( member4402528950554000163at_nat @ X4 @ A2 )
=> ( member4402528950554000163at_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_850_in__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,X4: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_851_in__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,X4: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ X4 @ A2 )
=> ( member952132173341509300at_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_852_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X4: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_853_subsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_854_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_855_subsetD,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A2 @ B2 )
=> ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_856_subsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_857_subsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_858_subsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_859_Diff__mono,axiom,
! [A2: set_nat_nat,C4: set_nat_nat,D3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C4 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ ( minus_8121590178497047118at_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_860_equalityE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_861_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A6 )
=> ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_862_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_863_subset__eq,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A6: set_na6626867396258451522at_nat,B6: set_na6626867396258451522at_nat] :
! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A6 )
=> ( member4402528950554000163at_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_864_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A6 )
=> ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_865_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A6 )
=> ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_866_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A6 )
=> ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_867_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_868_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_869_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A6 )
=> ( member_nat_nat_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_870_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_871_subset__iff,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A6: set_na6626867396258451522at_nat,B6: set_na6626867396258451522at_nat] :
! [T2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ T2 @ A6 )
=> ( member4402528950554000163at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_872_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A6 )
=> ( member_nat_nat_nat2 @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_873_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
! [T2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T2 @ A6 )
=> ( member952132173341509300at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_874_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A6 )
=> ( member_nat_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_875_Diff__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_876_double__diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ( minus_8121590178497047118at_nat @ B2 @ ( minus_8121590178497047118at_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_877_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_878_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_879_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_880_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_881_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_882_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_nat,Z: set_nat_nat] : ( Y3 = Z ) )
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_883_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_884_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_885_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_886_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_887_psubsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_888_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_889_psubsetD,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le2785809691299232406at_nat @ A2 @ B2 )
=> ( ( member4402528950554000163at_nat @ C @ A2 )
=> ( member4402528950554000163at_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_890_psubsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_891_psubsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_892_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ? [B5: ( nat > nat ) > nat] : ( member_nat_nat_nat @ B5 @ ( minus_1221035652888719293at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_893_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ? [B5: nat > nat] : ( member_nat_nat @ B5 @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_894_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_895_psubset__imp__ex__mem,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( ord_le2785809691299232406at_nat @ A2 @ B2 )
=> ? [B5: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ B5 @ ( minus_5225787954611647771at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_896_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ? [B5: nat > nat > nat] : ( member_nat_nat_nat2 @ B5 @ ( minus_7721066311745899709at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_897_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ? [B5: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ B5 @ ( minus_4646100876039749548at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_898_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S3 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_899_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_900_tfl__some,axiom,
! [P3: ( nat > nat ) > $o,X: nat > nat] :
( ( P3 @ X )
=> ( P3 @ ( fChoice_nat_nat @ P3 ) ) ) ).
% tfl_some
thf(fact_901_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_902_someI,axiom,
! [P: ( nat > nat ) > $o,X4: nat > nat] :
( ( P @ X4 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI
thf(fact_903_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_904_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_905_less__eq__set__def,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A6: set_na6626867396258451522at_nat,B6: set_na6626867396258451522at_nat] :
( ord_le319988079983864419_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_906_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A6 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_907_less__eq__set__def,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( ord_le5430825838364970130_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_908_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A6 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_909_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_910_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_911_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_912_Collect__subset,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_913_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_914_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_915_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_916_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_917_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A6 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_918_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_919_less__set__def,axiom,
( ord_le2785809691299232406at_nat
= ( ^ [A6: set_na6626867396258451522at_nat,B6: set_na6626867396258451522at_nat] :
( ord_le6599672692516096367_nat_o
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_920_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A6 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_921_less__set__def,axiom,
( ord_le6177938698872215975at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( ord_le4961065272816086430_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A6 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_922_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_923_someI__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ? [X_12: nat > nat] : ( P @ X_12 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI_ex
thf(fact_924_someI2__ex,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X_12: nat > nat] : ( P @ X_12 )
=> ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_925_someI2__bex,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( ! [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_926_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) )
=> ( ! [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_927_someI2__bex,axiom,
! [A2: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ? [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A2 )
& ( P @ X ) )
=> ( ! [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_928_someI2__bex,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
& ( P @ X ) )
=> ( ! [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_929_someI2__bex,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ? [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
& ( P @ X ) )
=> ( ! [X3: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoic52552927678224201at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_930_someI2__bex,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( ! [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_931_some__eq__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
% some_eq_ex
thf(fact_932_some1__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ? [X: nat > nat] :
( ( P @ X )
& ! [Y4: nat > nat] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( ( P @ A )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_933_psubsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_934_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_935_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_936_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_937_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ~ ( ord_le9059583361652607317at_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_938_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_939_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_less_set_nat_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_940_line__is__dim1__subspace,axiom,
! [N: nat,T: nat,L3: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_line @ L3 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L3 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_941_line__is__dim1__subspace__t__1,axiom,
! [N: nat,L3: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( hales_is_line @ L3 @ N @ one_one_nat )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L3 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ one_one_nat ) )
@ one_one_nat
@ N
@ one_one_nat ) ) ) ).
% line_is_dim1_subspace_t_1
thf(fact_942_line__is__dim1__subspace__t__ge__1,axiom,
! [N: nat,T: nat,L3: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ T )
=> ( ( hales_is_line @ L3 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y: nat > nat] : ( L3 @ ( Y @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_943_classes__subset__cube,axiom,
! [N: nat,T: nat,I3: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I3 ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_944_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat,X4: nat > nat,Y2: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( plus_plus_nat @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_945_sum_Ofinite__Collect__op,axiom,
! [I2: set_int,X4: int > nat,Y2: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I2 )
& ( ( plus_plus_nat @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_946_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat,X4: nat > int,Y2: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( plus_plus_int @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_947_sum_Ofinite__Collect__op,axiom,
! [I2: set_int,X4: int > int,Y2: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I2 )
& ( ( plus_plus_int @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_948_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat_nat,X4: ( nat > nat ) > nat,Y2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I2 )
& ( ( plus_plus_nat @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_949_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat_nat,X4: ( nat > nat ) > int,Y2: ( nat > nat ) > int] :
( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_int ) ) ) )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [I: nat > nat] :
( ( member_nat_nat @ I @ I2 )
& ( ( plus_plus_int @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_950_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat_nat_nat2,X4: ( ( nat > nat ) > nat ) > nat,Y2: ( ( nat > nat ) > nat ) > nat] :
( ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I2 )
& ( ( plus_plus_nat @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_951_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat_nat_nat,X4: ( nat > nat > nat ) > nat,Y2: ( nat > nat > nat ) > nat] :
( ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_nat ) ) ) )
=> ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I2 )
& ( ( plus_plus_nat @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_952_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat_nat_nat2,X4: ( ( nat > nat ) > nat ) > int,Y2: ( ( nat > nat ) > nat ) > int] :
( ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_int ) ) ) )
=> ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [I: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ I2 )
& ( ( plus_plus_int @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_953_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat_nat_nat,X4: ( nat > nat > nat ) > int,Y2: ( nat > nat > nat ) > int] :
( ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_int ) ) ) )
=> ( ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I2 )
& ( ( Y2 @ I )
!= zero_zero_int ) ) ) )
=> ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [I: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ I2 )
& ( ( plus_plus_int @ ( X4 @ I ) @ ( Y2 @ I ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_954_hj__def,axiom,
( hales_hj
= ( ^ [R3: 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 )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [L4: nat > nat > nat,C3: nat] :
( ( ord_less_nat @ C3 @ R3 )
& ( hales_is_line @ L4 @ N8 @ T2 )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( Chi3 @ X2 )
= C3 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_955_image__eqI,axiom,
! [B: nat,F: nat > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_956_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_957_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_958_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_959_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_960_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_961_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X4: nat,A2: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_962_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X4: nat > nat > nat,A2: set_nat_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_963_image__eqI,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_964_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat,X4: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_965_image__ident,axiom,
! [Y6: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_966_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_967_finite__imageI,axiom,
! [F4: set_nat_nat,H2: ( nat > nat ) > nat > nat] :
( ( finite2115694454571419734at_nat @ F4 )
=> ( finite2115694454571419734at_nat @ ( image_3205354838064109189at_nat @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_968_finite__imageI,axiom,
! [F4: set_nat,H2: nat > nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_969_finite__imageI,axiom,
! [F4: set_nat,H2: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_970_finite__imageI,axiom,
! [F4: set_nat,H2: nat > int] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_int @ ( image_nat_int @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_971_finite__imageI,axiom,
! [F4: set_int,H2: int > nat] :
( ( finite_finite_int @ F4 )
=> ( finite_finite_nat @ ( image_int_nat @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_972_finite__imageI,axiom,
! [F4: set_int,H2: int > int] :
( ( finite_finite_int @ F4 )
=> ( finite_finite_int @ ( image_int_int @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_973_image__restrict__eq,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_1991755285388994676at_nat @ ( restri6011711336257459485at_nat @ F @ A2 ) @ A2 )
= ( image_1991755285388994676at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_974_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_975_image__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_976_image__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_977_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_978_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_979_image__add__0,axiom,
! [S: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S )
= S ) ).
% image_add_0
thf(fact_980_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_981_subset__image__iff,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_982_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B2
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_983_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_984_image__subset__iff,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_985_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_986_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_987_subset__imageE,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_988_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B2
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_989_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_990_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_991_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_992_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B2: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_993_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_994_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_995_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B2: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_996_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_997_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_998_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_3101123049818244468at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_999_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1000_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1001_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1002_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1003_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1004_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1005_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1006_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1007_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1008_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1009_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1010_imageE,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1011_imageE,axiom,
! [B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_1012_image__image,axiom,
! [F: ( nat > nat ) > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1013_image__image,axiom,
! [F: nat > nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1014_image__image,axiom,
! [F: nat > nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1015_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1016_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_3205354838064109189at_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1017_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_1018_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1019_Compr__image__eq,axiom,
! [F: int > nat,A2: set_int,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_int_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_nat @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1020_Compr__image__eq,axiom,
! [F: nat > int,A2: set_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1021_Compr__image__eq,axiom,
! [F: int > int,A2: set_int,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_int_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_int @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1022_Compr__image__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1023_Compr__image__eq,axiom,
! [F: int > nat > nat,A2: set_int,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_int_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_nat_nat @ F
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1024_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1025_Compr__image__eq,axiom,
! [F: ( nat > nat ) > int,A2: set_nat_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_nat_int @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_int @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1026_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1027_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1028_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1029_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1030_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1031_rev__image__eqI,axiom,
! [X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1032_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1033_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1034_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1035_rev__image__eqI,axiom,
! [X4: nat > nat > nat,A2: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1036_rev__image__eqI,axiom,
! [X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_1262493855416953332at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1037_rev__image__eqI,axiom,
! [X4: nat > nat,A2: set_nat_nat,B: ( nat > nat ) > nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat @ B @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1038_ball__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_1039_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_1040_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_1041_image__cong,axiom,
! [M4: set_nat_nat,N6: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M4 = N6 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M4 )
= ( image_3205354838064109189at_nat @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_1042_image__cong,axiom,
! [M4: set_nat,N6: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M4 = N6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M4 )
= ( image_nat_nat_nat2 @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_1043_image__cong,axiom,
! [M4: set_nat,N6: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M4 )
= ( image_nat_nat @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_1044_bex__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1045_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1046_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1047_image__iff,axiom,
! [Z2: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ Z2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1048_image__iff,axiom,
! [Z2: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1049_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1050_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1051_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1052_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_1053_imageI,axiom,
! [X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1054_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1055_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1056_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1057_imageI,axiom,
! [X4: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1058_imageI,axiom,
! [X4: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_1262493855416953332at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1059_imageI,axiom,
! [X4: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( image_1991755285388994676at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_1060_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1061_all__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1062_all__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ A2 )
=> ( P @ ( image_3205354838064109189at_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1063_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1064_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1065_pigeonhole__infinite,axiom,
! [A2: set_int,F: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1066_pigeonhole__infinite,axiom,
! [A2: set_int,F: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ ( image_int_int @ F @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1067_pigeonhole__infinite,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat_nat @ F @ A2 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1068_pigeonhole__infinite,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > int] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite_finite_int @ ( image_nat_nat_int @ F @ A2 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1069_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1070_pigeonhole__infinite,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ~ ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [A3: nat > nat] :
( ( member_nat_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1071_pigeonhole__infinite,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ~ ( finite3753911285555252421at_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ~ ( finite3753911285555252421at_nat
@ ( collect_nat_nat_nat
@ ^ [A3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1072_pigeonhole__infinite,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ~ ( finite4863279049984502213at_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ~ ( finite4863279049984502213at_nat
@ ( collect_nat_nat_nat2
@ ^ [A3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1073_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1074_finite__surj,axiom,
! [A2: set_nat,B2: set_int,F: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
=> ( finite_finite_int @ B2 ) ) ) ).
% finite_surj
thf(fact_1075_finite__surj,axiom,
! [A2: set_int,B2: set_nat,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1076_finite__surj,axiom,
! [A2: set_int,B2: set_int,F: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A2 ) )
=> ( finite_finite_int @ B2 ) ) ) ).
% finite_surj
thf(fact_1077_finite__surj,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1078_finite__surj,axiom,
! [A2: set_nat,B2: set_nat_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1079_finite__surj,axiom,
! [A2: set_int,B2: set_nat_nat,F: int > nat > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_int_nat_nat @ F @ A2 ) )
=> ( finite2115694454571419734at_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1080_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1081_finite__subset__image,axiom,
! [B2: set_nat,F: int > nat,A2: set_int] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
=> ? [C5: set_int] :
( ( ord_less_eq_set_int @ C5 @ A2 )
& ( finite_finite_int @ C5 )
& ( B2
= ( image_int_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1082_finite__subset__image,axiom,
! [B2: set_int,F: nat > int,A2: set_nat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_nat_int @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1083_finite__subset__image,axiom,
! [B2: set_int,F: int > int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_int_int @ F @ A2 ) )
=> ? [C5: set_int] :
( ( ord_less_eq_set_int @ C5 @ A2 )
& ( finite_finite_int @ C5 )
& ( B2
= ( image_int_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1084_finite__subset__image,axiom,
! [B2: set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) )
=> ? [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
& ( finite2115694454571419734at_nat @ C5 )
& ( B2
= ( image_nat_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1085_finite__subset__image,axiom,
! [B2: set_int,F: ( nat > nat ) > int,A2: set_nat_nat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ ( image_nat_nat_int @ F @ A2 ) )
=> ? [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
& ( finite2115694454571419734at_nat @ C5 )
& ( B2
= ( image_nat_nat_int @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1086_finite__subset__image,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
& ( finite_finite_nat @ C5 )
& ( B2
= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1087_finite__subset__image,axiom,
! [B2: set_nat_nat,F: int > nat > nat,A2: set_int] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_int_nat_nat @ F @ A2 ) )
=> ? [C5: set_int] :
( ( ord_less_eq_set_int @ C5 @ A2 )
& ( finite_finite_int @ C5 )
& ( B2
= ( image_int_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1088_finite__subset__image,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ? [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
& ( finite2115694454571419734at_nat @ C5 )
& ( B2
= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1089_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1090_ex__finite__subset__image,axiom,
! [F: int > nat,A2: set_int,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A2 )
& ( P @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1091_ex__finite__subset__image,axiom,
! [F: nat > int,A2: set_nat,P: set_int > $o] :
( ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1092_ex__finite__subset__image,axiom,
! [F: int > int,A2: set_int,P: set_int > $o] :
( ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A2 )
& ( P @ ( image_int_int @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1093_ex__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1094_ex__finite__subset__image,axiom,
! [F: ( nat > nat ) > int,A2: set_nat_nat,P: set_int > $o] :
( ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_nat_nat_int @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat_int @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1095_ex__finite__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 )
& ( P @ ( image_nat_nat_nat2 @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1096_ex__finite__subset__image,axiom,
! [F: int > nat > nat,A2: set_int,P: set_nat_nat > $o] :
( ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_int_nat_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A2 )
& ( P @ ( image_int_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1097_ex__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 )
& ( P @ ( image_3205354838064109189at_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1098_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1099_all__finite__subset__image,axiom,
! [F: int > nat,A2: set_int,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A2 ) )
=> ( P @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1100_all__finite__subset__image,axiom,
! [F: nat > int,A2: set_nat,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1101_all__finite__subset__image,axiom,
! [F: int > int,A2: set_int,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A2 ) )
=> ( P @ ( image_int_int @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1102_all__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1103_all__finite__subset__image,axiom,
! [F: ( nat > nat ) > int,A2: set_nat_nat,P: set_int > $o] :
( ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_nat_nat_int @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat_int @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1104_all__finite__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A2 ) )
=> ( P @ ( image_nat_nat_nat2 @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1105_all__finite__subset__image,axiom,
! [F: int > nat > nat,A2: set_int,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_int_nat_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A2 ) )
=> ( P @ ( image_int_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1106_all__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ ( image_3205354838064109189at_nat @ F @ A2 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A2 ) )
=> ( P @ ( image_3205354838064109189at_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1107_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1108_image__diff__subset,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) @ ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1109_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) @ ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1110_finite__conv__nat__seg__image,axiom,
( finite2115694454571419734at_nat
= ( ^ [A6: set_nat_nat] :
? [N2: nat,F2: nat > nat > nat] :
( A6
= ( image_nat_nat_nat2 @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1111_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
? [N2: nat,F2: nat > nat] :
( A6
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1112_finite__conv__nat__seg__image,axiom,
( finite_finite_int
= ( ^ [A6: set_int] :
? [N2: nat,F2: nat > int] :
( A6
= ( image_nat_int @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1113_nat__seg__image__imp__finite,axiom,
! [A2: set_nat_nat,F: nat > nat > nat,N: nat] :
( ( A2
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) )
=> ( finite2115694454571419734at_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_1114_nat__seg__image__imp__finite,axiom,
! [A2: set_nat,F: nat > nat,N: nat] :
( ( A2
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) )
=> ( finite_finite_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_1115_nat__seg__image__imp__finite,axiom,
! [A2: set_int,F: nat > int,N: nat] :
( ( A2
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) )
=> ( finite_finite_int @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_1116_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1117_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_1991755285388994676at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member4402528950554000163at_nat @ Y5
@ ( piE_na7569501297962130601at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1118_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1119_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_3521005150465447523at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat] :
( ( member4107745174335074322at_nat @ X3
@ ( piE_na2138371880555796248at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat] :
( ( ( member4107745174335074322at_nat @ Y5
@ ( piE_na2138371880555796248at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1120_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( image_5175309785689899713at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member3693257457796161904at_nat @ X3
@ ( piE_na4170927303951785078at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat > nat] :
( ( ( member3693257457796161904at_nat @ Y5
@ ( piE_na4170927303951785078at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1121_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( image_4065942021260649921at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member6416598835793757296at_nat @ X3
@ ( piE_na3061559539522535286at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( ( member6416598835793757296at_nat @ Y5
@ ( piE_na3061559539522535286at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1122_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat,A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( ord_le973658574027395234at_nat @ ( image_323718453976782111at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat] :
( ( member6105598001968527566at_nat @ X3
@ ( piE_na799184809307736020at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > ( nat > nat ) > ( nat > nat ) > nat] :
( ( ( member6105598001968527566at_nat @ Y5
@ ( piE_na799184809307736020at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1123_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y5
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1124_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1125_PiE__uniqueness,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat,A2: set_na6626867396258451522at_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_722231358656203602at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat] :
( ( member1174580258192983937at_nat @ X3
@ ( piE_na5629913657871898759at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat > nat] :
( ( ( member1174580258192983937at_nat @ Y5
@ ( piE_na5629913657871898759at_nat @ A2
@ ^ [I: ( nat > nat ) > ( nat > nat ) > nat] : B2 ) )
& ! [Xa2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_1126_card__image__le,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1127_card__image__le,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_nat @ F @ A2 ) ) @ ( finite_card_int @ A2 ) ) ) ).
% card_image_le
thf(fact_1128_card__image__le,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1129_card__image__le,axiom,
! [A2: set_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1130_card__image__le,axiom,
! [A2: set_int,F: int > nat > nat] :
( ( finite_finite_int @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( image_int_nat_nat @ F @ A2 ) ) @ ( finite_card_int @ A2 ) ) ) ).
% card_image_le
thf(fact_1131_card__image__le,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1132_card__image__le,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) ) @ ( finite1794908990118856198at_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1133_card__image__le,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1134_card__image__le,axiom,
! [A2: set_int,F: int > ( nat > nat ) > nat] :
( ( finite_finite_int @ A2 )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ ( image_5036371617510403937at_nat @ F @ A2 ) ) @ ( finite_card_int @ A2 ) ) ) ).
% card_image_le
thf(fact_1135_card__image__le,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ ( image_1991755285388994676at_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1136_surj__card__le,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1137_surj__card__le,axiom,
! [A2: set_int,B2: set_nat,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_int_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_int @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1138_surj__card__le,axiom,
! [A2: set_nat_nat,B2: set_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1139_surj__card__le,axiom,
! [A2: set_nat,B2: set_nat_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1140_surj__card__le,axiom,
! [A2: set_int,B2: set_nat_nat,F: int > nat > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_int_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_int @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1141_surj__card__le,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( finite3753911285555252421at_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite1794908990118856198at_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1142_surj__card__le,axiom,
! [A2: set_nat,B2: set_nat_nat_nat2,F: nat > ( nat > nat ) > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1143_surj__card__le,axiom,
! [A2: set_int,B2: set_nat_nat_nat2,F: int > ( nat > nat ) > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ ( image_5036371617510403937at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite_card_int @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1144_surj__card__le,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1145_surj__card__le,axiom,
! [A2: set_nat_nat,B2: set_nat_nat_nat2,F: ( nat > nat ) > ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ ( image_1991755285388994676at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1794908990118856198at_nat @ B2 ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1146_bij__betw__add,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ ( plus_plus_nat @ A ) @ A2 @ B2 )
= ( ( image_nat_nat @ ( plus_plus_nat @ A ) @ A2 )
= B2 ) ) ).
% bij_betw_add
thf(fact_1147_bij__betw__add,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( bij_betw_int_int @ ( plus_plus_int @ A ) @ A2 @ B2 )
= ( ( image_int_int @ ( plus_plus_int @ A ) @ A2 )
= B2 ) ) ).
% bij_betw_add
thf(fact_1148_translation__subtract__diff,axiom,
! [A: int,S2: set_int,T: set_int] :
( ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A )
@ ( minus_minus_set_int @ S2 @ T ) )
= ( minus_minus_set_int
@ ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A )
@ S2 )
@ ( image_int_int
@ ^ [X2: int] : ( minus_minus_int @ X2 @ A )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_1149_bij__betw__subset,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,A7: set_nat,B2: set_nat_nat_nat2,B9: set_nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ A7 )
=> ( ( ord_le5934964663421696068at_nat @ B2 @ A2 )
=> ( ( ( image_7809927846809980933at_nat @ F @ B2 )
= B9 )
=> ( bij_be1059735840858801910at_nat @ F @ B2 @ B9 ) ) ) ) ).
% bij_betw_subset
thf(fact_1150_bij__betw__subset,axiom,
! [F: nat > nat > nat,A2: set_nat,A7: set_nat_nat,B2: set_nat,B9: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat_nat2 @ F @ B2 )
= B9 )
=> ( bij_betw_nat_nat_nat2 @ F @ B2 @ B9 ) ) ) ) ).
% bij_betw_subset
thf(fact_1151_bij__betw__subset,axiom,
! [F: nat > nat,A2: set_nat,A7: set_nat,B2: set_nat,B9: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ A7 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat @ F @ B2 )
= B9 )
=> ( bij_betw_nat_nat @ F @ B2 @ B9 ) ) ) ) ).
% bij_betw_subset
thf(fact_1152_bij__betw__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,A7: set_nat_nat,B2: set_nat_nat,B9: set_nat_nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ A7 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ( image_3205354838064109189at_nat @ F @ B2 )
= B9 )
=> ( bij_be5678534868967705974at_nat @ F @ B2 @ B9 ) ) ) ) ).
% bij_betw_subset
thf(fact_1153_bij__betw__subset,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,A7: set_nat,B2: set_nat_nat,B9: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ A7 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ( image_nat_nat_nat @ F @ B2 )
= B9 )
=> ( bij_betw_nat_nat_nat @ F @ B2 @ B9 ) ) ) ) ).
% bij_betw_subset
thf(fact_1154_bij__betw__byWitness,axiom,
! [A2: set_nat_nat_nat2,F5: nat > ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > nat,A7: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be1059735840858801910at_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1155_bij__betw__byWitness,axiom,
! [A2: set_nat,F5: nat > nat,F: nat > nat,A7: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A7 ) @ A2 )
=> ( bij_betw_nat_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1156_bij__betw__byWitness,axiom,
! [A2: set_nat_nat,F5: nat > nat > nat,F: ( nat > nat ) > nat,A7: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F5 @ A7 ) @ A2 )
=> ( bij_betw_nat_nat_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1157_bij__betw__byWitness,axiom,
! [A2: set_nat,F5: ( nat > nat ) > nat,F: nat > nat > nat,A7: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ A7 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F5 @ A7 ) @ A2 )
=> ( bij_betw_nat_nat_nat2 @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1158_bij__betw__byWitness,axiom,
! [A2: set_nat_nat,F5: ( nat > nat ) > nat > nat,F: ( nat > nat ) > nat > nat,A7: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F5 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A7 )
=> ( ( F @ ( F5 @ X3 ) )
= X3 ) )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ A7 )
=> ( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F5 @ A7 ) @ A2 )
=> ( bij_be5678534868967705974at_nat @ F @ A2 @ A7 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1159_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_1160_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A6: set_nat,B6: set_nat] :
? [G3: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( member_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1161_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat_nat
= ( ^ [F2: ( nat > nat ) > nat,A6: set_nat_nat,B6: set_nat] :
? [G3: nat > nat > nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A6 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( member_nat_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1162_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat_nat2
= ( ^ [F2: nat > nat > nat,A6: set_nat,B6: set_nat_nat] :
? [G3: ( nat > nat ) > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( ( member_nat_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B6 )
=> ( ( member_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1163_bij__betw__iff__bijections,axiom,
( bij_be8282881169987224566at_nat
= ( ^ [F2: nat > ( nat > nat ) > nat,A6: set_nat,B6: set_nat_nat_nat2] :
? [G3: ( ( nat > nat ) > nat ) > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( ( member_nat_nat_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ B6 )
=> ( ( member_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1164_bij__betw__iff__bijections,axiom,
( bij_be5678534868967705974at_nat
= ( ^ [F2: ( nat > nat ) > nat > nat,A6: set_nat_nat,B6: set_nat_nat] :
? [G3: ( nat > nat ) > nat > nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A6 )
=> ( ( member_nat_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B6 )
=> ( ( member_nat_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1165_bij__betw__iff__bijections,axiom,
( bij_be3386790225224311798at_nat
= ( ^ [F2: ( nat > nat > nat ) > nat,A6: set_nat_nat_nat,B6: set_nat] :
? [G3: nat > nat > nat > nat] :
( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A6 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( member_nat_nat_nat2 @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1166_bij__betw__iff__bijections,axiom,
( bij_be168876897561698550at_nat
= ( ^ [F2: nat > nat > nat > nat,A6: set_nat,B6: set_nat_nat_nat] :
? [G3: ( nat > nat > nat ) > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( ( member_nat_nat_nat2 @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ B6 )
=> ( ( member_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1167_bij__betw__iff__bijections,axiom,
( bij_be1059735840858801910at_nat
= ( ^ [F2: ( ( nat > nat ) > nat ) > nat,A6: set_nat_nat_nat2,B6: set_nat] :
? [G3: nat > ( nat > nat ) > nat] :
( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A6 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( member_nat_nat_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1168_bij__betw__iff__bijections,axiom,
( bij_be5311014265664741861at_nat
= ( ^ [F2: ( nat > nat ) > ( nat > nat ) > nat,A6: set_nat_nat,B6: set_nat_nat_nat2] :
? [G3: ( ( nat > nat ) > nat ) > nat > nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A6 )
=> ( ( member_nat_nat_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ B6 )
=> ( ( member_nat_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1169_bij__betw__iff__bijections,axiom,
( bij_be4581752835692700517at_nat
= ( ^ [F2: ( ( nat > nat ) > nat ) > nat > nat,A6: set_nat_nat_nat2,B6: set_nat_nat] :
? [G3: ( nat > nat ) > ( nat > nat ) > nat] :
( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A6 )
=> ( ( member_nat_nat @ ( F2 @ X2 ) @ B6 )
& ( ( G3 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B6 )
=> ( ( member_nat_nat_nat @ ( G3 @ X2 ) @ A6 )
& ( ( F2 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1170_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1171_bij__betw__apply,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat,A: nat > nat] :
( ( bij_betw_nat_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1172_bij__betw__apply,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat,A: nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1173_bij__betw__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat,A: nat > nat] :
( ( bij_be5678534868967705974at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1174_bij__betw__apply,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,B2: set_nat_nat_nat2,A: nat] :
( ( bij_be8282881169987224566at_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1175_bij__betw__apply,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,B2: set_nat_nat_nat,A: nat] :
( ( bij_be168876897561698550at_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1176_bij__betw__apply,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,B2: set_nat,A: nat > nat > nat] :
( ( bij_be3386790225224311798at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1177_bij__betw__apply,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat,A: ( nat > nat ) > nat] :
( ( bij_be1059735840858801910at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1178_bij__betw__apply,axiom,
! [F: ( ( nat > nat ) > nat ) > nat > nat,A2: set_nat_nat_nat2,B2: set_nat_nat,A: ( nat > nat ) > nat] :
( ( bij_be4581752835692700517at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ A @ A2 )
=> ( member_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1179_bij__betw__apply,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat_nat2,A: nat > nat] :
( ( bij_be5311014265664741861at_nat @ F @ A2 @ B2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ( member_nat_nat_nat @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1180_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat,A7: set_nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A2 @ A7 )
= ( bij_betw_nat_nat @ G @ A2 @ A7 ) ) ) ).
% bij_betw_cong
thf(fact_1181_subspace__elems__embed,axiom,
! [S: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S @ K @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S @ ( hales_cube @ K @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_1182_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1183_some__inv__into__2,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ 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 )
= S2 ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S2 ) ) ) ).
% some_inv_into_2
thf(fact_1184_some__inv__into,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( fChoice_nat_nat
@ ^ [P2: nat > nat] :
( ( member_nat_nat @ P2 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P2 @ zero_zero_nat )
= S2 ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S2 ) ) ) ).
% some_inv_into
thf(fact_1185_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K4 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K4 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1186_inv__into__cube__props_I2_J,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S2
@ zero_zero_nat )
= S2 ) ) ).
% inv_into_cube_props(2)
thf(fact_1187_inv__into__cube__props_I1_J,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( member_nat_nat
@ ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S2 )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% inv_into_cube_props(1)
thf(fact_1188_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
= ( P @ B5 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1189_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
& ( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1190_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N5 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% pos_int_cases
thf(fact_1191_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1192_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_1193_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_1194_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1195_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1196_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1197_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1198_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1199_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_1200_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_1201_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_1202_zdiff__int__split,axiom,
! [P: int > $o,X4: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X4 @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1203_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_1204_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_1205_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_1206_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1207_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1208_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_1209_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_1210_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( K
!= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% nonneg_int_cases
thf(fact_1211_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1212_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1213_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N5: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% int_diff_cases
thf(fact_1214_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1215_imp__le__cong,axiom,
! [X4: int,X7: int,P: $o,P4: $o] :
( ( X4 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1216_conj__le__cong,axiom,
! [X4: int,X7: int,P: $o,P4: $o] :
( ( X4 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1217_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1218_int__induct,axiom,
! [P: int > $o,K: int,I3: int] :
( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_induct
thf(fact_1219_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_1220_int__ge__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1221_int__gr__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1222_int__le__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_1223_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_1224_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1225_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1226_int__less__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I4: int] :
( ( ord_less_int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_1227_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_1228_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_1229_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1230_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1231_verit__la__generic,axiom,
! [A: int,X4: int] :
( ( ord_less_eq_int @ A @ X4 )
| ( A = X4 )
| ( ord_less_eq_int @ X4 @ A ) ) ).
% verit_la_generic
thf(fact_1232_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1233_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1234_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1235_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1236_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1237_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_1238_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1239_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1240_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1241_bij__betw__Suc,axiom,
! [M4: set_nat,N6: set_nat] :
( ( bij_betw_nat_nat @ suc @ M4 @ N6 )
= ( ( image_nat_nat @ suc @ M4 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_1242_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1243_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1244_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_1245_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_1246_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_1247_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_1248_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_1249_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1250_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1251_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1252_nat__power__eq__Suc__0__iff,axiom,
! [X4: nat,M: nat] :
( ( ( power_power_nat @ X4 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X4
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1253_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1254_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_1255_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1256_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_1257_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1258_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1259_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_1260_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_1261_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1262_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1263_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
% Helper facts (6)
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_T,axiom,
! [P: ( ( nat > nat ) > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat @ P ) )
= ( ? [X6: ( nat > nat ) > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( nat > nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat2 @ P ) )
= ( ? [X6: nat > nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( P @ ( fChoic52552927678224201at_nat @ P ) )
= ( ? [X6: ( nat > nat ) > nat > nat] : ( P @ X6 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( P @ ( fChoic2516396905127217208at_nat @ P ) )
= ( ? [X6: ( nat > nat ) > ( nat > nat ) > nat] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_nat_nat_nat
@ ( restri6011711336257459485at_nat
@ ^ [X2: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y: nat > nat] : ( chi @ ( hales_join_nat @ X2 @ Y @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ x )
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
%------------------------------------------------------------------------------