TPTP Problem File: SCT171^2.p
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%------------------------------------------------------------------------------
% File : SCT171^2 : TPTP v9.0.0. Released v5.3.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 309, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : arrow_500_thf_l309 [Bla11]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0, 0.29 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.80 v5.3.0
% Syntax : Number of formulae : 870 ( 91 unt; 166 typ; 0 def)
% Number of atoms : 2239 ( 359 equ; 6 cnn)
% Maximal formula atoms : 19 ( 3 avg)
% Number of connectives : 5454 ( 280 ~; 32 |; 60 &;4072 @)
% ( 179 <=>; 831 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 2346 (2346 >; 0 *; 0 +; 0 <<)
% Number of symbols : 165 ( 161 usr; 12 con; 0-4 aty)
% Number of variables : 1714 ( 84 ^;1614 !; 16 ?;1714 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:35:00
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
thf(ty_ty_tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
arrow_475358991le_alt: $tType ).
thf(ty_ty_tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi,type,
arrow_1429601828e_indi: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_ty_tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_Mtc__Arrow__,type,
produc1501160679le_alt: $tType ).
%----Explicit typings (161)
thf(sy_c_All,type,
all: ( produc1501160679le_alt > $o ) > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_OIIA,type,
arrow_797024463le_IIA: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_OLin,type,
arrow_823908191le_Lin: ( produc1501160679le_alt > $o ) > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_OProf,type,
arrow_734252939e_Prof: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_Oabove,type,
arrow_789600939_above: ( produc1501160679le_alt > $o ) > arrow_475358991le_alt > arrow_475358991le_alt > produc1501160679le_alt > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_Obelow,type,
arrow_2098199487_below: ( produc1501160679le_alt > $o ) > arrow_475358991le_alt > arrow_475358991le_alt > produc1501160679le_alt > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_Odictator,type,
arrow_1212662430ctator: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_Omkbot,type,
arrow_2054445623_mkbot: ( produc1501160679le_alt > $o ) > arrow_475358991le_alt > produc1501160679le_alt > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_Omktop,type,
arrow_55669061_mktop: ( produc1501160679le_alt > $o ) > arrow_475358991le_alt > produc1501160679le_alt > $o ).
thf(sy_c_Arrow__Order__Mirabelle__lcilvlkkzv_Ounanimity,type,
arrow_1706409458nimity: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_Finite__Set_Ocard_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__O,type,
finite120663670_alt_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lci,type,
finite28306938_alt_o: ( ( produc1501160679le_alt > $o ) > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000_Eo,type,
finite_card_o: ( $o > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi,type,
finite97476818e_indi: ( arrow_1429601828e_indi > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
finite_card_nat: ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Product____Type__Ounit,type,
finite1949902593t_unit: ( product_unit > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkz,type,
finite537683861le_alt: ( produc1501160679le_alt > $o ) > nat ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv_,type,
finite1956767223_alt_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____l,type,
finite2112685307_alt_o: ( ( produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_Eo,type,
finite_finite_o: ( $o > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi,type,
finite664979089e_indi: ( arrow_1429601828e_indi > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlk,type,
finite449174868le_alt: ( produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_000t,type,
in_rel1252994498le_alt: ( produc1501160679le_alt > $o ) > arrow_475358991le_alt > arrow_475358991le_alt > $o ).
thf(sy_c_Fun_Oinj__on_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_,type,
inj_on1284293749_alt_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkk,type,
inj_on743426285_alt_o: ( ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( ( produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkk_001,type,
inj_on1877294875e_indi: ( ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > ( ( produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000_Eo_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlk,type,
inj_on867909093le_alt: ( $o > produc1501160679le_alt ) > ( $o > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000_06,type,
inj_on1190919077_alt_o: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( arrow_1429601828e_indi > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000tc_,type,
inj_on978774663di_nat: ( arrow_1429601828e_indi > nat ) > ( arrow_1429601828e_indi > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000tc__Nat__Onat_000tc__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > ( nat > $o ) > $o ).
thf(sy_c_Fun_Oinj__on_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oa,type,
inj_on1911943593_alt_o: ( produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__O,type,
pi_Arr195212324lt_o_o: ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o > $o ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__O_002,type,
pi_Arr338314351e_indi: ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__O_003,type,
pi_Arr830584606t_unit: ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > product_unit > $o ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > product_unit ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M,type,
pi_Arr1304755663_alt_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M_004,type,
pi_Arr952516694lt_o_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M_005,type,
pi_Arr1232280765e_indi: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M_006,type,
pi_Arr1963174508t_unit: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > product_unit > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > product_unit ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkz,type,
pi_Pro763888199_alt_o: ( ( produc1501160679le_alt > $o ) > $o ) > ( ( produc1501160679le_alt > $o ) > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkz_007,type,
pi_Pro422690258lt_o_o: ( ( produc1501160679le_alt > $o ) > $o ) > ( ( produc1501160679le_alt > $o ) > $o > $o ) > ( ( produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkz_008,type,
pi_Pro468373057e_indi: ( ( produc1501160679le_alt > $o ) > $o ) > ( ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o ) > ( ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > $o ).
thf(sy_c_FuncSet_OPi_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkz_009,type,
pi_Pro1306850800t_unit: ( ( produc1501160679le_alt > $o ) > $o ) > ( ( produc1501160679le_alt > $o ) > product_unit > $o ) > ( ( produc1501160679le_alt > $o ) > product_unit ) > $o ).
thf(sy_c_FuncSet_OPi_000_Eo_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvl,type,
pi_o_A1186128886_alt_o: ( $o > $o ) > ( $o > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > ( $o > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_Eo_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__,type,
pi_o_A1182933120_alt_o: ( $o > $o ) > ( $o > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( $o > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_Eo_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lc,type,
pi_o_P553196292_alt_o: ( $o > $o ) > ( $o > ( produc1501160679le_alt > $o ) > $o ) > ( $o > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000_Eo_000tc__Nat__Onat,type,
pi_o_nat: ( $o > $o ) > ( $o > nat > $o ) > ( $o > nat ) > $o ).
thf(sy_c_FuncSet_OPi_000_Eo_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkk,type,
pi_o_P657324555le_alt: ( $o > $o ) > ( $o > produc1501160679le_alt > $o ) > ( $o > produc1501160679le_alt ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000_062,type,
pi_Arr1564509167_alt_o: ( arrow_1429601828e_indi > $o ) > ( arrow_1429601828e_indi > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > ( arrow_1429601828e_indi > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000_062_010,type,
pi_Arr1060328391_alt_o: ( arrow_1429601828e_indi > $o ) > ( arrow_1429601828e_indi > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( arrow_1429601828e_indi > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000_062_011,type,
pi_Arr1929480907_alt_o: ( arrow_1429601828e_indi > $o ) > ( arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o ) > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000tc__,type,
pi_Arr251692973di_nat: ( arrow_1429601828e_indi > $o ) > ( arrow_1429601828e_indi > nat > $o ) > ( arrow_1429601828e_indi > nat ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000tc___012,type,
pi_Arr329216900le_alt: ( arrow_1429601828e_indi > $o ) > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( arrow_1429601828e_indi > produc1501160679le_alt ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Nat__Onat_000_Eo,type,
pi_nat_o: ( nat > $o ) > ( nat > $o > $o ) > ( nat > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____lcilvlkk,type,
pi_nat1219304995e_indi: ( nat > $o ) > ( nat > arrow_1429601828e_indi > $o ) > ( nat > arrow_1429601828e_indi ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Nat__Onat_000tc__Nat__Onat,type,
pi_nat_nat: ( nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Nat__Onat_000tc__Product____Type__Ounit,type,
pi_nat_Product_unit: ( nat > $o ) > ( nat > product_unit > $o ) > ( nat > product_unit ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Product____Type__Ounit_000_062_I_062_Itc__Arrow____Order,type,
pi_Pro1782982558_alt_o: ( product_unit > $o ) > ( product_unit > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > ( product_unit > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Product____Type__Ounit_000_062_Itc__Arrow____Order____Mi,type,
pi_Pro1662176984_alt_o: ( product_unit > $o ) > ( product_unit > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( product_unit > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Product____Type__Ounit_000_062_Itc__prod_Itc__Arrow____O,type,
pi_Pro1312660828_alt_o: ( product_unit > $o ) > ( product_unit > ( produc1501160679le_alt > $o ) > $o ) > ( product_unit > produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Product____Type__Ounit_000tc__Nat__Onat,type,
pi_Product_unit_nat: ( product_unit > $o ) > ( product_unit > nat > $o ) > ( product_unit > nat ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__Product____Type__Ounit_000tc__prod_Itc__Arrow____Order__,type,
pi_Pro701847987le_alt: ( product_unit > $o ) > ( product_unit > produc1501160679le_alt > $o ) > ( product_unit > produc1501160679le_alt ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oal,type,
pi_Pro1701359055_alt_o: ( produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o > $o ) > ( produc1501160679le_alt > $o ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oal_013,type,
pi_Pro1767455108e_indi: ( produc1501160679le_alt > $o ) > ( produc1501160679le_alt > arrow_1429601828e_indi > $o ) > ( produc1501160679le_alt > arrow_1429601828e_indi ) > $o ).
thf(sy_c_FuncSet_OPi_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oal_014,type,
pi_Pro1475896499t_unit: ( produc1501160679le_alt > $o ) > ( produc1501160679le_alt > product_unit > $o ) > ( produc1501160679le_alt > product_unit ) > $o ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_000tc__Arrow____Order____Mirabelle____lcilvlkkzv,type,
hilber598459244di_nat: ( arrow_1429601828e_indi > $o ) > ( arrow_1429601828e_indi > nat ) > nat > arrow_1429601828e_indi ).
thf(sy_c_Hilbert__Choice_Oinv__into_000tc__Nat__Onat_000tc__Nat__Onat,type,
hilber195283148at_nat: ( nat > $o ) > ( nat > nat ) > nat > nat ).
thf(sy_c_If_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_M,type,
if_Pro1561232536_alt_o: $o > ( produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_I_062_I_062_Itc__Arrow____Order____Mira,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_I_062_Itc__Arrow____Order____Mirabelle_,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_I_062_Itc__prod_Itc__Arrow____Order____,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Arrow____Order____Mirabelle____lci,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Product____Type__Ounit_M_Eo_J,type,
ord_le232288914unit_o: ( product_unit > $o ) > ( product_unit > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__prod_Itc__Arrow____Order____Mirabe,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_I_062_Itc__Arrow____Order____,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Arrow____Order____Mirabe,type,
ord_le1992928527lt_o_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__prod_Itc__Arrow____Order,type,
ord_le1063113995lt_o_o: ( ( produc1501160679le_alt > $o ) > $o ) > ( ( produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Arrow____Order____Mirabelle___,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Arrow____Order____Mirabelle____015,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Product____Type__Ounit_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__prod_Itc__Arrow____Order____Mi,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_000_062_I_062_I_062_Itc__Arrow____Order____Mirab,type,
top_to1969627639lt_o_o: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ).
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thf(sy_c_Orderings_Otop__class_Otop_000_062_I_062_Itc__prod_Itc__Arrow____Order____M,type,
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thf(sy_c_Orderings_Otop__class_Otop_000_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_000_062_Itc__Arrow____Order____Mirabelle____lcil,type,
top_to988227749indi_o: arrow_1429601828e_indi > $o ).
thf(sy_c_Orderings_Otop__class_Otop_000_062_Itc__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_000_062_Itc__Product____Type__Ounit_M_Eo_J,type,
top_to1984820022unit_o: product_unit > $o ).
thf(sy_c_Orderings_Otop__class_Otop_000_062_Itc__prod_Itc__Arrow____Order____Mirabel,type,
top_to1841428258_alt_o: produc1501160679le_alt > $o ).
thf(sy_c_Orderings_Otop__class_Otop_000_Eo,type,
top_top_o: $o ).
thf(sy_c_Product__Type_OPair_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_,type,
produc1347929815le_alt: arrow_475358991le_alt > arrow_475358991le_alt > produc1501160679le_alt ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Nat__Onat,type,
ord_at4362885an_nat: nat > nat > nat > $o ).
thf(sy_c_Set_OCollect_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__,type,
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thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_OCollect_000tc__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > product_unit > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oa,type,
collec869865362le_alt: ( produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ).
thf(sy_c_Set_Oimage_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_000tc__N,type,
image_484224243di_nat: ( arrow_1429601828e_indi > nat ) > ( arrow_1429601828e_indi > $o ) > nat > $o ).
thf(sy_c_member_000_062_I_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__,type,
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thf(sy_c_member_000_062_I_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv___016,type,
member1452482393e_indi: ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > ( ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > $o ) > $o ).
thf(sy_c_member_000_062_I_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv___017,type,
member1924666376t_unit: ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > product_unit ) > ( ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > product_unit ) > $o ) > $o ).
thf(sy_c_member_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_,type,
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thf(sy_c_member_000_062_I_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi__018,type,
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member44294883e_indi: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > ( ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > arrow_1429601828e_indi ) > $o ) > $o ).
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thf(sy_c_member_000_062_I_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkk,type,
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thf(sy_c_member_000_062_I_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkk_021,type,
member1961363906lt_o_o: ( ( produc1501160679le_alt > $o ) > $o ) > ( ( ( produc1501160679le_alt > $o ) > $o ) > $o ) > $o ).
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member221730070t_unit: ( ( produc1501160679le_alt > $o ) > product_unit ) > ( ( ( produc1501160679le_alt > $o ) > product_unit ) > $o ) > $o ).
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member1957863580_alt_o: ( $o > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > ( ( $o > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_I_Eo_M_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__O,type,
member1394214384_alt_o: ( $o > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( ( $o > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_I_Eo_M_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lci,type,
member1862122484_alt_o: ( $o > produc1501160679le_alt > $o ) > ( ( $o > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_I_Eo_Mtc__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > ( ( $o > nat ) > $o ) > $o ).
thf(sy_c_member_000_062_I_Eo_Mtc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkz,type,
member492167345le_alt: ( $o > produc1501160679le_alt ) > ( ( $o > produc1501160679le_alt ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M_062_,type,
member811956313_alt_o: ( arrow_1429601828e_indi > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > ( ( arrow_1429601828e_indi > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M_062__024,type,
member1234151027_alt_o: ( arrow_1429601828e_indi > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( ( arrow_1429601828e_indi > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_M_062__025,type,
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thf(sy_c_member_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_Mtc__N,type,
member1315464153di_nat: ( arrow_1429601828e_indi > nat ) > ( ( arrow_1429601828e_indi > nat ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi_Mtc__p,type,
member351225838le_alt: ( arrow_1429601828e_indi > produc1501160679le_alt ) > ( ( arrow_1429601828e_indi > produc1501160679le_alt ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Nat__Onat_Mtc__Arrow____Order____Mirabelle____lcilvlkkz,type,
member1391860553e_indi: ( nat > arrow_1429601828e_indi ) > ( ( nat > arrow_1429601828e_indi ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Nat__Onat_Mtc__Product____Type__Ounit_J,type,
member616671224t_unit: ( nat > product_unit ) > ( ( nat > product_unit ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Product____Type__Ounit_M_062_I_062_Itc__Arrow____Order_,type,
member1536989448_alt_o: ( product_unit > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > ( ( product_unit > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Product____Type__Ounit_M_062_Itc__Arrow____Order____Mir,type,
member283501700_alt_o: ( product_unit > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( ( product_unit > arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Product____Type__Ounit_M_062_Itc__prod_Itc__Arrow____Or,type,
member1661784200_alt_o: ( product_unit > produc1501160679le_alt > $o ) > ( ( product_unit > produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Product____Type__Ounit_Mtc__Nat__Onat_J,type,
member1827227242it_nat: ( product_unit > nat ) > ( ( product_unit > nat ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Product____Type__Ounit_Mtc__prod_Itc__Arrow____Order___,type,
member495332125le_alt: ( product_unit > produc1501160679le_alt ) > ( ( product_unit > produc1501160679le_alt ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oa,type,
member377231867_alt_o: ( produc1501160679le_alt > $o ) > ( ( produc1501160679le_alt > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oa_026,type,
member1640632174e_indi: ( produc1501160679le_alt > arrow_1429601828e_indi ) > ( ( produc1501160679le_alt > arrow_1429601828e_indi ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oa_027,type,
member593902749t_unit: ( produc1501160679le_alt > product_unit ) > ( ( produc1501160679le_alt > product_unit ) > $o ) > $o ).
thf(sy_c_member_000_Eo,type,
member_o: $o > ( $o > $o ) > $o ).
thf(sy_c_member_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oindi,type,
member2052026769e_indi: arrow_1429601828e_indi > ( arrow_1429601828e_indi > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_c_member_000tc__Product____Type__Ounit,type,
member_Product_unit: product_unit > ( product_unit > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_Mtc,type,
member214075476le_alt: produc1501160679le_alt > ( produc1501160679le_alt > $o ) > $o ).
thf(sy_v_F,type,
f: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ).
thf(sy_v_Lab____,type,
lab: produc1501160679le_alt > $o ).
thf(sy_v_Lba____,type,
lba: produc1501160679le_alt > $o ).
thf(sy_v_P____,type,
p: arrow_1429601828e_indi > produc1501160679le_alt > $o ).
thf(sy_v_a____,type,
a: arrow_475358991le_alt ).
thf(sy_v_b____,type,
b: arrow_475358991le_alt ).
thf(sy_v_c____,type,
c: arrow_475358991le_alt ).
thf(sy_v_d____,type,
d: arrow_475358991le_alt ).
thf(sy_v_e____,type,
e: arrow_475358991le_alt ).
thf(sy_v_h____,type,
h: arrow_1429601828e_indi > nat ).
thf(sy_v_n____,type,
n: nat ).
%----Relevant facts (700)
thf(fact_0_assms_I3_J,axiom,
arrow_797024463le_IIA @ f ).
thf(fact_1_u,axiom,
arrow_1706409458nimity @ f ).
thf(fact_2__096c_A_126_061_Ad_096,axiom,
c != d ).
thf(fact_3__096P_A_058_AProf_096,axiom,
member526088951_alt_o @ p @ arrow_734252939e_Prof ).
thf(fact_4_in__mkbot,axiom,
! [X: arrow_475358991le_alt,Y: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o,Z: arrow_475358991le_alt] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ ( arrow_2054445623_mkbot @ L_2 @ Z ) )
<=> ( ( Y != Z )
& ( ( X = Z )
=> ( X != Y ) )
& ( ( X != Z )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ L_2 ) ) ) ) ).
thf(fact_5_in__mktop,axiom,
! [X: arrow_475358991le_alt,Y: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o,Z: arrow_475358991le_alt] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ ( arrow_55669061_mktop @ L_2 @ Z ) )
<=> ( ( X != Z )
& ( ( Y = Z )
=> ( X != Y ) )
& ( ( Y != Z )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ L_2 ) ) ) ) ).
thf(fact_6__C2_C,axiom,
! [P_8: arrow_1429601828e_indi > produc1501160679le_alt > $o,P_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_94: arrow_475358991le_alt,B_73: arrow_475358991le_alt,A_93: arrow_475358991le_alt,B_72: arrow_475358991le_alt] :
( ( A_93 != B_72 )
=> ( ( A_94 != B_73 )
=> ( ( A_93 != B_73 )
=> ( ( B_72 != A_94 )
=> ( ( member526088951_alt_o @ P_7 @ arrow_734252939e_Prof )
=> ( ( member526088951_alt_o @ P_8 @ arrow_734252939e_Prof )
=> ( ! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( P_7 @ I_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_94 @ B_73 ) @ ( P_8 @ I_1 ) ) )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( f @ P_7 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_94 @ B_73 ) @ ( f @ P_8 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_7__C1_C,axiom,
! [P_8: arrow_1429601828e_indi > produc1501160679le_alt > $o,P_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_94: arrow_475358991le_alt,B_73: arrow_475358991le_alt,A_93: arrow_475358991le_alt,B_72: arrow_475358991le_alt] :
( ( A_93 != B_72 )
=> ( ( A_94 != B_73 )
=> ( ( A_93 != B_73 )
=> ( ( B_72 != A_94 )
=> ( ( member526088951_alt_o @ P_7 @ arrow_734252939e_Prof )
=> ( ( member526088951_alt_o @ P_8 @ arrow_734252939e_Prof )
=> ( ! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( P_7 @ I_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_94 @ B_73 ) @ ( P_8 @ I_1 ) ) )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( f @ P_7 ) )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_94 @ B_73 ) @ ( f @ P_8 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_8__C4_C,axiom,
! [P_8: arrow_1429601828e_indi > produc1501160679le_alt > $o,P_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,C_37: arrow_475358991le_alt,A_93: arrow_475358991le_alt,B_72: arrow_475358991le_alt] :
( ( A_93 != B_72 )
=> ( ( B_72 != C_37 )
=> ( ( A_93 != C_37 )
=> ( ( member526088951_alt_o @ P_7 @ arrow_734252939e_Prof )
=> ( ( member526088951_alt_o @ P_8 @ arrow_734252939e_Prof )
=> ( ! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( P_7 @ I_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ B_72 @ C_37 ) @ ( P_8 @ I_1 ) ) )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( f @ P_7 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ B_72 @ C_37 ) @ ( f @ P_8 ) ) ) ) ) ) ) ) ) ).
thf(fact_9_pairwise__neutrality,axiom,
! [P_8: arrow_1429601828e_indi > produc1501160679le_alt > $o,P_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_94: arrow_475358991le_alt,B_73: arrow_475358991le_alt,A_93: arrow_475358991le_alt,B_72: arrow_475358991le_alt] :
( ( A_93 != B_72 )
=> ( ( A_94 != B_73 )
=> ( ( member526088951_alt_o @ P_7 @ arrow_734252939e_Prof )
=> ( ( member526088951_alt_o @ P_8 @ arrow_734252939e_Prof )
=> ( ! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( P_7 @ I_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_94 @ B_73 ) @ ( P_8 @ I_1 ) ) )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( f @ P_7 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_94 @ B_73 ) @ ( f @ P_8 ) ) ) ) ) ) ) ) ).
thf(fact_10__C3_C,axiom,
! [P_8: arrow_1429601828e_indi > produc1501160679le_alt > $o,P_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_93: arrow_475358991le_alt,B_72: arrow_475358991le_alt] :
( ( A_93 != B_72 )
=> ( ( member526088951_alt_o @ P_7 @ arrow_734252939e_Prof )
=> ( ( member526088951_alt_o @ P_8 @ arrow_734252939e_Prof )
=> ( ! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( P_7 @ I_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ B_72 @ A_93 ) @ ( P_8 @ I_1 ) ) )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_93 @ B_72 ) @ ( f @ P_7 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ B_72 @ A_93 ) @ ( f @ P_8 ) ) ) ) ) ) ) ).
thf(fact_11__096ALL_Ai_O_A_Ic_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ad_J,axiom,
! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( p @ I_1 ) )
<=> ( ( ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e ) ) )
& ( ~ ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( ( ( ( h @ I_1 )
= n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e ) ) )
& ( ( ( h @ I_1 )
!= n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) ) ) ).
thf(fact_12__096c_A_060_092_060_094bsub_062P_A_Iinv_Ah_An_J_092_060_094esub_062_Ad_0,axiom,
member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( p @ ( hilber598459244di_nat @ top_to988227749indi_o @ h @ n ) ) ).
thf(fact_13__096c_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amkto,axiom,
( member214075476le_alt @ ( produc1347929815le_alt @ c @ d )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) ).
thf(fact_14_PW,axiom,
( ( member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( f @ p ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ d )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) ) ).
thf(fact_15_split__paired__All,axiom,
! [P_6: produc1501160679le_alt > $o] :
( ( all @ P_6 )
<=> ! [A_3: arrow_475358991le_alt,B_61: arrow_475358991le_alt] : ( P_6 @ ( produc1347929815le_alt @ A_3 @ B_61 ) ) ) ).
thf(fact_16__096_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai,axiom,
( member526088951_alt_o
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) )
@ arrow_734252939e_Prof ) ).
thf(fact_17__096c_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amkto,axiom,
( member214075476le_alt @ ( produc1347929815le_alt @ c @ e )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) ).
thf(fact_18__096e_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amkto,axiom,
( member214075476le_alt @ ( produc1347929815le_alt @ e @ d )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) ).
thf(fact_19_n_I1_J,axiom,
ord_less_nat @ n @ ( finite97476818e_indi @ top_to988227749indi_o ) ).
thf(fact_20_injh,axiom,
inj_on978774663di_nat @ h @ top_to988227749indi_o ).
thf(fact_21_Pair__inject,axiom,
! [A_92: arrow_475358991le_alt,B_71: arrow_475358991le_alt,A_91: arrow_475358991le_alt,B_70: arrow_475358991le_alt] :
( ( ( produc1347929815le_alt @ A_92 @ B_71 )
= ( produc1347929815le_alt @ A_91 @ B_70 ) )
=> ~ ( ( A_92 = A_91 )
=> ( B_71 != B_70 ) ) ) ).
thf(fact_22_Pair__eq,axiom,
! [A_90: arrow_475358991le_alt,B_69: arrow_475358991le_alt,A_89: arrow_475358991le_alt,B_68: arrow_475358991le_alt] :
( ( ( produc1347929815le_alt @ A_90 @ B_69 )
= ( produc1347929815le_alt @ A_89 @ B_68 ) )
<=> ( ( A_90 = A_89 )
& ( B_69 = B_68 ) ) ) ).
thf(fact_23_IIA__def,axiom,
! [F_18: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] :
( ( arrow_797024463le_IIA @ F_18 )
<=> ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ arrow_734252939e_Prof )
=> ! [Xa: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ Xa @ arrow_734252939e_Prof )
=> ! [A_3: arrow_475358991le_alt,B_61: arrow_475358991le_alt] :
( ! [I_1: arrow_1429601828e_indi] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( X_1 @ I_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( Xa @ I_1 ) ) )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( F_18 @ X_1 ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( F_18 @ Xa ) ) ) ) ) ) ) ).
thf(fact_24_unanimity__def,axiom,
! [F_18: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] :
( ( arrow_1706409458nimity @ F_18 )
<=> ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ arrow_734252939e_Prof )
=> ! [A_3: arrow_475358991le_alt,B_61: arrow_475358991le_alt] :
( ! [I_1: arrow_1429601828e_indi] : ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( X_1 @ I_1 ) )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( F_18 @ X_1 ) ) ) ) ) ).
thf(fact_25_top1I,axiom,
! [X_70: produc1501160679le_alt > $o] : ( top_to1842727771lt_o_o @ X_70 ) ).
thf(fact_26_top1I,axiom,
! [X_70: arrow_1429601828e_indi > produc1501160679le_alt > $o] : ( top_to2122763103lt_o_o @ X_70 ) ).
thf(fact_27_top1I,axiom,
! [X_70: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] : ( top_to1969627639lt_o_o @ X_70 ) ).
thf(fact_28_top1I,axiom,
! [X_70: produc1501160679le_alt] : ( top_to1841428258_alt_o @ X_70 ) ).
thf(fact_29_top1I,axiom,
! [X_70: arrow_1429601828e_indi] : ( top_to988227749indi_o @ X_70 ) ).
thf(fact_30_top1I,axiom,
! [X_70: product_unit] : ( top_to1984820022unit_o @ X_70 ) ).
thf(fact_31_top1I,axiom,
! [X_70: nat] : ( top_top_nat_o @ X_70 ) ).
thf(fact_32_UNIV__I,axiom,
! [X_69: $o] : ( member_o @ X_69 @ top_top_o_o ) ).
thf(fact_33_UNIV__I,axiom,
! [X_69: arrow_1429601828e_indi] : ( member2052026769e_indi @ X_69 @ top_to988227749indi_o ) ).
thf(fact_34_UNIV__I,axiom,
! [X_69: product_unit] : ( member_Product_unit @ X_69 @ top_to1984820022unit_o ) ).
thf(fact_35_UNIV__I,axiom,
! [X_69: produc1501160679le_alt] : ( member214075476le_alt @ X_69 @ top_to1841428258_alt_o ) ).
thf(fact_36_UNIV__I,axiom,
! [X_69: arrow_1429601828e_indi > produc1501160679le_alt > $o] : ( member526088951_alt_o @ X_69 @ top_to2122763103lt_o_o ) ).
thf(fact_37_UNIV__I,axiom,
! [X_69: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] : ( member616898751_alt_o @ X_69 @ top_to1969627639lt_o_o ) ).
thf(fact_38_UNIV__I,axiom,
! [X_69: produc1501160679le_alt > $o] : ( member377231867_alt_o @ X_69 @ top_to1842727771lt_o_o ) ).
thf(fact_39_UNIV__I,axiom,
! [X_69: nat] : ( member_nat @ X_69 @ top_top_nat_o ) ).
thf(fact_40_iso__tuple__UNIV__I,axiom,
! [X_68: $o] : ( member_o @ X_68 @ top_top_o_o ) ).
thf(fact_41_iso__tuple__UNIV__I,axiom,
! [X_68: arrow_1429601828e_indi] : ( member2052026769e_indi @ X_68 @ top_to988227749indi_o ) ).
thf(fact_42_iso__tuple__UNIV__I,axiom,
! [X_68: product_unit] : ( member_Product_unit @ X_68 @ top_to1984820022unit_o ) ).
thf(fact_43_iso__tuple__UNIV__I,axiom,
! [X_68: produc1501160679le_alt] : ( member214075476le_alt @ X_68 @ top_to1841428258_alt_o ) ).
thf(fact_44_iso__tuple__UNIV__I,axiom,
! [X_68: arrow_1429601828e_indi > produc1501160679le_alt > $o] : ( member526088951_alt_o @ X_68 @ top_to2122763103lt_o_o ) ).
thf(fact_45_iso__tuple__UNIV__I,axiom,
! [X_68: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] : ( member616898751_alt_o @ X_68 @ top_to1969627639lt_o_o ) ).
thf(fact_46_iso__tuple__UNIV__I,axiom,
! [X_68: produc1501160679le_alt > $o] : ( member377231867_alt_o @ X_68 @ top_to1842727771lt_o_o ) ).
thf(fact_47_iso__tuple__UNIV__I,axiom,
! [X_68: nat] : ( member_nat @ X_68 @ top_top_nat_o ) ).
thf(fact_48_top__apply,axiom,
! [X_67: produc1501160679le_alt > $o] :
( ( top_to1842727771lt_o_o @ X_67 )
<=> top_top_o ) ).
thf(fact_49_top__apply,axiom,
! [X_67: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( top_to2122763103lt_o_o @ X_67 )
<=> top_top_o ) ).
thf(fact_50_top__apply,axiom,
! [X_67: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] :
( ( top_to1969627639lt_o_o @ X_67 )
<=> top_top_o ) ).
thf(fact_51_top__apply,axiom,
! [X_67: produc1501160679le_alt] :
( ( top_to1841428258_alt_o @ X_67 )
<=> top_top_o ) ).
thf(fact_52_top__apply,axiom,
! [X_67: arrow_1429601828e_indi] :
( ( top_to988227749indi_o @ X_67 )
<=> top_top_o ) ).
thf(fact_53_top__apply,axiom,
! [X_67: product_unit] :
( ( top_to1984820022unit_o @ X_67 )
<=> top_top_o ) ).
thf(fact_54_top__apply,axiom,
! [X_67: nat] :
( ( top_top_nat_o @ X_67 )
<=> top_top_o ) ).
thf(fact_55_not__top__less,axiom,
! [A_88: ( produc1501160679le_alt > $o ) > $o] :
~ ( ord_le910298367lt_o_o @ top_to1842727771lt_o_o @ A_88 ) ).
thf(fact_56_not__top__less,axiom,
! [A_88: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
~ ( ord_le157835011lt_o_o @ top_to2122763103lt_o_o @ A_88 ) ).
thf(fact_57_not__top__less,axiom,
! [A_88: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
~ ( ord_le1859604819lt_o_o @ top_to1969627639lt_o_o @ A_88 ) ).
thf(fact_58_not__top__less,axiom,
! [A_88: $o] :
~ ( ord_less_o @ top_top_o @ A_88 ) ).
thf(fact_59_not__top__less,axiom,
! [A_88: produc1501160679le_alt > $o] :
~ ( ord_le988258430_alt_o @ top_to1841428258_alt_o @ A_88 ) ).
thf(fact_60_not__top__less,axiom,
! [A_88: arrow_1429601828e_indi > $o] :
~ ( ord_le777687553indi_o @ top_to988227749indi_o @ A_88 ) ).
thf(fact_61_not__top__less,axiom,
! [A_88: product_unit > $o] :
~ ( ord_le232288914unit_o @ top_to1984820022unit_o @ A_88 ) ).
thf(fact_62_not__top__less,axiom,
! [A_88: nat > $o] :
~ ( ord_less_nat_o @ top_top_nat_o @ A_88 ) ).
thf(fact_63_less__top,axiom,
! [A_87: ( produc1501160679le_alt > $o ) > $o] :
( ( A_87 != top_to1842727771lt_o_o )
<=> ( ord_le910298367lt_o_o @ A_87 @ top_to1842727771lt_o_o ) ) ).
thf(fact_64_less__top,axiom,
! [A_87: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( A_87 != top_to2122763103lt_o_o )
<=> ( ord_le157835011lt_o_o @ A_87 @ top_to2122763103lt_o_o ) ) ).
thf(fact_65_less__top,axiom,
! [A_87: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( A_87 != top_to1969627639lt_o_o )
<=> ( ord_le1859604819lt_o_o @ A_87 @ top_to1969627639lt_o_o ) ) ).
thf(fact_66_less__top,axiom,
! [A_87: $o] :
( ~ ( A_87
<=> top_top_o )
<=> ( ord_less_o @ A_87 @ top_top_o ) ) ).
thf(fact_67_less__top,axiom,
! [A_87: produc1501160679le_alt > $o] :
( ( A_87 != top_to1841428258_alt_o )
<=> ( ord_le988258430_alt_o @ A_87 @ top_to1841428258_alt_o ) ) ).
thf(fact_68_less__top,axiom,
! [A_87: arrow_1429601828e_indi > $o] :
( ( A_87 != top_to988227749indi_o )
<=> ( ord_le777687553indi_o @ A_87 @ top_to988227749indi_o ) ) ).
thf(fact_69_less__top,axiom,
! [A_87: product_unit > $o] :
( ( A_87 != top_to1984820022unit_o )
<=> ( ord_le232288914unit_o @ A_87 @ top_to1984820022unit_o ) ) ).
thf(fact_70_less__top,axiom,
! [A_87: nat > $o] :
( ( A_87 != top_top_nat_o )
<=> ( ord_less_nat_o @ A_87 @ top_top_nat_o ) ) ).
thf(fact_71_assms_I1_J,axiom,
( member616898751_alt_o @ f
@ ( pi_Arr1304755663_alt_o @ arrow_734252939e_Prof
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : arrow_823908191le_Lin ) ) ).
thf(fact_72__096_Ie_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amk,axiom,
( ( member214075476le_alt @ ( produc1347929815le_alt @ e @ d )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ lab @ lba ) ) ) ) ).
thf(fact_73__096Lab_A_058_ALin_096,axiom,
member377231867_alt_o @ lab @ arrow_823908191le_Lin ).
thf(fact_74__096Lba_A_058_ALin_096,axiom,
member377231867_alt_o @ lba @ arrow_823908191le_Lin ).
thf(fact_75__096a_A_126_061_Ab_096,axiom,
a != b ).
thf(fact_76__096a_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_096,axiom,
member214075476le_alt @ ( produc1347929815le_alt @ a @ b ) @ lab ).
thf(fact_77__096b_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_096,axiom,
member214075476le_alt @ ( produc1347929815le_alt @ b @ a ) @ lba ).
thf(fact_78__096_Ia_M_Ab_J_A_126_058_ALba_096,axiom,
~ ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b ) @ lba ) ).
thf(fact_79__096_Ib_M_Aa_J_A_126_058_ALab_096,axiom,
~ ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a ) @ lab ) ).
thf(fact_80_PiProf,axiom,
! [N: nat] :
( member526088951_alt_o
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ N ) @ lab @ lba )
@ arrow_734252939e_Prof ) ).
thf(fact_81__096_B_Bthesis_O_A_I_B_BLab_O_A_091_124_Aa_A_060_092_060_094bsub_062Lab_,axiom,
~ ! [Lab: produc1501160679le_alt > $o] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b ) @ Lab )
=> ~ ( member377231867_alt_o @ Lab @ arrow_823908191le_Lin ) ) ).
thf(fact_82__096_B_Bthesis_O_A_I_B_BLba_O_A_091_124_Ab_A_060_092_060_094bsub_062Lba_,axiom,
~ ! [Lba: produc1501160679le_alt > $o] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a ) @ Lba )
=> ~ ( member377231867_alt_o @ Lba @ arrow_823908191le_Lin ) ) ).
thf(fact_83__096ALL_Ai_O_A_Ie_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_Amk,axiom,
! [I_1: arrow_1429601828e_indi] :
( ( ( ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ e @ d ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e ) ) )
& ( ~ ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( ( ( ( h @ I_1 )
= n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ e @ d ) @ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e ) ) )
& ( ( ( h @ I_1 )
!= n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ e @ d ) @ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) )
<=> ( ( ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a ) @ lab ) )
& ( ~ ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a ) @ lba ) ) ) ) ).
thf(fact_84_n_I2_J,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ M @ n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ M ) @ lab @ lba ) ) ) ) ).
thf(fact_85_notin__Lin__iff,axiom,
! [X: arrow_475358991le_alt,Y: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( ( X != Y )
=> ( ~ ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ L_2 )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ Y @ X ) @ L_2 ) ) ) ) ).
thf(fact_86_Lin__irrefl,axiom,
! [A_9: arrow_475358991le_alt,B_5: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_9 @ B_5 ) @ L_2 )
=> ~ ( member214075476le_alt @ ( produc1347929815le_alt @ B_5 @ A_9 ) @ L_2 ) ) ) ).
thf(fact_87_mktop__Lin,axiom,
! [X: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( member377231867_alt_o @ ( arrow_55669061_mktop @ L_2 @ X ) @ arrow_823908191le_Lin ) ) ).
thf(fact_88_mkbot__Lin,axiom,
! [X: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( member377231867_alt_o @ ( arrow_2054445623_mkbot @ L_2 @ X ) @ arrow_823908191le_Lin ) ) ).
thf(fact_89_above__Lin,axiom,
! [L_2: produc1501160679le_alt > $o,X: arrow_475358991le_alt,Y: arrow_475358991le_alt] :
( ( X != Y )
=> ( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( member377231867_alt_o @ ( arrow_789600939_above @ L_2 @ X @ Y ) @ arrow_823908191le_Lin ) ) ) ).
thf(fact_90_const__Lin__Prof,axiom,
! [L_2: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( member526088951_alt_o
@ ^ [P_5: arrow_1429601828e_indi] : L_2
@ arrow_734252939e_Prof ) ) ).
thf(fact_91_linorder__cases,axiom,
! [X_66: nat,Y_50: nat] :
( ~ ( ord_less_nat @ X_66 @ Y_50 )
=> ( ( X_66 != Y_50 )
=> ( ord_less_nat @ Y_50 @ X_66 ) ) ) ).
thf(fact_92_order__less__asym,axiom,
! [X_65: nat > $o,Y_49: nat > $o] :
( ( ord_less_nat_o @ X_65 @ Y_49 )
=> ~ ( ord_less_nat_o @ Y_49 @ X_65 ) ) ).
thf(fact_93_order__less__asym,axiom,
! [X_65: product_unit > $o,Y_49: product_unit > $o] :
( ( ord_le232288914unit_o @ X_65 @ Y_49 )
=> ~ ( ord_le232288914unit_o @ Y_49 @ X_65 ) ) ).
thf(fact_94_order__less__asym,axiom,
! [X_65: arrow_1429601828e_indi > $o,Y_49: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_65 @ Y_49 )
=> ~ ( ord_le777687553indi_o @ Y_49 @ X_65 ) ) ).
thf(fact_95_order__less__asym,axiom,
! [X_65: nat,Y_49: nat] :
( ( ord_less_nat @ X_65 @ Y_49 )
=> ~ ( ord_less_nat @ Y_49 @ X_65 ) ) ).
thf(fact_96_xt1_I10_J,axiom,
! [Z_8: nat > $o,Y_48: nat > $o,X_64: nat > $o] :
( ( ord_less_nat_o @ Y_48 @ X_64 )
=> ( ( ord_less_nat_o @ Z_8 @ Y_48 )
=> ( ord_less_nat_o @ Z_8 @ X_64 ) ) ) ).
thf(fact_97_xt1_I10_J,axiom,
! [Z_8: product_unit > $o,Y_48: product_unit > $o,X_64: product_unit > $o] :
( ( ord_le232288914unit_o @ Y_48 @ X_64 )
=> ( ( ord_le232288914unit_o @ Z_8 @ Y_48 )
=> ( ord_le232288914unit_o @ Z_8 @ X_64 ) ) ) ).
thf(fact_98_xt1_I10_J,axiom,
! [Z_8: arrow_1429601828e_indi > $o,Y_48: arrow_1429601828e_indi > $o,X_64: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ Y_48 @ X_64 )
=> ( ( ord_le777687553indi_o @ Z_8 @ Y_48 )
=> ( ord_le777687553indi_o @ Z_8 @ X_64 ) ) ) ).
thf(fact_99_xt1_I10_J,axiom,
! [Z_8: nat,Y_48: nat,X_64: nat] :
( ( ord_less_nat @ Y_48 @ X_64 )
=> ( ( ord_less_nat @ Z_8 @ Y_48 )
=> ( ord_less_nat @ Z_8 @ X_64 ) ) ) ).
thf(fact_100_order__less__trans,axiom,
! [Z_7: nat > $o,X_63: nat > $o,Y_47: nat > $o] :
( ( ord_less_nat_o @ X_63 @ Y_47 )
=> ( ( ord_less_nat_o @ Y_47 @ Z_7 )
=> ( ord_less_nat_o @ X_63 @ Z_7 ) ) ) ).
thf(fact_101_order__less__trans,axiom,
! [Z_7: product_unit > $o,X_63: product_unit > $o,Y_47: product_unit > $o] :
( ( ord_le232288914unit_o @ X_63 @ Y_47 )
=> ( ( ord_le232288914unit_o @ Y_47 @ Z_7 )
=> ( ord_le232288914unit_o @ X_63 @ Z_7 ) ) ) ).
thf(fact_102_order__less__trans,axiom,
! [Z_7: arrow_1429601828e_indi > $o,X_63: arrow_1429601828e_indi > $o,Y_47: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_63 @ Y_47 )
=> ( ( ord_le777687553indi_o @ Y_47 @ Z_7 )
=> ( ord_le777687553indi_o @ X_63 @ Z_7 ) ) ) ).
thf(fact_103_order__less__trans,axiom,
! [Z_7: nat,X_63: nat,Y_47: nat] :
( ( ord_less_nat @ X_63 @ Y_47 )
=> ( ( ord_less_nat @ Y_47 @ Z_7 )
=> ( ord_less_nat @ X_63 @ Z_7 ) ) ) ).
thf(fact_104_xt1_I2_J,axiom,
! [C_36: nat > $o,B_67: nat > $o,A_86: nat > $o] :
( ( ord_less_nat_o @ B_67 @ A_86 )
=> ( ( B_67 = C_36 )
=> ( ord_less_nat_o @ C_36 @ A_86 ) ) ) ).
thf(fact_105_xt1_I2_J,axiom,
! [C_36: product_unit > $o,B_67: product_unit > $o,A_86: product_unit > $o] :
( ( ord_le232288914unit_o @ B_67 @ A_86 )
=> ( ( B_67 = C_36 )
=> ( ord_le232288914unit_o @ C_36 @ A_86 ) ) ) ).
thf(fact_106_xt1_I2_J,axiom,
! [C_36: arrow_1429601828e_indi > $o,B_67: arrow_1429601828e_indi > $o,A_86: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ B_67 @ A_86 )
=> ( ( B_67 = C_36 )
=> ( ord_le777687553indi_o @ C_36 @ A_86 ) ) ) ).
thf(fact_107_xt1_I2_J,axiom,
! [C_36: nat,B_67: nat,A_86: nat] :
( ( ord_less_nat @ B_67 @ A_86 )
=> ( ( B_67 = C_36 )
=> ( ord_less_nat @ C_36 @ A_86 ) ) ) ).
thf(fact_108_ord__less__eq__trans,axiom,
! [C_35: nat > $o,A_85: nat > $o,B_66: nat > $o] :
( ( ord_less_nat_o @ A_85 @ B_66 )
=> ( ( B_66 = C_35 )
=> ( ord_less_nat_o @ A_85 @ C_35 ) ) ) ).
thf(fact_109_ord__less__eq__trans,axiom,
! [C_35: product_unit > $o,A_85: product_unit > $o,B_66: product_unit > $o] :
( ( ord_le232288914unit_o @ A_85 @ B_66 )
=> ( ( B_66 = C_35 )
=> ( ord_le232288914unit_o @ A_85 @ C_35 ) ) ) ).
thf(fact_110_ord__less__eq__trans,axiom,
! [C_35: arrow_1429601828e_indi > $o,A_85: arrow_1429601828e_indi > $o,B_66: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_85 @ B_66 )
=> ( ( B_66 = C_35 )
=> ( ord_le777687553indi_o @ A_85 @ C_35 ) ) ) ).
thf(fact_111_ord__less__eq__trans,axiom,
! [C_35: nat,A_85: nat,B_66: nat] :
( ( ord_less_nat @ A_85 @ B_66 )
=> ( ( B_66 = C_35 )
=> ( ord_less_nat @ A_85 @ C_35 ) ) ) ).
thf(fact_112_xt1_I1_J,axiom,
! [C_34: nat > $o,A_84: nat > $o,B_65: nat > $o] :
( ( A_84 = B_65 )
=> ( ( ord_less_nat_o @ C_34 @ B_65 )
=> ( ord_less_nat_o @ C_34 @ A_84 ) ) ) ).
thf(fact_113_xt1_I1_J,axiom,
! [C_34: product_unit > $o,A_84: product_unit > $o,B_65: product_unit > $o] :
( ( A_84 = B_65 )
=> ( ( ord_le232288914unit_o @ C_34 @ B_65 )
=> ( ord_le232288914unit_o @ C_34 @ A_84 ) ) ) ).
thf(fact_114_xt1_I1_J,axiom,
! [C_34: arrow_1429601828e_indi > $o,A_84: arrow_1429601828e_indi > $o,B_65: arrow_1429601828e_indi > $o] :
( ( A_84 = B_65 )
=> ( ( ord_le777687553indi_o @ C_34 @ B_65 )
=> ( ord_le777687553indi_o @ C_34 @ A_84 ) ) ) ).
thf(fact_115_xt1_I1_J,axiom,
! [C_34: nat,A_84: nat,B_65: nat] :
( ( A_84 = B_65 )
=> ( ( ord_less_nat @ C_34 @ B_65 )
=> ( ord_less_nat @ C_34 @ A_84 ) ) ) ).
thf(fact_116_ord__eq__less__trans,axiom,
! [C_33: nat > $o,A_83: nat > $o,B_64: nat > $o] :
( ( A_83 = B_64 )
=> ( ( ord_less_nat_o @ B_64 @ C_33 )
=> ( ord_less_nat_o @ A_83 @ C_33 ) ) ) ).
thf(fact_117_ord__eq__less__trans,axiom,
! [C_33: product_unit > $o,A_83: product_unit > $o,B_64: product_unit > $o] :
( ( A_83 = B_64 )
=> ( ( ord_le232288914unit_o @ B_64 @ C_33 )
=> ( ord_le232288914unit_o @ A_83 @ C_33 ) ) ) ).
thf(fact_118_ord__eq__less__trans,axiom,
! [C_33: arrow_1429601828e_indi > $o,A_83: arrow_1429601828e_indi > $o,B_64: arrow_1429601828e_indi > $o] :
( ( A_83 = B_64 )
=> ( ( ord_le777687553indi_o @ B_64 @ C_33 )
=> ( ord_le777687553indi_o @ A_83 @ C_33 ) ) ) ).
thf(fact_119_ord__eq__less__trans,axiom,
! [C_33: nat,A_83: nat,B_64: nat] :
( ( A_83 = B_64 )
=> ( ( ord_less_nat @ B_64 @ C_33 )
=> ( ord_less_nat @ A_83 @ C_33 ) ) ) ).
thf(fact_120_xt1_I9_J,axiom,
! [B_63: nat > $o,A_82: nat > $o] :
( ( ord_less_nat_o @ B_63 @ A_82 )
=> ~ ( ord_less_nat_o @ A_82 @ B_63 ) ) ).
thf(fact_121_xt1_I9_J,axiom,
! [B_63: product_unit > $o,A_82: product_unit > $o] :
( ( ord_le232288914unit_o @ B_63 @ A_82 )
=> ~ ( ord_le232288914unit_o @ A_82 @ B_63 ) ) ).
thf(fact_122_xt1_I9_J,axiom,
! [B_63: arrow_1429601828e_indi > $o,A_82: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ B_63 @ A_82 )
=> ~ ( ord_le777687553indi_o @ A_82 @ B_63 ) ) ).
thf(fact_123_xt1_I9_J,axiom,
! [B_63: nat,A_82: nat] :
( ( ord_less_nat @ B_63 @ A_82 )
=> ~ ( ord_less_nat @ A_82 @ B_63 ) ) ).
thf(fact_124_order__less__asym_H,axiom,
! [A_81: nat > $o,B_62: nat > $o] :
( ( ord_less_nat_o @ A_81 @ B_62 )
=> ~ ( ord_less_nat_o @ B_62 @ A_81 ) ) ).
thf(fact_125_order__less__asym_H,axiom,
! [A_81: product_unit > $o,B_62: product_unit > $o] :
( ( ord_le232288914unit_o @ A_81 @ B_62 )
=> ~ ( ord_le232288914unit_o @ B_62 @ A_81 ) ) ).
thf(fact_126_order__less__asym_H,axiom,
! [A_81: arrow_1429601828e_indi > $o,B_62: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_81 @ B_62 )
=> ~ ( ord_le777687553indi_o @ B_62 @ A_81 ) ) ).
thf(fact_127_order__less__asym_H,axiom,
! [A_81: nat,B_62: nat] :
( ( ord_less_nat @ A_81 @ B_62 )
=> ~ ( ord_less_nat @ B_62 @ A_81 ) ) ).
thf(fact_128_order__less__imp__triv,axiom,
! [P_4: $o,X_62: nat > $o,Y_46: nat > $o] :
( ( ord_less_nat_o @ X_62 @ Y_46 )
=> ( ( ord_less_nat_o @ Y_46 @ X_62 )
=> P_4 ) ) ).
thf(fact_129_order__less__imp__triv,axiom,
! [P_4: $o,X_62: product_unit > $o,Y_46: product_unit > $o] :
( ( ord_le232288914unit_o @ X_62 @ Y_46 )
=> ( ( ord_le232288914unit_o @ Y_46 @ X_62 )
=> P_4 ) ) ).
thf(fact_130_order__less__imp__triv,axiom,
! [P_4: $o,X_62: arrow_1429601828e_indi > $o,Y_46: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_62 @ Y_46 )
=> ( ( ord_le777687553indi_o @ Y_46 @ X_62 )
=> P_4 ) ) ).
thf(fact_131_order__less__imp__triv,axiom,
! [P_4: $o,X_62: nat,Y_46: nat] :
( ( ord_less_nat @ X_62 @ Y_46 )
=> ( ( ord_less_nat @ Y_46 @ X_62 )
=> P_4 ) ) ).
thf(fact_132_order__less__imp__not__eq2,axiom,
! [X_61: nat > $o,Y_45: nat > $o] :
( ( ord_less_nat_o @ X_61 @ Y_45 )
=> ( Y_45 != X_61 ) ) ).
thf(fact_133_order__less__imp__not__eq2,axiom,
! [X_61: product_unit > $o,Y_45: product_unit > $o] :
( ( ord_le232288914unit_o @ X_61 @ Y_45 )
=> ( Y_45 != X_61 ) ) ).
thf(fact_134_order__less__imp__not__eq2,axiom,
! [X_61: arrow_1429601828e_indi > $o,Y_45: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_61 @ Y_45 )
=> ( Y_45 != X_61 ) ) ).
thf(fact_135_order__less__imp__not__eq2,axiom,
! [X_61: nat,Y_45: nat] :
( ( ord_less_nat @ X_61 @ Y_45 )
=> ( Y_45 != X_61 ) ) ).
thf(fact_136_order__less__imp__not__eq,axiom,
! [X_60: nat > $o,Y_44: nat > $o] :
( ( ord_less_nat_o @ X_60 @ Y_44 )
=> ( X_60 != Y_44 ) ) ).
thf(fact_137_order__less__imp__not__eq,axiom,
! [X_60: product_unit > $o,Y_44: product_unit > $o] :
( ( ord_le232288914unit_o @ X_60 @ Y_44 )
=> ( X_60 != Y_44 ) ) ).
thf(fact_138_order__less__imp__not__eq,axiom,
! [X_60: arrow_1429601828e_indi > $o,Y_44: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_60 @ Y_44 )
=> ( X_60 != Y_44 ) ) ).
thf(fact_139_order__less__imp__not__eq,axiom,
! [X_60: nat,Y_44: nat] :
( ( ord_less_nat @ X_60 @ Y_44 )
=> ( X_60 != Y_44 ) ) ).
thf(fact_140_order__less__imp__not__less,axiom,
! [X_59: nat > $o,Y_43: nat > $o] :
( ( ord_less_nat_o @ X_59 @ Y_43 )
=> ~ ( ord_less_nat_o @ Y_43 @ X_59 ) ) ).
thf(fact_141_order__less__imp__not__less,axiom,
! [X_59: product_unit > $o,Y_43: product_unit > $o] :
( ( ord_le232288914unit_o @ X_59 @ Y_43 )
=> ~ ( ord_le232288914unit_o @ Y_43 @ X_59 ) ) ).
thf(fact_142_order__less__imp__not__less,axiom,
! [X_59: arrow_1429601828e_indi > $o,Y_43: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_59 @ Y_43 )
=> ~ ( ord_le777687553indi_o @ Y_43 @ X_59 ) ) ).
thf(fact_143_order__less__imp__not__less,axiom,
! [X_59: nat,Y_43: nat] :
( ( ord_less_nat @ X_59 @ Y_43 )
=> ~ ( ord_less_nat @ Y_43 @ X_59 ) ) ).
thf(fact_144_order__less__not__sym,axiom,
! [X_58: nat > $o,Y_42: nat > $o] :
( ( ord_less_nat_o @ X_58 @ Y_42 )
=> ~ ( ord_less_nat_o @ Y_42 @ X_58 ) ) ).
thf(fact_145_order__less__not__sym,axiom,
! [X_58: product_unit > $o,Y_42: product_unit > $o] :
( ( ord_le232288914unit_o @ X_58 @ Y_42 )
=> ~ ( ord_le232288914unit_o @ Y_42 @ X_58 ) ) ).
thf(fact_146_order__less__not__sym,axiom,
! [X_58: arrow_1429601828e_indi > $o,Y_42: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_58 @ Y_42 )
=> ~ ( ord_le777687553indi_o @ Y_42 @ X_58 ) ) ).
thf(fact_147_order__less__not__sym,axiom,
! [X_58: nat,Y_42: nat] :
( ( ord_less_nat @ X_58 @ Y_42 )
=> ~ ( ord_less_nat @ Y_42 @ X_58 ) ) ).
thf(fact_148_less__imp__neq,axiom,
! [X_57: nat > $o,Y_41: nat > $o] :
( ( ord_less_nat_o @ X_57 @ Y_41 )
=> ( X_57 != Y_41 ) ) ).
thf(fact_149_less__imp__neq,axiom,
! [X_57: product_unit > $o,Y_41: product_unit > $o] :
( ( ord_le232288914unit_o @ X_57 @ Y_41 )
=> ( X_57 != Y_41 ) ) ).
thf(fact_150_less__imp__neq,axiom,
! [X_57: arrow_1429601828e_indi > $o,Y_41: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_57 @ Y_41 )
=> ( X_57 != Y_41 ) ) ).
thf(fact_151_less__imp__neq,axiom,
! [X_57: nat,Y_41: nat] :
( ( ord_less_nat @ X_57 @ Y_41 )
=> ( X_57 != Y_41 ) ) ).
thf(fact_152_linorder__neqE,axiom,
! [X_56: nat,Y_40: nat] :
( ( X_56 != Y_40 )
=> ( ~ ( ord_less_nat @ X_56 @ Y_40 )
=> ( ord_less_nat @ Y_40 @ X_56 ) ) ) ).
thf(fact_153_linorder__antisym__conv3,axiom,
! [Y_39: nat,X_55: nat] :
( ~ ( ord_less_nat @ Y_39 @ X_55 )
=> ( ~ ( ord_less_nat @ X_55 @ Y_39 )
<=> ( X_55 = Y_39 ) ) ) ).
thf(fact_154_linorder__less__linear,axiom,
! [X_54: nat,Y_38: nat] :
( ( ord_less_nat @ X_54 @ Y_38 )
| ( X_54 = Y_38 )
| ( ord_less_nat @ Y_38 @ X_54 ) ) ).
thf(fact_155_not__less__iff__gr__or__eq,axiom,
! [X_53: nat,Y_37: nat] :
( ~ ( ord_less_nat @ X_53 @ Y_37 )
<=> ( ( ord_less_nat @ Y_37 @ X_53 )
| ( X_53 = Y_37 ) ) ) ).
thf(fact_156_linorder__neq__iff,axiom,
! [X_52: nat,Y_36: nat] :
( ( X_52 != Y_36 )
<=> ( ( ord_less_nat @ X_52 @ Y_36 )
| ( ord_less_nat @ Y_36 @ X_52 ) ) ) ).
thf(fact_157_order__less__irrefl,axiom,
! [X_51: nat > $o] :
~ ( ord_less_nat_o @ X_51 @ X_51 ) ).
thf(fact_158_order__less__irrefl,axiom,
! [X_51: product_unit > $o] :
~ ( ord_le232288914unit_o @ X_51 @ X_51 ) ).
thf(fact_159_order__less__irrefl,axiom,
! [X_51: arrow_1429601828e_indi > $o] :
~ ( ord_le777687553indi_o @ X_51 @ X_51 ) ).
thf(fact_160_order__less__irrefl,axiom,
! [X_51: nat] :
~ ( ord_less_nat @ X_51 @ X_51 ) ).
thf(fact_161_UNIV__def,axiom,
( top_to1842727771lt_o_o
= ( collec94295101_alt_o
@ ^ [X_1: produc1501160679le_alt > $o] : $true ) ) ).
thf(fact_162_UNIV__def,axiom,
( top_to2122763103lt_o_o
= ( collec682858041_alt_o
@ ^ [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] : $true ) ) ).
thf(fact_163_UNIV__def,axiom,
( top_to1969627639lt_o_o
= ( collec2009291517_alt_o
@ ^ [X_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] : $true ) ) ).
thf(fact_164_UNIV__def,axiom,
( top_to1841428258_alt_o
= ( collec869865362le_alt
@ ^ [X_1: produc1501160679le_alt] : $true ) ) ).
thf(fact_165_UNIV__def,axiom,
( top_to988227749indi_o
= ( collec22405327e_indi
@ ^ [X_1: arrow_1429601828e_indi] : $true ) ) ).
thf(fact_166_UNIV__def,axiom,
( top_to1984820022unit_o
= ( collect_Product_unit
@ ^ [X_1: product_unit] : $true ) ) ).
thf(fact_167_UNIV__def,axiom,
( top_top_nat_o
= ( collect_nat
@ ^ [X_1: nat] : $true ) ) ).
thf(fact_168_mem__def,axiom,
! [X_50: arrow_1429601828e_indi,A_80: arrow_1429601828e_indi > $o] :
( ( member2052026769e_indi @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_169_mem__def,axiom,
! [X_50: $o,A_80: $o > $o] :
( ( member_o @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_170_mem__def,axiom,
! [X_50: product_unit,A_80: product_unit > $o] :
( ( member_Product_unit @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_171_mem__def,axiom,
! [X_50: produc1501160679le_alt,A_80: produc1501160679le_alt > $o] :
( ( member214075476le_alt @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_172_mem__def,axiom,
! [X_50: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_80: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_173_mem__def,axiom,
! [X_50: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_80: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_174_mem__def,axiom,
! [X_50: produc1501160679le_alt > $o,A_80: ( produc1501160679le_alt > $o ) > $o] :
( ( member377231867_alt_o @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_175_mem__def,axiom,
! [X_50: nat,A_80: nat > $o] :
( ( member_nat @ X_50 @ A_80 )
<=> ( A_80 @ X_50 ) ) ).
thf(fact_176_Collect__def,axiom,
! [P_3: product_unit > $o] :
( ( collect_Product_unit @ P_3 )
= P_3 ) ).
thf(fact_177_Collect__def,axiom,
! [P_3: arrow_1429601828e_indi > $o] :
( ( collec22405327e_indi @ P_3 )
= P_3 ) ).
thf(fact_178_Collect__def,axiom,
! [P_3: nat > $o] :
( ( collect_nat @ P_3 )
= P_3 ) ).
thf(fact_179_in__above,axiom,
! [X: arrow_475358991le_alt,Y: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o,A_9: arrow_475358991le_alt,B_5: arrow_475358991le_alt] :
( ( A_9 != B_5 )
=> ( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ ( arrow_789600939_above @ L_2 @ A_9 @ B_5 ) )
<=> ( ( X != Y )
& ( ( X = B_5 )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_9 @ Y ) @ L_2 ) )
& ( ( X != B_5 )
=> ( ( ( Y = B_5 )
=> ( ( X = A_9 )
| ( member214075476le_alt @ ( produc1347929815le_alt @ X @ A_9 ) @ L_2 ) ) )
& ( ( Y != B_5 )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ L_2 ) ) ) ) ) ) ) ) ).
thf(fact_180_pred__equals__eq2,axiom,
! [S_3: produc1501160679le_alt > $o,R_3: produc1501160679le_alt > $o] :
( ! [X_1: arrow_475358991le_alt,Xa: arrow_475358991le_alt] :
( ( member214075476le_alt @ ( produc1347929815le_alt @ X_1 @ Xa ) @ R_3 )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ X_1 @ Xa ) @ S_3 ) )
<=> ( R_3 = S_3 ) ) ).
thf(fact_181_n_I3_J,axiom,
( member214075476le_alt @ ( produc1347929815le_alt @ a @ b )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ ( plus_plus_nat @ n @ one_one_nat ) ) @ lab @ lba ) ) ) ).
thf(fact_182__096_Ic_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Am,axiom,
( ( member214075476le_alt @ ( produc1347929815le_alt @ c @ e )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] :
( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ n ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e )
@ ( if_Pro1561232536_alt_o
@ ( ( h @ I_1 )
= n )
@ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e )
@ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ ( plus_plus_nat @ n @ one_one_nat ) ) @ lab @ lba ) ) ) ) ).
thf(fact_183__096ALL_Ai_O_A_Ic_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_Am,axiom,
! [I_1: arrow_1429601828e_indi] :
( ( ( ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ e ) @ ( arrow_55669061_mktop @ ( p @ I_1 ) @ e ) ) )
& ( ~ ( ord_less_nat @ ( h @ I_1 ) @ n )
=> ( ( ( ( h @ I_1 )
= n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ e ) @ ( arrow_789600939_above @ ( p @ I_1 ) @ c @ e ) ) )
& ( ( ( h @ I_1 )
!= n )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ c @ e ) @ ( arrow_2054445623_mkbot @ ( p @ I_1 ) @ e ) ) ) ) ) )
<=> ( ( ( ord_less_nat @ ( h @ I_1 ) @ ( plus_plus_nat @ n @ one_one_nat ) )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b ) @ lab ) )
& ( ~ ( ord_less_nat @ ( h @ I_1 ) @ ( plus_plus_nat @ n @ one_one_nat ) )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b ) @ lba ) ) ) ) ).
thf(fact_184_inv__f__eq,axiom,
! [X_49: arrow_1429601828e_indi,Y_35: nat,F_30: arrow_1429601828e_indi > nat] :
( ( inj_on978774663di_nat @ F_30 @ top_to988227749indi_o )
=> ( ( ( F_30 @ X_49 )
= Y_35 )
=> ( ( hilber598459244di_nat @ top_to988227749indi_o @ F_30 @ Y_35 )
= X_49 ) ) ) ).
thf(fact_185_inv__f__eq,axiom,
! [X_49: nat,Y_35: nat,F_30: nat > nat] :
( ( inj_on_nat_nat @ F_30 @ top_top_nat_o )
=> ( ( ( F_30 @ X_49 )
= Y_35 )
=> ( ( hilber195283148at_nat @ top_top_nat_o @ F_30 @ Y_35 )
= X_49 ) ) ) ).
thf(fact_186_inv__f__f,axiom,
! [X_48: arrow_1429601828e_indi,F_29: arrow_1429601828e_indi > nat] :
( ( inj_on978774663di_nat @ F_29 @ top_to988227749indi_o )
=> ( ( hilber598459244di_nat @ top_to988227749indi_o @ F_29 @ ( F_29 @ X_48 ) )
= X_48 ) ) ).
thf(fact_187_inv__f__f,axiom,
! [X_48: nat,F_29: nat > nat] :
( ( inj_on_nat_nat @ F_29 @ top_top_nat_o )
=> ( ( hilber195283148at_nat @ top_top_nat_o @ F_29 @ ( F_29 @ X_48 ) )
= X_48 ) ) ).
thf(fact_188_dictatorI,axiom,
! [I: arrow_1429601828e_indi,F_18: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] :
( ( member616898751_alt_o @ F_18
@ ( pi_Arr1304755663_alt_o @ arrow_734252939e_Prof
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : arrow_823908191le_Lin ) )
=> ( ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ arrow_734252939e_Prof )
=> ! [A_3: arrow_475358991le_alt,B_61: arrow_475358991le_alt] :
( ( A_3 != B_61 )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( X_1 @ I ) )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ A_3 @ B_61 ) @ ( F_18 @ X_1 ) ) ) ) )
=> ( arrow_1212662430ctator @ F_18 @ I ) ) ) ).
thf(fact_189_PiE,axiom,
! [X_47: produc1501160679le_alt,F_28: produc1501160679le_alt > $o,A_79: produc1501160679le_alt > $o,B_60: produc1501160679le_alt > $o > $o] :
( ( member377231867_alt_o @ F_28 @ ( pi_Pro1701359055_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member214075476le_alt @ X_47 @ A_79 ) ) ) ).
thf(fact_190_PiE,axiom,
! [X_47: arrow_1429601828e_indi,F_28: arrow_1429601828e_indi > produc1501160679le_alt,A_79: arrow_1429601828e_indi > $o,B_60: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member351225838le_alt @ F_28 @ ( pi_Arr329216900le_alt @ A_79 @ B_60 ) )
=> ( ~ ( member214075476le_alt @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member2052026769e_indi @ X_47 @ A_79 ) ) ) ).
thf(fact_191_PiE,axiom,
! [X_47: $o,F_28: $o > produc1501160679le_alt,A_79: $o > $o,B_60: $o > produc1501160679le_alt > $o] :
( ( member492167345le_alt @ F_28 @ ( pi_o_P657324555le_alt @ A_79 @ B_60 ) )
=> ( ~ ( member214075476le_alt @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_o @ X_47 @ A_79 ) ) ) ).
thf(fact_192_PiE,axiom,
! [X_47: product_unit,F_28: product_unit > produc1501160679le_alt,A_79: product_unit > $o,B_60: product_unit > produc1501160679le_alt > $o] :
( ( member495332125le_alt @ F_28 @ ( pi_Pro701847987le_alt @ A_79 @ B_60 ) )
=> ( ~ ( member214075476le_alt @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_Product_unit @ X_47 @ A_79 ) ) ) ).
thf(fact_193_PiE,axiom,
! [X_47: arrow_1429601828e_indi,F_28: arrow_1429601828e_indi > arrow_1429601828e_indi > produc1501160679le_alt > $o,A_79: arrow_1429601828e_indi > $o,B_60: arrow_1429601828e_indi > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( member1234151027_alt_o @ F_28 @ ( pi_Arr1060328391_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member526088951_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member2052026769e_indi @ X_47 @ A_79 ) ) ) ).
thf(fact_194_PiE,axiom,
! [X_47: $o,F_28: $o > arrow_1429601828e_indi > produc1501160679le_alt > $o,A_79: $o > $o,B_60: $o > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( member1394214384_alt_o @ F_28 @ ( pi_o_A1182933120_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member526088951_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_o @ X_47 @ A_79 ) ) ) ).
thf(fact_195_PiE,axiom,
! [X_47: product_unit,F_28: product_unit > arrow_1429601828e_indi > produc1501160679le_alt > $o,A_79: product_unit > $o,B_60: product_unit > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( member283501700_alt_o @ F_28 @ ( pi_Pro1662176984_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member526088951_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_Product_unit @ X_47 @ A_79 ) ) ) ).
thf(fact_196_PiE,axiom,
! [X_47: arrow_1429601828e_indi,F_28: arrow_1429601828e_indi > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_79: arrow_1429601828e_indi > $o,B_60: arrow_1429601828e_indi > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( member811956313_alt_o @ F_28 @ ( pi_Arr1564509167_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member616898751_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member2052026769e_indi @ X_47 @ A_79 ) ) ) ).
thf(fact_197_PiE,axiom,
! [X_47: $o,F_28: $o > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_79: $o > $o,B_60: $o > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( member1957863580_alt_o @ F_28 @ ( pi_o_A1186128886_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member616898751_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_o @ X_47 @ A_79 ) ) ) ).
thf(fact_198_PiE,axiom,
! [X_47: product_unit,F_28: product_unit > ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_79: product_unit > $o,B_60: product_unit > ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( member1536989448_alt_o @ F_28 @ ( pi_Pro1782982558_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member616898751_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_Product_unit @ X_47 @ A_79 ) ) ) ).
thf(fact_199_PiE,axiom,
! [X_47: $o,F_28: $o > produc1501160679le_alt > $o,A_79: $o > $o,B_60: $o > ( produc1501160679le_alt > $o ) > $o] :
( ( member1862122484_alt_o @ F_28 @ ( pi_o_P553196292_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member377231867_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_o @ X_47 @ A_79 ) ) ) ).
thf(fact_200_PiE,axiom,
! [X_47: product_unit,F_28: product_unit > produc1501160679le_alt > $o,A_79: product_unit > $o,B_60: product_unit > ( produc1501160679le_alt > $o ) > $o] :
( ( member1661784200_alt_o @ F_28 @ ( pi_Pro1312660828_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member377231867_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_Product_unit @ X_47 @ A_79 ) ) ) ).
thf(fact_201_PiE,axiom,
! [X_47: arrow_1429601828e_indi,F_28: arrow_1429601828e_indi > nat,A_79: arrow_1429601828e_indi > $o,B_60: arrow_1429601828e_indi > nat > $o] :
( ( member1315464153di_nat @ F_28 @ ( pi_Arr251692973di_nat @ A_79 @ B_60 ) )
=> ( ~ ( member_nat @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member2052026769e_indi @ X_47 @ A_79 ) ) ) ).
thf(fact_202_PiE,axiom,
! [X_47: $o,F_28: $o > nat,A_79: $o > $o,B_60: $o > nat > $o] :
( ( member_o_nat @ F_28 @ ( pi_o_nat @ A_79 @ B_60 ) )
=> ( ~ ( member_nat @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_o @ X_47 @ A_79 ) ) ) ).
thf(fact_203_PiE,axiom,
! [X_47: product_unit,F_28: product_unit > nat,A_79: product_unit > $o,B_60: product_unit > nat > $o] :
( ( member1827227242it_nat @ F_28 @ ( pi_Product_unit_nat @ A_79 @ B_60 ) )
=> ( ~ ( member_nat @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_Product_unit @ X_47 @ A_79 ) ) ) ).
thf(fact_204_PiE,axiom,
! [X_47: produc1501160679le_alt,F_28: produc1501160679le_alt > arrow_1429601828e_indi,A_79: produc1501160679le_alt > $o,B_60: produc1501160679le_alt > arrow_1429601828e_indi > $o] :
( ( member1640632174e_indi @ F_28 @ ( pi_Pro1767455108e_indi @ A_79 @ B_60 ) )
=> ( ~ ( member2052026769e_indi @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member214075476le_alt @ X_47 @ A_79 ) ) ) ).
thf(fact_205_PiE,axiom,
! [X_47: produc1501160679le_alt,F_28: produc1501160679le_alt > product_unit,A_79: produc1501160679le_alt > $o,B_60: produc1501160679le_alt > product_unit > $o] :
( ( member593902749t_unit @ F_28 @ ( pi_Pro1475896499t_unit @ A_79 @ B_60 ) )
=> ( ~ ( member_Product_unit @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member214075476le_alt @ X_47 @ A_79 ) ) ) ).
thf(fact_206_PiE,axiom,
! [X_47: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_28: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > arrow_1429601828e_indi,A_79: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_60: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o] :
( ( member44294883e_indi @ F_28 @ ( pi_Arr1232280765e_indi @ A_79 @ B_60 ) )
=> ( ~ ( member2052026769e_indi @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member526088951_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_207_PiE,axiom,
! [X_47: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_28: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,A_79: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_60: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o > $o] :
( ( member939334982lt_o_o @ F_28 @ ( pi_Arr952516694lt_o_o @ A_79 @ B_60 ) )
=> ( ~ ( member_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member526088951_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_208_PiE,axiom,
! [X_47: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_28: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > product_unit,A_79: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_60: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > product_unit > $o] :
( ( member843528338t_unit @ F_28 @ ( pi_Arr1963174508t_unit @ A_79 @ B_60 ) )
=> ( ~ ( member_Product_unit @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member526088951_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_209_PiE,axiom,
! [X_47: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,F_28: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi,A_79: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_60: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o] :
( ( member1452482393e_indi @ F_28 @ ( pi_Arr338314351e_indi @ A_79 @ B_60 ) )
=> ( ~ ( member2052026769e_indi @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member616898751_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_210_PiE,axiom,
! [X_47: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,F_28: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,A_79: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_60: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o > $o] :
( ( member1823529808lt_o_o @ F_28 @ ( pi_Arr195212324lt_o_o @ A_79 @ B_60 ) )
=> ( ~ ( member_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member616898751_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_211_PiE,axiom,
! [X_47: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,F_28: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > product_unit,A_79: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_60: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > product_unit > $o] :
( ( member1924666376t_unit @ F_28 @ ( pi_Arr830584606t_unit @ A_79 @ B_60 ) )
=> ( ~ ( member_Product_unit @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member616898751_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_212_PiE,axiom,
! [X_47: produc1501160679le_alt > $o,F_28: ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi,A_79: ( produc1501160679le_alt > $o ) > $o,B_60: ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi > $o] :
( ( member304866663e_indi @ F_28 @ ( pi_Pro468373057e_indi @ A_79 @ B_60 ) )
=> ( ~ ( member2052026769e_indi @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member377231867_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_213_PiE,axiom,
! [X_47: produc1501160679le_alt > $o,F_28: ( produc1501160679le_alt > $o ) > $o,A_79: ( produc1501160679le_alt > $o ) > $o,B_60: ( produc1501160679le_alt > $o ) > $o > $o] :
( ( member1961363906lt_o_o @ F_28 @ ( pi_Pro422690258lt_o_o @ A_79 @ B_60 ) )
=> ( ~ ( member_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member377231867_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_214_PiE,axiom,
! [X_47: produc1501160679le_alt > $o,F_28: ( produc1501160679le_alt > $o ) > product_unit,A_79: ( produc1501160679le_alt > $o ) > $o,B_60: ( produc1501160679le_alt > $o ) > product_unit > $o] :
( ( member221730070t_unit @ F_28 @ ( pi_Pro1306850800t_unit @ A_79 @ B_60 ) )
=> ( ~ ( member_Product_unit @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member377231867_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_215_PiE,axiom,
! [X_47: nat,F_28: nat > arrow_1429601828e_indi,A_79: nat > $o,B_60: nat > arrow_1429601828e_indi > $o] :
( ( member1391860553e_indi @ F_28 @ ( pi_nat1219304995e_indi @ A_79 @ B_60 ) )
=> ( ~ ( member2052026769e_indi @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_nat @ X_47 @ A_79 ) ) ) ).
thf(fact_216_PiE,axiom,
! [X_47: nat,F_28: nat > $o,A_79: nat > $o,B_60: nat > $o > $o] :
( ( member_nat_o @ F_28 @ ( pi_nat_o @ A_79 @ B_60 ) )
=> ( ~ ( member_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_nat @ X_47 @ A_79 ) ) ) ).
thf(fact_217_PiE,axiom,
! [X_47: nat,F_28: nat > product_unit,A_79: nat > $o,B_60: nat > product_unit > $o] :
( ( member616671224t_unit @ F_28 @ ( pi_nat_Product_unit @ A_79 @ B_60 ) )
=> ( ~ ( member_Product_unit @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member_nat @ X_47 @ A_79 ) ) ) ).
thf(fact_218_PiE,axiom,
! [X_47: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_28: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_79: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_60: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ F_28 @ ( pi_Arr1304755663_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member377231867_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member526088951_alt_o @ X_47 @ A_79 ) ) ) ).
thf(fact_219_PiE,axiom,
! [X_47: arrow_1429601828e_indi,F_28: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_79: arrow_1429601828e_indi > $o,B_60: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ F_28 @ ( pi_Arr1929480907_alt_o @ A_79 @ B_60 ) )
=> ( ~ ( member377231867_alt_o @ ( F_28 @ X_47 ) @ ( B_60 @ X_47 ) )
=> ~ ( member2052026769e_indi @ X_47 @ A_79 ) ) ) ).
thf(fact_220_complete__Lin,axiom,
! [A_9: arrow_475358991le_alt,B_5: arrow_475358991le_alt] :
( ( A_9 != B_5 )
=> ? [X_1: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ X_1 @ arrow_823908191le_Lin )
& ( member214075476le_alt @ ( produc1347929815le_alt @ A_9 @ B_5 ) @ X_1 ) ) ) ).
thf(fact_221_Pi__UNIV,axiom,
! [A_78: produc1501160679le_alt > $o] :
( ( pi_Pro1701359055_alt_o @ A_78
@ ^ [Uu: produc1501160679le_alt] : top_top_o_o )
= top_to1842727771lt_o_o ) ).
thf(fact_222_Pi__UNIV,axiom,
! [A_78: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( pi_Arr1304755663_alt_o @ A_78
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : top_to1842727771lt_o_o )
= top_to1969627639lt_o_o ) ).
thf(fact_223_Pi__UNIV,axiom,
! [A_78: arrow_1429601828e_indi > $o] :
( ( pi_Arr1929480907_alt_o @ A_78
@ ^ [Uu: arrow_1429601828e_indi] : top_to1842727771lt_o_o )
= top_to2122763103lt_o_o ) ).
thf(fact_224_order__refl,axiom,
! [X_46: $o] : ( ord_less_eq_o @ X_46 @ X_46 ) ).
thf(fact_225_order__refl,axiom,
! [X_46: nat] : ( ord_less_eq_nat @ X_46 @ X_46 ) ).
thf(fact_226_order__refl,axiom,
! [X_46: nat > $o] : ( ord_less_eq_nat_o @ X_46 @ X_46 ) ).
thf(fact_227__096EX_An_060N_O_A_IALL_Am_060_061n_O_Ab_A_060_092_060_094bsub_062F_A_I,axiom,
? [N_1: nat] :
( ( ord_less_nat @ N_1 @ ( finite97476818e_indi @ top_to988227749indi_o ) )
& ! [M: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ M ) @ lab @ lba ) ) ) )
& ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ ( plus_plus_nat @ N_1 @ one_one_nat ) ) @ lab @ lba ) ) ) ) ).
thf(fact_228__096_B_Bthesis_O_A_I_B_Bn_O_A_091_124_An_A_060_AN_059_AALL_Am_060_061n_,axiom,
~ ! [N_1: nat] :
( ( ord_less_nat @ N_1 @ ( finite97476818e_indi @ top_to988227749indi_o ) )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ b @ a )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ M ) @ lab @ lba ) ) ) )
=> ~ ( member214075476le_alt @ ( produc1347929815le_alt @ a @ b )
@ ( f
@ ^ [I_1: arrow_1429601828e_indi] : ( if_Pro1561232536_alt_o @ ( ord_less_nat @ ( h @ I_1 ) @ ( plus_plus_nat @ N_1 @ one_one_nat ) ) @ lab @ lba ) ) ) ) ) ).
thf(fact_229_linorder__le__cases,axiom,
! [X_45: nat,Y_34: nat] :
( ~ ( ord_less_eq_nat @ X_45 @ Y_34 )
=> ( ord_less_eq_nat @ Y_34 @ X_45 ) ) ).
thf(fact_230_le__funE,axiom,
! [X_44: nat,F_27: nat > $o,G_8: nat > $o] :
( ( ord_less_eq_nat_o @ F_27 @ G_8 )
=> ( ord_less_eq_o @ ( F_27 @ X_44 ) @ ( G_8 @ X_44 ) ) ) ).
thf(fact_231_xt1_I6_J,axiom,
! [Z_6: $o,Y_33: $o,X_43: $o] :
( ( ord_less_eq_o @ Y_33 @ X_43 )
=> ( ( ord_less_eq_o @ Z_6 @ Y_33 )
=> ( ord_less_eq_o @ Z_6 @ X_43 ) ) ) ).
thf(fact_232_xt1_I6_J,axiom,
! [Z_6: nat,Y_33: nat,X_43: nat] :
( ( ord_less_eq_nat @ Y_33 @ X_43 )
=> ( ( ord_less_eq_nat @ Z_6 @ Y_33 )
=> ( ord_less_eq_nat @ Z_6 @ X_43 ) ) ) ).
thf(fact_233_xt1_I6_J,axiom,
! [Z_6: nat > $o,Y_33: nat > $o,X_43: nat > $o] :
( ( ord_less_eq_nat_o @ Y_33 @ X_43 )
=> ( ( ord_less_eq_nat_o @ Z_6 @ Y_33 )
=> ( ord_less_eq_nat_o @ Z_6 @ X_43 ) ) ) ).
thf(fact_234_xt1_I5_J,axiom,
! [Y_32: $o,X_42: $o] :
( ( ord_less_eq_o @ Y_32 @ X_42 )
=> ( ( ord_less_eq_o @ X_42 @ Y_32 )
=> ( X_42
<=> Y_32 ) ) ) ).
thf(fact_235_xt1_I5_J,axiom,
! [Y_32: nat,X_42: nat] :
( ( ord_less_eq_nat @ Y_32 @ X_42 )
=> ( ( ord_less_eq_nat @ X_42 @ Y_32 )
=> ( X_42 = Y_32 ) ) ) ).
thf(fact_236_xt1_I5_J,axiom,
! [Y_32: nat > $o,X_42: nat > $o] :
( ( ord_less_eq_nat_o @ Y_32 @ X_42 )
=> ( ( ord_less_eq_nat_o @ X_42 @ Y_32 )
=> ( X_42 = Y_32 ) ) ) ).
thf(fact_237_order__trans,axiom,
! [Z_5: $o,X_41: $o,Y_31: $o] :
( ( ord_less_eq_o @ X_41 @ Y_31 )
=> ( ( ord_less_eq_o @ Y_31 @ Z_5 )
=> ( ord_less_eq_o @ X_41 @ Z_5 ) ) ) ).
thf(fact_238_order__trans,axiom,
! [Z_5: nat,X_41: nat,Y_31: nat] :
( ( ord_less_eq_nat @ X_41 @ Y_31 )
=> ( ( ord_less_eq_nat @ Y_31 @ Z_5 )
=> ( ord_less_eq_nat @ X_41 @ Z_5 ) ) ) ).
thf(fact_239_order__trans,axiom,
! [Z_5: nat > $o,X_41: nat > $o,Y_31: nat > $o] :
( ( ord_less_eq_nat_o @ X_41 @ Y_31 )
=> ( ( ord_less_eq_nat_o @ Y_31 @ Z_5 )
=> ( ord_less_eq_nat_o @ X_41 @ Z_5 ) ) ) ).
thf(fact_240_order__antisym,axiom,
! [X_40: $o,Y_30: $o] :
( ( ord_less_eq_o @ X_40 @ Y_30 )
=> ( ( ord_less_eq_o @ Y_30 @ X_40 )
=> ( X_40
<=> Y_30 ) ) ) ).
thf(fact_241_order__antisym,axiom,
! [X_40: nat,Y_30: nat] :
( ( ord_less_eq_nat @ X_40 @ Y_30 )
=> ( ( ord_less_eq_nat @ Y_30 @ X_40 )
=> ( X_40 = Y_30 ) ) ) ).
thf(fact_242_order__antisym,axiom,
! [X_40: nat > $o,Y_30: nat > $o] :
( ( ord_less_eq_nat_o @ X_40 @ Y_30 )
=> ( ( ord_less_eq_nat_o @ Y_30 @ X_40 )
=> ( X_40 = Y_30 ) ) ) ).
thf(fact_243_xt1_I4_J,axiom,
! [C_32: $o,B_59: $o,A_77: $o] :
( ( ord_less_eq_o @ B_59 @ A_77 )
=> ( ( B_59
<=> C_32 )
=> ( ord_less_eq_o @ C_32 @ A_77 ) ) ) ).
thf(fact_244_xt1_I4_J,axiom,
! [C_32: nat,B_59: nat,A_77: nat] :
( ( ord_less_eq_nat @ B_59 @ A_77 )
=> ( ( B_59 = C_32 )
=> ( ord_less_eq_nat @ C_32 @ A_77 ) ) ) ).
thf(fact_245_xt1_I4_J,axiom,
! [C_32: nat > $o,B_59: nat > $o,A_77: nat > $o] :
( ( ord_less_eq_nat_o @ B_59 @ A_77 )
=> ( ( B_59 = C_32 )
=> ( ord_less_eq_nat_o @ C_32 @ A_77 ) ) ) ).
thf(fact_246_ord__le__eq__trans,axiom,
! [C_31: $o,A_76: $o,B_58: $o] :
( ( ord_less_eq_o @ A_76 @ B_58 )
=> ( ( B_58
<=> C_31 )
=> ( ord_less_eq_o @ A_76 @ C_31 ) ) ) ).
thf(fact_247_ord__le__eq__trans,axiom,
! [C_31: nat,A_76: nat,B_58: nat] :
( ( ord_less_eq_nat @ A_76 @ B_58 )
=> ( ( B_58 = C_31 )
=> ( ord_less_eq_nat @ A_76 @ C_31 ) ) ) ).
thf(fact_248_ord__le__eq__trans,axiom,
! [C_31: nat > $o,A_76: nat > $o,B_58: nat > $o] :
( ( ord_less_eq_nat_o @ A_76 @ B_58 )
=> ( ( B_58 = C_31 )
=> ( ord_less_eq_nat_o @ A_76 @ C_31 ) ) ) ).
thf(fact_249_xt1_I3_J,axiom,
! [C_30: $o,B_57: $o,A_75: $o] :
( ( A_75
<=> B_57 )
=> ( ( ord_less_eq_o @ C_30 @ B_57 )
=> ( ord_less_eq_o @ C_30 @ A_75 ) ) ) ).
thf(fact_250_xt1_I3_J,axiom,
! [C_30: nat,A_75: nat,B_57: nat] :
( ( A_75 = B_57 )
=> ( ( ord_less_eq_nat @ C_30 @ B_57 )
=> ( ord_less_eq_nat @ C_30 @ A_75 ) ) ) ).
thf(fact_251_xt1_I3_J,axiom,
! [C_30: nat > $o,A_75: nat > $o,B_57: nat > $o] :
( ( A_75 = B_57 )
=> ( ( ord_less_eq_nat_o @ C_30 @ B_57 )
=> ( ord_less_eq_nat_o @ C_30 @ A_75 ) ) ) ).
thf(fact_252_ord__eq__le__trans,axiom,
! [C_29: $o,B_56: $o,A_74: $o] :
( ( A_74
<=> B_56 )
=> ( ( ord_less_eq_o @ B_56 @ C_29 )
=> ( ord_less_eq_o @ A_74 @ C_29 ) ) ) ).
thf(fact_253_ord__eq__le__trans,axiom,
! [C_29: nat,A_74: nat,B_56: nat] :
( ( A_74 = B_56 )
=> ( ( ord_less_eq_nat @ B_56 @ C_29 )
=> ( ord_less_eq_nat @ A_74 @ C_29 ) ) ) ).
thf(fact_254_ord__eq__le__trans,axiom,
! [C_29: nat > $o,A_74: nat > $o,B_56: nat > $o] :
( ( A_74 = B_56 )
=> ( ( ord_less_eq_nat_o @ B_56 @ C_29 )
=> ( ord_less_eq_nat_o @ A_74 @ C_29 ) ) ) ).
thf(fact_255_order__antisym__conv,axiom,
! [Y_29: $o,X_39: $o] :
( ( ord_less_eq_o @ Y_29 @ X_39 )
=> ( ( ord_less_eq_o @ X_39 @ Y_29 )
<=> ( X_39
<=> Y_29 ) ) ) ).
thf(fact_256_order__antisym__conv,axiom,
! [Y_29: nat,X_39: nat] :
( ( ord_less_eq_nat @ Y_29 @ X_39 )
=> ( ( ord_less_eq_nat @ X_39 @ Y_29 )
<=> ( X_39 = Y_29 ) ) ) ).
thf(fact_257_order__antisym__conv,axiom,
! [Y_29: nat > $o,X_39: nat > $o] :
( ( ord_less_eq_nat_o @ Y_29 @ X_39 )
=> ( ( ord_less_eq_nat_o @ X_39 @ Y_29 )
<=> ( X_39 = Y_29 ) ) ) ).
thf(fact_258_le__funD,axiom,
! [X_38: nat,F_26: nat > $o,G_7: nat > $o] :
( ( ord_less_eq_nat_o @ F_26 @ G_7 )
=> ( ord_less_eq_o @ ( F_26 @ X_38 ) @ ( G_7 @ X_38 ) ) ) ).
thf(fact_259_order__eq__refl,axiom,
! [Y_28: $o,X_37: $o] :
( ( X_37
<=> Y_28 )
=> ( ord_less_eq_o @ X_37 @ Y_28 ) ) ).
thf(fact_260_order__eq__refl,axiom,
! [X_37: nat,Y_28: nat] :
( ( X_37 = Y_28 )
=> ( ord_less_eq_nat @ X_37 @ Y_28 ) ) ).
thf(fact_261_order__eq__refl,axiom,
! [X_37: nat > $o,Y_28: nat > $o] :
( ( X_37 = Y_28 )
=> ( ord_less_eq_nat_o @ X_37 @ Y_28 ) ) ).
thf(fact_262_order__eq__iff,axiom,
! [Y_27: $o,X_36: $o] :
( ( X_36
<=> Y_27 )
<=> ( ( ord_less_eq_o @ X_36 @ Y_27 )
& ( ord_less_eq_o @ Y_27 @ X_36 ) ) ) ).
thf(fact_263_order__eq__iff,axiom,
! [X_36: nat,Y_27: nat] :
( ( X_36 = Y_27 )
<=> ( ( ord_less_eq_nat @ X_36 @ Y_27 )
& ( ord_less_eq_nat @ Y_27 @ X_36 ) ) ) ).
thf(fact_264_order__eq__iff,axiom,
! [X_36: nat > $o,Y_27: nat > $o] :
( ( X_36 = Y_27 )
<=> ( ( ord_less_eq_nat_o @ X_36 @ Y_27 )
& ( ord_less_eq_nat_o @ Y_27 @ X_36 ) ) ) ).
thf(fact_265_linorder__linear,axiom,
! [X_35: nat,Y_26: nat] :
( ( ord_less_eq_nat @ X_35 @ Y_26 )
| ( ord_less_eq_nat @ Y_26 @ X_35 ) ) ).
thf(fact_266_le__fun__def,axiom,
! [F_25: nat > $o,G_6: nat > $o] :
( ( ord_less_eq_nat_o @ F_25 @ G_6 )
<=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_25 @ X_1 ) @ ( G_6 @ X_1 ) ) ) ).
thf(fact_267_linorder__not__less,axiom,
! [X_34: nat,Y_25: nat] :
( ~ ( ord_less_nat @ X_34 @ Y_25 )
<=> ( ord_less_eq_nat @ Y_25 @ X_34 ) ) ).
thf(fact_268_linorder__not__le,axiom,
! [X_33: nat,Y_24: nat] :
( ~ ( ord_less_eq_nat @ X_33 @ Y_24 )
<=> ( ord_less_nat @ Y_24 @ X_33 ) ) ).
thf(fact_269_linorder__le__less__linear,axiom,
! [X_32: nat,Y_23: nat] :
( ( ord_less_eq_nat @ X_32 @ Y_23 )
| ( ord_less_nat @ Y_23 @ X_32 ) ) ).
thf(fact_270_order__less__le,axiom,
! [X_31: product_unit > $o,Y_22: product_unit > $o] :
( ( ord_le232288914unit_o @ X_31 @ Y_22 )
<=> ( ( ord_le1511552390unit_o @ X_31 @ Y_22 )
& ( X_31 != Y_22 ) ) ) ).
thf(fact_271_order__less__le,axiom,
! [X_31: arrow_1429601828e_indi > $o,Y_22: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_31 @ Y_22 )
<=> ( ( ord_le1799070453indi_o @ X_31 @ Y_22 )
& ( X_31 != Y_22 ) ) ) ).
thf(fact_272_order__less__le,axiom,
! [X_31: $o,Y_22: $o] :
( ( ord_less_o @ X_31 @ Y_22 )
<=> ( ( ord_less_eq_o @ X_31 @ Y_22 )
& ~ ( X_31
<=> Y_22 ) ) ) ).
thf(fact_273_order__less__le,axiom,
! [X_31: nat > $o,Y_22: nat > $o] :
( ( ord_less_nat_o @ X_31 @ Y_22 )
<=> ( ( ord_less_eq_nat_o @ X_31 @ Y_22 )
& ( X_31 != Y_22 ) ) ) ).
thf(fact_274_order__less__le,axiom,
! [X_31: nat,Y_22: nat] :
( ( ord_less_nat @ X_31 @ Y_22 )
<=> ( ( ord_less_eq_nat @ X_31 @ Y_22 )
& ( X_31 != Y_22 ) ) ) ).
thf(fact_275_less__le__not__le,axiom,
! [X_30: product_unit > $o,Y_21: product_unit > $o] :
( ( ord_le232288914unit_o @ X_30 @ Y_21 )
<=> ( ( ord_le1511552390unit_o @ X_30 @ Y_21 )
& ~ ( ord_le1511552390unit_o @ Y_21 @ X_30 ) ) ) ).
thf(fact_276_less__le__not__le,axiom,
! [X_30: arrow_1429601828e_indi > $o,Y_21: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_30 @ Y_21 )
<=> ( ( ord_le1799070453indi_o @ X_30 @ Y_21 )
& ~ ( ord_le1799070453indi_o @ Y_21 @ X_30 ) ) ) ).
thf(fact_277_less__le__not__le,axiom,
! [X_30: $o,Y_21: $o] :
( ( ord_less_o @ X_30 @ Y_21 )
<=> ( ( ord_less_eq_o @ X_30 @ Y_21 )
& ~ ( ord_less_eq_o @ Y_21 @ X_30 ) ) ) ).
thf(fact_278_less__le__not__le,axiom,
! [X_30: nat > $o,Y_21: nat > $o] :
( ( ord_less_nat_o @ X_30 @ Y_21 )
<=> ( ( ord_less_eq_nat_o @ X_30 @ Y_21 )
& ~ ( ord_less_eq_nat_o @ Y_21 @ X_30 ) ) ) ).
thf(fact_279_less__le__not__le,axiom,
! [X_30: nat,Y_21: nat] :
( ( ord_less_nat @ X_30 @ Y_21 )
<=> ( ( ord_less_eq_nat @ X_30 @ Y_21 )
& ~ ( ord_less_eq_nat @ Y_21 @ X_30 ) ) ) ).
thf(fact_280_order__le__less,axiom,
! [X_29: product_unit > $o,Y_20: product_unit > $o] :
( ( ord_le1511552390unit_o @ X_29 @ Y_20 )
<=> ( ( ord_le232288914unit_o @ X_29 @ Y_20 )
| ( X_29 = Y_20 ) ) ) ).
thf(fact_281_order__le__less,axiom,
! [X_29: arrow_1429601828e_indi > $o,Y_20: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ X_29 @ Y_20 )
<=> ( ( ord_le777687553indi_o @ X_29 @ Y_20 )
| ( X_29 = Y_20 ) ) ) ).
thf(fact_282_order__le__less,axiom,
! [X_29: $o,Y_20: $o] :
( ( ord_less_eq_o @ X_29 @ Y_20 )
<=> ( ( ord_less_o @ X_29 @ Y_20 )
| ( X_29
<=> Y_20 ) ) ) ).
thf(fact_283_order__le__less,axiom,
! [X_29: nat > $o,Y_20: nat > $o] :
( ( ord_less_eq_nat_o @ X_29 @ Y_20 )
<=> ( ( ord_less_nat_o @ X_29 @ Y_20 )
| ( X_29 = Y_20 ) ) ) ).
thf(fact_284_order__le__less,axiom,
! [X_29: nat,Y_20: nat] :
( ( ord_less_eq_nat @ X_29 @ Y_20 )
<=> ( ( ord_less_nat @ X_29 @ Y_20 )
| ( X_29 = Y_20 ) ) ) ).
thf(fact_285_leI,axiom,
! [X_28: nat,Y_19: nat] :
( ~ ( ord_less_nat @ X_28 @ Y_19 )
=> ( ord_less_eq_nat @ Y_19 @ X_28 ) ) ).
thf(fact_286_not__leE,axiom,
! [Y_18: nat,X_27: nat] :
( ~ ( ord_less_eq_nat @ Y_18 @ X_27 )
=> ( ord_less_nat @ X_27 @ Y_18 ) ) ).
thf(fact_287_linorder__antisym__conv1,axiom,
! [X_26: nat,Y_17: nat] :
( ~ ( ord_less_nat @ X_26 @ Y_17 )
=> ( ( ord_less_eq_nat @ X_26 @ Y_17 )
<=> ( X_26 = Y_17 ) ) ) ).
thf(fact_288_order__neq__le__trans,axiom,
! [A_73: product_unit > $o,B_55: product_unit > $o] :
( ( A_73 != B_55 )
=> ( ( ord_le1511552390unit_o @ A_73 @ B_55 )
=> ( ord_le232288914unit_o @ A_73 @ B_55 ) ) ) ).
thf(fact_289_order__neq__le__trans,axiom,
! [A_73: arrow_1429601828e_indi > $o,B_55: arrow_1429601828e_indi > $o] :
( ( A_73 != B_55 )
=> ( ( ord_le1799070453indi_o @ A_73 @ B_55 )
=> ( ord_le777687553indi_o @ A_73 @ B_55 ) ) ) ).
thf(fact_290_order__neq__le__trans,axiom,
! [B_55: $o,A_73: $o] :
( ~ ( A_73
<=> B_55 )
=> ( ( ord_less_eq_o @ A_73 @ B_55 )
=> ( ord_less_o @ A_73 @ B_55 ) ) ) ).
thf(fact_291_order__neq__le__trans,axiom,
! [A_73: nat > $o,B_55: nat > $o] :
( ( A_73 != B_55 )
=> ( ( ord_less_eq_nat_o @ A_73 @ B_55 )
=> ( ord_less_nat_o @ A_73 @ B_55 ) ) ) ).
thf(fact_292_order__neq__le__trans,axiom,
! [A_73: nat,B_55: nat] :
( ( A_73 != B_55 )
=> ( ( ord_less_eq_nat @ A_73 @ B_55 )
=> ( ord_less_nat @ A_73 @ B_55 ) ) ) ).
thf(fact_293_xt1_I12_J,axiom,
! [A_72: product_unit > $o,B_54: product_unit > $o] :
( ( A_72 != B_54 )
=> ( ( ord_le1511552390unit_o @ B_54 @ A_72 )
=> ( ord_le232288914unit_o @ B_54 @ A_72 ) ) ) ).
thf(fact_294_xt1_I12_J,axiom,
! [A_72: arrow_1429601828e_indi > $o,B_54: arrow_1429601828e_indi > $o] :
( ( A_72 != B_54 )
=> ( ( ord_le1799070453indi_o @ B_54 @ A_72 )
=> ( ord_le777687553indi_o @ B_54 @ A_72 ) ) ) ).
thf(fact_295_xt1_I12_J,axiom,
! [B_54: $o,A_72: $o] :
( ~ ( A_72
<=> B_54 )
=> ( ( ord_less_eq_o @ B_54 @ A_72 )
=> ( ord_less_o @ B_54 @ A_72 ) ) ) ).
thf(fact_296_xt1_I12_J,axiom,
! [A_72: nat > $o,B_54: nat > $o] :
( ( A_72 != B_54 )
=> ( ( ord_less_eq_nat_o @ B_54 @ A_72 )
=> ( ord_less_nat_o @ B_54 @ A_72 ) ) ) ).
thf(fact_297_xt1_I12_J,axiom,
! [A_72: nat,B_54: nat] :
( ( A_72 != B_54 )
=> ( ( ord_less_eq_nat @ B_54 @ A_72 )
=> ( ord_less_nat @ B_54 @ A_72 ) ) ) ).
thf(fact_298_leD,axiom,
! [Y_16: nat,X_25: nat] :
( ( ord_less_eq_nat @ Y_16 @ X_25 )
=> ~ ( ord_less_nat @ X_25 @ Y_16 ) ) ).
thf(fact_299_order__less__imp__le,axiom,
! [X_24: product_unit > $o,Y_15: product_unit > $o] :
( ( ord_le232288914unit_o @ X_24 @ Y_15 )
=> ( ord_le1511552390unit_o @ X_24 @ Y_15 ) ) ).
thf(fact_300_order__less__imp__le,axiom,
! [X_24: arrow_1429601828e_indi > $o,Y_15: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_24 @ Y_15 )
=> ( ord_le1799070453indi_o @ X_24 @ Y_15 ) ) ).
thf(fact_301_order__less__imp__le,axiom,
! [X_24: $o,Y_15: $o] :
( ( ord_less_o @ X_24 @ Y_15 )
=> ( ord_less_eq_o @ X_24 @ Y_15 ) ) ).
thf(fact_302_order__less__imp__le,axiom,
! [X_24: nat > $o,Y_15: nat > $o] :
( ( ord_less_nat_o @ X_24 @ Y_15 )
=> ( ord_less_eq_nat_o @ X_24 @ Y_15 ) ) ).
thf(fact_303_order__less__imp__le,axiom,
! [X_24: nat,Y_15: nat] :
( ( ord_less_nat @ X_24 @ Y_15 )
=> ( ord_less_eq_nat @ X_24 @ Y_15 ) ) ).
thf(fact_304_linorder__antisym__conv2,axiom,
! [X_23: nat,Y_14: nat] :
( ( ord_less_eq_nat @ X_23 @ Y_14 )
=> ( ~ ( ord_less_nat @ X_23 @ Y_14 )
<=> ( X_23 = Y_14 ) ) ) ).
thf(fact_305_order__le__imp__less__or__eq,axiom,
! [X_22: product_unit > $o,Y_13: product_unit > $o] :
( ( ord_le1511552390unit_o @ X_22 @ Y_13 )
=> ( ( ord_le232288914unit_o @ X_22 @ Y_13 )
| ( X_22 = Y_13 ) ) ) ).
thf(fact_306_order__le__imp__less__or__eq,axiom,
! [X_22: arrow_1429601828e_indi > $o,Y_13: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ X_22 @ Y_13 )
=> ( ( ord_le777687553indi_o @ X_22 @ Y_13 )
| ( X_22 = Y_13 ) ) ) ).
thf(fact_307_order__le__imp__less__or__eq,axiom,
! [X_22: $o,Y_13: $o] :
( ( ord_less_eq_o @ X_22 @ Y_13 )
=> ( ( ord_less_o @ X_22 @ Y_13 )
| ( X_22
<=> Y_13 ) ) ) ).
thf(fact_308_order__le__imp__less__or__eq,axiom,
! [X_22: nat > $o,Y_13: nat > $o] :
( ( ord_less_eq_nat_o @ X_22 @ Y_13 )
=> ( ( ord_less_nat_o @ X_22 @ Y_13 )
| ( X_22 = Y_13 ) ) ) ).
thf(fact_309_order__le__imp__less__or__eq,axiom,
! [X_22: nat,Y_13: nat] :
( ( ord_less_eq_nat @ X_22 @ Y_13 )
=> ( ( ord_less_nat @ X_22 @ Y_13 )
| ( X_22 = Y_13 ) ) ) ).
thf(fact_310_order__le__neq__trans,axiom,
! [A_71: product_unit > $o,B_53: product_unit > $o] :
( ( ord_le1511552390unit_o @ A_71 @ B_53 )
=> ( ( A_71 != B_53 )
=> ( ord_le232288914unit_o @ A_71 @ B_53 ) ) ) ).
thf(fact_311_order__le__neq__trans,axiom,
! [A_71: arrow_1429601828e_indi > $o,B_53: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_71 @ B_53 )
=> ( ( A_71 != B_53 )
=> ( ord_le777687553indi_o @ A_71 @ B_53 ) ) ) ).
thf(fact_312_order__le__neq__trans,axiom,
! [A_71: $o,B_53: $o] :
( ( ord_less_eq_o @ A_71 @ B_53 )
=> ( ~ ( A_71
<=> B_53 )
=> ( ord_less_o @ A_71 @ B_53 ) ) ) ).
thf(fact_313_order__le__neq__trans,axiom,
! [A_71: nat > $o,B_53: nat > $o] :
( ( ord_less_eq_nat_o @ A_71 @ B_53 )
=> ( ( A_71 != B_53 )
=> ( ord_less_nat_o @ A_71 @ B_53 ) ) ) ).
thf(fact_314_order__le__neq__trans,axiom,
! [A_71: nat,B_53: nat] :
( ( ord_less_eq_nat @ A_71 @ B_53 )
=> ( ( A_71 != B_53 )
=> ( ord_less_nat @ A_71 @ B_53 ) ) ) ).
thf(fact_315_xt1_I11_J,axiom,
! [B_52: nat > $o,A_70: nat > $o] :
( ( ord_less_eq_nat_o @ B_52 @ A_70 )
=> ( ( A_70 != B_52 )
=> ( ord_less_nat_o @ B_52 @ A_70 ) ) ) ).
thf(fact_316_xt1_I11_J,axiom,
! [B_52: product_unit > $o,A_70: product_unit > $o] :
( ( ord_le1511552390unit_o @ B_52 @ A_70 )
=> ( ( A_70 != B_52 )
=> ( ord_le232288914unit_o @ B_52 @ A_70 ) ) ) ).
thf(fact_317_xt1_I11_J,axiom,
! [B_52: arrow_1429601828e_indi > $o,A_70: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ B_52 @ A_70 )
=> ( ( A_70 != B_52 )
=> ( ord_le777687553indi_o @ B_52 @ A_70 ) ) ) ).
thf(fact_318_xt1_I11_J,axiom,
! [B_52: nat,A_70: nat] :
( ( ord_less_eq_nat @ B_52 @ A_70 )
=> ( ( A_70 != B_52 )
=> ( ord_less_nat @ B_52 @ A_70 ) ) ) ).
thf(fact_319_xt1_I11_J,axiom,
! [B_52: $o,A_70: $o] :
( ( ord_less_eq_o @ B_52 @ A_70 )
=> ( ~ ( A_70
<=> B_52 )
=> ( ord_less_o @ B_52 @ A_70 ) ) ) ).
thf(fact_320_order__less__le__trans,axiom,
! [Z_4: nat > $o,X_21: nat > $o,Y_12: nat > $o] :
( ( ord_less_nat_o @ X_21 @ Y_12 )
=> ( ( ord_less_eq_nat_o @ Y_12 @ Z_4 )
=> ( ord_less_nat_o @ X_21 @ Z_4 ) ) ) ).
thf(fact_321_order__less__le__trans,axiom,
! [Z_4: product_unit > $o,X_21: product_unit > $o,Y_12: product_unit > $o] :
( ( ord_le232288914unit_o @ X_21 @ Y_12 )
=> ( ( ord_le1511552390unit_o @ Y_12 @ Z_4 )
=> ( ord_le232288914unit_o @ X_21 @ Z_4 ) ) ) ).
thf(fact_322_order__less__le__trans,axiom,
! [Z_4: arrow_1429601828e_indi > $o,X_21: arrow_1429601828e_indi > $o,Y_12: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ X_21 @ Y_12 )
=> ( ( ord_le1799070453indi_o @ Y_12 @ Z_4 )
=> ( ord_le777687553indi_o @ X_21 @ Z_4 ) ) ) ).
thf(fact_323_order__less__le__trans,axiom,
! [Z_4: nat,X_21: nat,Y_12: nat] :
( ( ord_less_nat @ X_21 @ Y_12 )
=> ( ( ord_less_eq_nat @ Y_12 @ Z_4 )
=> ( ord_less_nat @ X_21 @ Z_4 ) ) ) ).
thf(fact_324_order__less__le__trans,axiom,
! [Z_4: $o,X_21: $o,Y_12: $o] :
( ( ord_less_o @ X_21 @ Y_12 )
=> ( ( ord_less_eq_o @ Y_12 @ Z_4 )
=> ( ord_less_o @ X_21 @ Z_4 ) ) ) ).
thf(fact_325_xt1_I7_J,axiom,
! [Z_3: nat > $o,Y_11: nat > $o,X_20: nat > $o] :
( ( ord_less_nat_o @ Y_11 @ X_20 )
=> ( ( ord_less_eq_nat_o @ Z_3 @ Y_11 )
=> ( ord_less_nat_o @ Z_3 @ X_20 ) ) ) ).
thf(fact_326_xt1_I7_J,axiom,
! [Z_3: product_unit > $o,Y_11: product_unit > $o,X_20: product_unit > $o] :
( ( ord_le232288914unit_o @ Y_11 @ X_20 )
=> ( ( ord_le1511552390unit_o @ Z_3 @ Y_11 )
=> ( ord_le232288914unit_o @ Z_3 @ X_20 ) ) ) ).
thf(fact_327_xt1_I7_J,axiom,
! [Z_3: arrow_1429601828e_indi > $o,Y_11: arrow_1429601828e_indi > $o,X_20: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ Y_11 @ X_20 )
=> ( ( ord_le1799070453indi_o @ Z_3 @ Y_11 )
=> ( ord_le777687553indi_o @ Z_3 @ X_20 ) ) ) ).
thf(fact_328_xt1_I7_J,axiom,
! [Z_3: nat,Y_11: nat,X_20: nat] :
( ( ord_less_nat @ Y_11 @ X_20 )
=> ( ( ord_less_eq_nat @ Z_3 @ Y_11 )
=> ( ord_less_nat @ Z_3 @ X_20 ) ) ) ).
thf(fact_329_xt1_I7_J,axiom,
! [Z_3: $o,Y_11: $o,X_20: $o] :
( ( ord_less_o @ Y_11 @ X_20 )
=> ( ( ord_less_eq_o @ Z_3 @ Y_11 )
=> ( ord_less_o @ Z_3 @ X_20 ) ) ) ).
thf(fact_330_order__le__less__trans,axiom,
! [Z_2: nat > $o,X_19: nat > $o,Y_10: nat > $o] :
( ( ord_less_eq_nat_o @ X_19 @ Y_10 )
=> ( ( ord_less_nat_o @ Y_10 @ Z_2 )
=> ( ord_less_nat_o @ X_19 @ Z_2 ) ) ) ).
thf(fact_331_order__le__less__trans,axiom,
! [Z_2: product_unit > $o,X_19: product_unit > $o,Y_10: product_unit > $o] :
( ( ord_le1511552390unit_o @ X_19 @ Y_10 )
=> ( ( ord_le232288914unit_o @ Y_10 @ Z_2 )
=> ( ord_le232288914unit_o @ X_19 @ Z_2 ) ) ) ).
thf(fact_332_order__le__less__trans,axiom,
! [Z_2: arrow_1429601828e_indi > $o,X_19: arrow_1429601828e_indi > $o,Y_10: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ X_19 @ Y_10 )
=> ( ( ord_le777687553indi_o @ Y_10 @ Z_2 )
=> ( ord_le777687553indi_o @ X_19 @ Z_2 ) ) ) ).
thf(fact_333_order__le__less__trans,axiom,
! [Z_2: nat,X_19: nat,Y_10: nat] :
( ( ord_less_eq_nat @ X_19 @ Y_10 )
=> ( ( ord_less_nat @ Y_10 @ Z_2 )
=> ( ord_less_nat @ X_19 @ Z_2 ) ) ) ).
thf(fact_334_order__le__less__trans,axiom,
! [Z_2: $o,X_19: $o,Y_10: $o] :
( ( ord_less_eq_o @ X_19 @ Y_10 )
=> ( ( ord_less_o @ Y_10 @ Z_2 )
=> ( ord_less_o @ X_19 @ Z_2 ) ) ) ).
thf(fact_335_xt1_I8_J,axiom,
! [Z_1: nat > $o,Y_9: nat > $o,X_18: nat > $o] :
( ( ord_less_eq_nat_o @ Y_9 @ X_18 )
=> ( ( ord_less_nat_o @ Z_1 @ Y_9 )
=> ( ord_less_nat_o @ Z_1 @ X_18 ) ) ) ).
thf(fact_336_xt1_I8_J,axiom,
! [Z_1: product_unit > $o,Y_9: product_unit > $o,X_18: product_unit > $o] :
( ( ord_le1511552390unit_o @ Y_9 @ X_18 )
=> ( ( ord_le232288914unit_o @ Z_1 @ Y_9 )
=> ( ord_le232288914unit_o @ Z_1 @ X_18 ) ) ) ).
thf(fact_337_xt1_I8_J,axiom,
! [Z_1: arrow_1429601828e_indi > $o,Y_9: arrow_1429601828e_indi > $o,X_18: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ Y_9 @ X_18 )
=> ( ( ord_le777687553indi_o @ Z_1 @ Y_9 )
=> ( ord_le777687553indi_o @ Z_1 @ X_18 ) ) ) ).
thf(fact_338_xt1_I8_J,axiom,
! [Z_1: nat,Y_9: nat,X_18: nat] :
( ( ord_less_eq_nat @ Y_9 @ X_18 )
=> ( ( ord_less_nat @ Z_1 @ Y_9 )
=> ( ord_less_nat @ Z_1 @ X_18 ) ) ) ).
thf(fact_339_xt1_I8_J,axiom,
! [Z_1: $o,Y_9: $o,X_18: $o] :
( ( ord_less_eq_o @ Y_9 @ X_18 )
=> ( ( ord_less_o @ Z_1 @ Y_9 )
=> ( ord_less_o @ Z_1 @ X_18 ) ) ) ).
thf(fact_340_top__greatest,axiom,
! [A_69: $o] : ( ord_less_eq_o @ A_69 @ top_top_o ) ).
thf(fact_341_top__greatest,axiom,
! [A_69: ( produc1501160679le_alt > $o ) > $o] : ( ord_le1063113995lt_o_o @ A_69 @ top_to1842727771lt_o_o ) ).
thf(fact_342_top__greatest,axiom,
! [A_69: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] : ( ord_le134800455lt_o_o @ A_69 @ top_to1969627639lt_o_o ) ).
thf(fact_343_top__greatest,axiom,
! [A_69: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] : ( ord_le1992928527lt_o_o @ A_69 @ top_to2122763103lt_o_o ) ).
thf(fact_344_top__greatest,axiom,
! [A_69: produc1501160679le_alt > $o] : ( ord_le97612146_alt_o @ A_69 @ top_to1841428258_alt_o ) ).
thf(fact_345_top__greatest,axiom,
! [A_69: nat > $o] : ( ord_less_eq_nat_o @ A_69 @ top_top_nat_o ) ).
thf(fact_346_top__greatest,axiom,
! [A_69: product_unit > $o] : ( ord_le1511552390unit_o @ A_69 @ top_to1984820022unit_o ) ).
thf(fact_347_top__greatest,axiom,
! [A_69: arrow_1429601828e_indi > $o] : ( ord_le1799070453indi_o @ A_69 @ top_to988227749indi_o ) ).
thf(fact_348_top__unique,axiom,
! [A_68: $o] :
( ( ord_less_eq_o @ top_top_o @ A_68 )
<=> ( A_68
<=> top_top_o ) ) ).
thf(fact_349_top__unique,axiom,
! [A_68: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le1063113995lt_o_o @ top_to1842727771lt_o_o @ A_68 )
<=> ( A_68 = top_to1842727771lt_o_o ) ) ).
thf(fact_350_top__unique,axiom,
! [A_68: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le134800455lt_o_o @ top_to1969627639lt_o_o @ A_68 )
<=> ( A_68 = top_to1969627639lt_o_o ) ) ).
thf(fact_351_top__unique,axiom,
! [A_68: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o @ top_to2122763103lt_o_o @ A_68 )
<=> ( A_68 = top_to2122763103lt_o_o ) ) ).
thf(fact_352_top__unique,axiom,
! [A_68: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o @ top_to1841428258_alt_o @ A_68 )
<=> ( A_68 = top_to1841428258_alt_o ) ) ).
thf(fact_353_top__unique,axiom,
! [A_68: nat > $o] :
( ( ord_less_eq_nat_o @ top_top_nat_o @ A_68 )
<=> ( A_68 = top_top_nat_o ) ) ).
thf(fact_354_top__unique,axiom,
! [A_68: product_unit > $o] :
( ( ord_le1511552390unit_o @ top_to1984820022unit_o @ A_68 )
<=> ( A_68 = top_to1984820022unit_o ) ) ).
thf(fact_355_top__unique,axiom,
! [A_68: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ top_to988227749indi_o @ A_68 )
<=> ( A_68 = top_to988227749indi_o ) ) ).
thf(fact_356_top__le,axiom,
! [A_67: $o] :
( ( ord_less_eq_o @ top_top_o @ A_67 )
=> ( A_67
<=> top_top_o ) ) ).
thf(fact_357_top__le,axiom,
! [A_67: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le1063113995lt_o_o @ top_to1842727771lt_o_o @ A_67 )
=> ( A_67 = top_to1842727771lt_o_o ) ) ).
thf(fact_358_top__le,axiom,
! [A_67: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le134800455lt_o_o @ top_to1969627639lt_o_o @ A_67 )
=> ( A_67 = top_to1969627639lt_o_o ) ) ).
thf(fact_359_top__le,axiom,
! [A_67: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o @ top_to2122763103lt_o_o @ A_67 )
=> ( A_67 = top_to2122763103lt_o_o ) ) ).
thf(fact_360_top__le,axiom,
! [A_67: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o @ top_to1841428258_alt_o @ A_67 )
=> ( A_67 = top_to1841428258_alt_o ) ) ).
thf(fact_361_top__le,axiom,
! [A_67: nat > $o] :
( ( ord_less_eq_nat_o @ top_top_nat_o @ A_67 )
=> ( A_67 = top_top_nat_o ) ) ).
thf(fact_362_top__le,axiom,
! [A_67: product_unit > $o] :
( ( ord_le1511552390unit_o @ top_to1984820022unit_o @ A_67 )
=> ( A_67 = top_to1984820022unit_o ) ) ).
thf(fact_363_top__le,axiom,
! [A_67: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ top_to988227749indi_o @ A_67 )
=> ( A_67 = top_to988227749indi_o ) ) ).
thf(fact_364_inj__on__def,axiom,
! [F_24: nat > nat,A_66: nat > $o] :
( ( inj_on_nat_nat @ F_24 @ A_66 )
<=> ! [X_1: nat] :
( ( member_nat @ X_1 @ A_66 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A_66 )
=> ( ( ( F_24 @ X_1 )
= ( F_24 @ Xa ) )
=> ( X_1 = Xa ) ) ) ) ) ).
thf(fact_365_inj__on__def,axiom,
! [F_24: arrow_1429601828e_indi > nat,A_66: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_24 @ A_66 )
<=> ! [X_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_66 )
=> ! [Xa: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ Xa @ A_66 )
=> ( ( ( F_24 @ X_1 )
= ( F_24 @ Xa ) )
=> ( X_1 = Xa ) ) ) ) ) ).
thf(fact_366_inj__on__contraD,axiom,
! [X_17: nat,Y_8: nat,F_23: nat > nat,A_65: nat > $o] :
( ( inj_on_nat_nat @ F_23 @ A_65 )
=> ( ( X_17 != Y_8 )
=> ( ( member_nat @ X_17 @ A_65 )
=> ( ( member_nat @ Y_8 @ A_65 )
=> ( ( F_23 @ X_17 )
!= ( F_23 @ Y_8 ) ) ) ) ) ) ).
thf(fact_367_inj__on__contraD,axiom,
! [X_17: arrow_1429601828e_indi,Y_8: arrow_1429601828e_indi,F_23: arrow_1429601828e_indi > nat,A_65: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_23 @ A_65 )
=> ( ( X_17 != Y_8 )
=> ( ( member2052026769e_indi @ X_17 @ A_65 )
=> ( ( member2052026769e_indi @ Y_8 @ A_65 )
=> ( ( F_23 @ X_17 )
!= ( F_23 @ Y_8 ) ) ) ) ) ) ).
thf(fact_368_inj__on__iff,axiom,
! [Y_7: nat,X_16: nat,F_22: nat > nat,A_64: nat > $o] :
( ( inj_on_nat_nat @ F_22 @ A_64 )
=> ( ( member_nat @ X_16 @ A_64 )
=> ( ( member_nat @ Y_7 @ A_64 )
=> ( ( ( F_22 @ X_16 )
= ( F_22 @ Y_7 ) )
<=> ( X_16 = Y_7 ) ) ) ) ) ).
thf(fact_369_inj__on__iff,axiom,
! [Y_7: arrow_1429601828e_indi,X_16: arrow_1429601828e_indi,F_22: arrow_1429601828e_indi > nat,A_64: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_22 @ A_64 )
=> ( ( member2052026769e_indi @ X_16 @ A_64 )
=> ( ( member2052026769e_indi @ Y_7 @ A_64 )
=> ( ( ( F_22 @ X_16 )
= ( F_22 @ Y_7 ) )
<=> ( X_16 = Y_7 ) ) ) ) ) ).
thf(fact_370_inj__onD,axiom,
! [X_15: nat,Y_6: nat,F_21: nat > nat,A_63: nat > $o] :
( ( inj_on_nat_nat @ F_21 @ A_63 )
=> ( ( ( F_21 @ X_15 )
= ( F_21 @ Y_6 ) )
=> ( ( member_nat @ X_15 @ A_63 )
=> ( ( member_nat @ Y_6 @ A_63 )
=> ( X_15 = Y_6 ) ) ) ) ) ).
thf(fact_371_inj__onD,axiom,
! [X_15: arrow_1429601828e_indi,Y_6: arrow_1429601828e_indi,F_21: arrow_1429601828e_indi > nat,A_63: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_21 @ A_63 )
=> ( ( ( F_21 @ X_15 )
= ( F_21 @ Y_6 ) )
=> ( ( member2052026769e_indi @ X_15 @ A_63 )
=> ( ( member2052026769e_indi @ Y_6 @ A_63 )
=> ( X_15 = Y_6 ) ) ) ) ) ).
thf(fact_372_Pi__mem,axiom,
! [X_14: produc1501160679le_alt,F_20: produc1501160679le_alt > $o,A_62: produc1501160679le_alt > $o,B_51: produc1501160679le_alt > $o > $o] :
( ( member377231867_alt_o @ F_20 @ ( pi_Pro1701359055_alt_o @ A_62 @ B_51 ) )
=> ( ( member214075476le_alt @ X_14 @ A_62 )
=> ( member_o @ ( F_20 @ X_14 ) @ ( B_51 @ X_14 ) ) ) ) ).
thf(fact_373_Pi__mem,axiom,
! [X_14: arrow_1429601828e_indi,F_20: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_62: arrow_1429601828e_indi > $o,B_51: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ F_20 @ ( pi_Arr1929480907_alt_o @ A_62 @ B_51 ) )
=> ( ( member2052026769e_indi @ X_14 @ A_62 )
=> ( member377231867_alt_o @ ( F_20 @ X_14 ) @ ( B_51 @ X_14 ) ) ) ) ).
thf(fact_374_Pi__mem,axiom,
! [X_14: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_20: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_62: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_51: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ F_20 @ ( pi_Arr1304755663_alt_o @ A_62 @ B_51 ) )
=> ( ( member526088951_alt_o @ X_14 @ A_62 )
=> ( member377231867_alt_o @ ( F_20 @ X_14 ) @ ( B_51 @ X_14 ) ) ) ) ).
thf(fact_375_Prof__def,axiom,
( arrow_734252939e_Prof
= ( pi_Arr1929480907_alt_o @ top_to988227749indi_o
@ ^ [Uu: arrow_1429601828e_indi] : arrow_823908191le_Lin ) ) ).
thf(fact_376_inj__on__id2,axiom,
! [A_61: nat > $o] :
( inj_on_nat_nat
@ ^ [X_1: nat] : X_1
@ A_61 ) ).
thf(fact_377_funcset__mem,axiom,
! [X_13: produc1501160679le_alt,F_19: produc1501160679le_alt > $o,A_60: produc1501160679le_alt > $o,B_50: $o > $o] :
( ( member377231867_alt_o @ F_19
@ ( pi_Pro1701359055_alt_o @ A_60
@ ^ [Uu: produc1501160679le_alt] : B_50 ) )
=> ( ( member214075476le_alt @ X_13 @ A_60 )
=> ( member_o @ ( F_19 @ X_13 ) @ B_50 ) ) ) ).
thf(fact_378_funcset__mem,axiom,
! [X_13: arrow_1429601828e_indi,F_19: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_60: arrow_1429601828e_indi > $o,B_50: ( produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ F_19
@ ( pi_Arr1929480907_alt_o @ A_60
@ ^ [Uu: arrow_1429601828e_indi] : B_50 ) )
=> ( ( member2052026769e_indi @ X_13 @ A_60 )
=> ( member377231867_alt_o @ ( F_19 @ X_13 ) @ B_50 ) ) ) ).
thf(fact_379_funcset__mem,axiom,
! [X_13: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_19: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_60: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_50: ( produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ F_19
@ ( pi_Arr1304755663_alt_o @ A_60
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : B_50 ) )
=> ( ( member526088951_alt_o @ X_13 @ A_60 )
=> ( member377231867_alt_o @ ( F_19 @ X_13 ) @ B_50 ) ) ) ).
thf(fact_380_dictator__def,axiom,
! [F_18: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,I: arrow_1429601828e_indi] :
( ( arrow_1212662430ctator @ F_18 @ I )
<=> ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ arrow_734252939e_Prof )
=> ( ( F_18 @ X_1 )
= ( X_1 @ I ) ) ) ) ).
thf(fact_381_inj__eq,axiom,
! [X_12: nat,Y_5: nat,F_17: nat > nat] :
( ( inj_on_nat_nat @ F_17 @ top_top_nat_o )
=> ( ( ( F_17 @ X_12 )
= ( F_17 @ Y_5 ) )
<=> ( X_12 = Y_5 ) ) ) ).
thf(fact_382_inj__eq,axiom,
! [X_12: arrow_1429601828e_indi,Y_5: arrow_1429601828e_indi,F_17: arrow_1429601828e_indi > nat] :
( ( inj_on978774663di_nat @ F_17 @ top_to988227749indi_o )
=> ( ( ( F_17 @ X_12 )
= ( F_17 @ Y_5 ) )
<=> ( X_12 = Y_5 ) ) ) ).
thf(fact_383_injD,axiom,
! [X_11: nat,Y_4: nat,F_16: nat > nat] :
( ( inj_on_nat_nat @ F_16 @ top_top_nat_o )
=> ( ( ( F_16 @ X_11 )
= ( F_16 @ Y_4 ) )
=> ( X_11 = Y_4 ) ) ) ).
thf(fact_384_injD,axiom,
! [X_11: arrow_1429601828e_indi,Y_4: arrow_1429601828e_indi,F_16: arrow_1429601828e_indi > nat] :
( ( inj_on978774663di_nat @ F_16 @ top_to988227749indi_o )
=> ( ( ( F_16 @ X_11 )
= ( F_16 @ Y_4 ) )
=> ( X_11 = Y_4 ) ) ) ).
thf(fact_385_inv__into__f__f,axiom,
! [X_10: nat,F_15: nat > nat,A_59: nat > $o] :
( ( inj_on_nat_nat @ F_15 @ A_59 )
=> ( ( member_nat @ X_10 @ A_59 )
=> ( ( hilber195283148at_nat @ A_59 @ F_15 @ ( F_15 @ X_10 ) )
= X_10 ) ) ) ).
thf(fact_386_inv__into__f__f,axiom,
! [X_10: arrow_1429601828e_indi,F_15: arrow_1429601828e_indi > nat,A_59: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_15 @ A_59 )
=> ( ( member2052026769e_indi @ X_10 @ A_59 )
=> ( ( hilber598459244di_nat @ A_59 @ F_15 @ ( F_15 @ X_10 ) )
= X_10 ) ) ) ).
thf(fact_387_inv__into__f__eq,axiom,
! [Y_3: nat,X_9: nat,F_14: nat > nat,A_58: nat > $o] :
( ( inj_on_nat_nat @ F_14 @ A_58 )
=> ( ( member_nat @ X_9 @ A_58 )
=> ( ( ( F_14 @ X_9 )
= Y_3 )
=> ( ( hilber195283148at_nat @ A_58 @ F_14 @ Y_3 )
= X_9 ) ) ) ) ).
thf(fact_388_inv__into__f__eq,axiom,
! [Y_3: nat,X_9: arrow_1429601828e_indi,F_14: arrow_1429601828e_indi > nat,A_58: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_14 @ A_58 )
=> ( ( member2052026769e_indi @ X_9 @ A_58 )
=> ( ( ( F_14 @ X_9 )
= Y_3 )
=> ( ( hilber598459244di_nat @ A_58 @ F_14 @ Y_3 )
= X_9 ) ) ) ) ).
thf(fact_389_Pi__I,axiom,
! [F_13: produc1501160679le_alt > $o,B_49: produc1501160679le_alt > $o > $o,A_57: produc1501160679le_alt > $o] :
( ! [X_1: produc1501160679le_alt] :
( ( member214075476le_alt @ X_1 @ A_57 )
=> ( member_o @ ( F_13 @ X_1 ) @ ( B_49 @ X_1 ) ) )
=> ( member377231867_alt_o @ F_13 @ ( pi_Pro1701359055_alt_o @ A_57 @ B_49 ) ) ) ).
thf(fact_390_Pi__I,axiom,
! [F_13: arrow_1429601828e_indi > produc1501160679le_alt > $o,B_49: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o,A_57: arrow_1429601828e_indi > $o] :
( ! [X_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_57 )
=> ( member377231867_alt_o @ ( F_13 @ X_1 ) @ ( B_49 @ X_1 ) ) )
=> ( member526088951_alt_o @ F_13 @ ( pi_Arr1929480907_alt_o @ A_57 @ B_49 ) ) ) ).
thf(fact_391_Pi__I,axiom,
! [F_13: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,B_49: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o,A_57: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ A_57 )
=> ( member377231867_alt_o @ ( F_13 @ X_1 ) @ ( B_49 @ X_1 ) ) )
=> ( member616898751_alt_o @ F_13 @ ( pi_Arr1304755663_alt_o @ A_57 @ B_49 ) ) ) ).
thf(fact_392_inj__imp__inv__eq,axiom,
! [G_5: nat > nat,F_12: nat > nat] :
( ( inj_on_nat_nat @ F_12 @ top_top_nat_o )
=> ( ! [X_1: nat] :
( ( F_12 @ ( G_5 @ X_1 ) )
= X_1 )
=> ( ( hilber195283148at_nat @ top_top_nat_o @ F_12 )
= G_5 ) ) ) ).
thf(fact_393_inj__imp__inv__eq,axiom,
! [G_5: nat > arrow_1429601828e_indi,F_12: arrow_1429601828e_indi > nat] :
( ( inj_on978774663di_nat @ F_12 @ top_to988227749indi_o )
=> ( ! [X_1: nat] :
( ( F_12 @ ( G_5 @ X_1 ) )
= X_1 )
=> ( ( hilber598459244di_nat @ top_to988227749indi_o @ F_12 )
= G_5 ) ) ) ).
thf(fact_394_less__add__one,axiom,
! [A_56: nat] : ( ord_less_nat @ A_56 @ ( plus_plus_nat @ A_56 @ one_one_nat ) ) ).
thf(fact_395_add__less__le__mono,axiom,
! [C_28: nat,D_3: nat,A_55: nat,B_48: nat] :
( ( ord_less_nat @ A_55 @ B_48 )
=> ( ( ord_less_eq_nat @ C_28 @ D_3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_55 @ C_28 ) @ ( plus_plus_nat @ B_48 @ D_3 ) ) ) ) ).
thf(fact_396_add__le__less__mono,axiom,
! [C_27: nat,D_2: nat,A_54: nat,B_47: nat] :
( ( ord_less_eq_nat @ A_54 @ B_47 )
=> ( ( ord_less_nat @ C_27 @ D_2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_54 @ C_27 ) @ ( plus_plus_nat @ B_47 @ D_2 ) ) ) ) ).
thf(fact_397_injI,axiom,
! [F_11: nat > nat] :
( ! [X_1: nat,Y_1: nat] :
( ( ( F_11 @ X_1 )
= ( F_11 @ Y_1 ) )
=> ( X_1 = Y_1 ) )
=> ( inj_on_nat_nat @ F_11 @ top_top_nat_o ) ) ).
thf(fact_398_injI,axiom,
! [F_11: arrow_1429601828e_indi > nat] :
( ! [X_1: arrow_1429601828e_indi,Y_1: arrow_1429601828e_indi] :
( ( ( F_11 @ X_1 )
= ( F_11 @ Y_1 ) )
=> ( X_1 = Y_1 ) )
=> ( inj_on978774663di_nat @ F_11 @ top_to988227749indi_o ) ) ).
thf(fact_399_less__fun__def,axiom,
! [F_10: nat > $o,G_4: nat > $o] :
( ( ord_less_nat_o @ F_10 @ G_4 )
<=> ( ( ord_less_eq_nat_o @ F_10 @ G_4 )
& ~ ( ord_less_eq_nat_o @ G_4 @ F_10 ) ) ) ).
thf(fact_400_less__fun__def,axiom,
! [F_10: product_unit > $o,G_4: product_unit > $o] :
( ( ord_le232288914unit_o @ F_10 @ G_4 )
<=> ( ( ord_le1511552390unit_o @ F_10 @ G_4 )
& ~ ( ord_le1511552390unit_o @ G_4 @ F_10 ) ) ) ).
thf(fact_401_less__fun__def,axiom,
! [F_10: arrow_1429601828e_indi > $o,G_4: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ F_10 @ G_4 )
<=> ( ( ord_le1799070453indi_o @ F_10 @ G_4 )
& ~ ( ord_le1799070453indi_o @ G_4 @ F_10 ) ) ) ).
thf(fact_402_Pi__anti__mono,axiom,
! [B_46: produc1501160679le_alt > $o > $o,A_53: produc1501160679le_alt > $o,A_52: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o @ A_53 @ A_52 )
=> ( ord_le1063113995lt_o_o @ ( pi_Pro1701359055_alt_o @ A_52 @ B_46 ) @ ( pi_Pro1701359055_alt_o @ A_53 @ B_46 ) ) ) ).
thf(fact_403_Pi__anti__mono,axiom,
! [B_46: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o,A_53: arrow_1429601828e_indi > $o,A_52: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_53 @ A_52 )
=> ( ord_le1992928527lt_o_o @ ( pi_Arr1929480907_alt_o @ A_52 @ B_46 ) @ ( pi_Arr1929480907_alt_o @ A_53 @ B_46 ) ) ) ).
thf(fact_404_Pi__anti__mono,axiom,
! [B_46: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o,A_53: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,A_52: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o @ A_53 @ A_52 )
=> ( ord_le134800455lt_o_o @ ( pi_Arr1304755663_alt_o @ A_52 @ B_46 ) @ ( pi_Arr1304755663_alt_o @ A_53 @ B_46 ) ) ) ).
thf(fact_405_pred__subset__eq2,axiom,
! [R_2: produc1501160679le_alt > $o,S_2: produc1501160679le_alt > $o] :
( ( ord_le2080035663_alt_o
@ ^ [X_1: arrow_475358991le_alt,Y_1: arrow_475358991le_alt] : ( member214075476le_alt @ ( produc1347929815le_alt @ X_1 @ Y_1 ) @ R_2 )
@ ^ [X_1: arrow_475358991le_alt,Y_1: arrow_475358991le_alt] : ( member214075476le_alt @ ( produc1347929815le_alt @ X_1 @ Y_1 ) @ S_2 ) )
<=> ( ord_le97612146_alt_o @ R_2 @ S_2 ) ) ).
thf(fact_406_subset__UNIV,axiom,
! [A_51: ( produc1501160679le_alt > $o ) > $o] : ( ord_le1063113995lt_o_o @ A_51 @ top_to1842727771lt_o_o ) ).
thf(fact_407_subset__UNIV,axiom,
! [A_51: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] : ( ord_le134800455lt_o_o @ A_51 @ top_to1969627639lt_o_o ) ).
thf(fact_408_subset__UNIV,axiom,
! [A_51: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] : ( ord_le1992928527lt_o_o @ A_51 @ top_to2122763103lt_o_o ) ).
thf(fact_409_subset__UNIV,axiom,
! [A_51: produc1501160679le_alt > $o] : ( ord_le97612146_alt_o @ A_51 @ top_to1841428258_alt_o ) ).
thf(fact_410_subset__UNIV,axiom,
! [A_51: nat > $o] : ( ord_less_eq_nat_o @ A_51 @ top_top_nat_o ) ).
thf(fact_411_subset__UNIV,axiom,
! [A_51: product_unit > $o] : ( ord_le1511552390unit_o @ A_51 @ top_to1984820022unit_o ) ).
thf(fact_412_subset__UNIV,axiom,
! [A_51: arrow_1429601828e_indi > $o] : ( ord_le1799070453indi_o @ A_51 @ top_to988227749indi_o ) ).
thf(fact_413_subset__inj__on,axiom,
! [A_50: nat > $o,F_9: nat > nat,B_45: nat > $o] :
( ( inj_on_nat_nat @ F_9 @ B_45 )
=> ( ( ord_less_eq_nat_o @ A_50 @ B_45 )
=> ( inj_on_nat_nat @ F_9 @ A_50 ) ) ) ).
thf(fact_414_subset__inj__on,axiom,
! [A_50: arrow_1429601828e_indi > $o,F_9: arrow_1429601828e_indi > nat,B_45: arrow_1429601828e_indi > $o] :
( ( inj_on978774663di_nat @ F_9 @ B_45 )
=> ( ( ord_le1799070453indi_o @ A_50 @ B_45 )
=> ( inj_on978774663di_nat @ F_9 @ A_50 ) ) ) ).
thf(fact_415_add__right__imp__eq,axiom,
! [B_44: nat,A_49: nat,C_26: nat] :
( ( ( plus_plus_nat @ B_44 @ A_49 )
= ( plus_plus_nat @ C_26 @ A_49 ) )
=> ( B_44 = C_26 ) ) ).
thf(fact_416_add__imp__eq,axiom,
! [A_48: nat,B_43: nat,C_25: nat] :
( ( ( plus_plus_nat @ A_48 @ B_43 )
= ( plus_plus_nat @ A_48 @ C_25 ) )
=> ( B_43 = C_25 ) ) ).
thf(fact_417_add__left__imp__eq,axiom,
! [A_47: nat,B_42: nat,C_24: nat] :
( ( ( plus_plus_nat @ A_47 @ B_42 )
= ( plus_plus_nat @ A_47 @ C_24 ) )
=> ( B_42 = C_24 ) ) ).
thf(fact_418_add__right__cancel,axiom,
! [B_41: nat,A_46: nat,C_23: nat] :
( ( ( plus_plus_nat @ B_41 @ A_46 )
= ( plus_plus_nat @ C_23 @ A_46 ) )
<=> ( B_41 = C_23 ) ) ).
thf(fact_419_add__left__cancel,axiom,
! [A_45: nat,B_40: nat,C_22: nat] :
( ( ( plus_plus_nat @ A_45 @ B_40 )
= ( plus_plus_nat @ A_45 @ C_22 ) )
<=> ( B_40 = C_22 ) ) ).
thf(fact_420_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_44: nat,B_39: nat,C_21: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_44 @ B_39 ) @ C_21 )
= ( plus_plus_nat @ A_44 @ ( plus_plus_nat @ B_39 @ C_21 ) ) ) ).
thf(fact_421_one__reorient,axiom,
! [X_8: nat] :
( ( one_one_nat = X_8 )
<=> ( X_8 = one_one_nat ) ) ).
thf(fact_422_add__le__imp__le__left,axiom,
! [C_20: nat,A_43: nat,B_38: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_20 @ A_43 ) @ ( plus_plus_nat @ C_20 @ B_38 ) )
=> ( ord_less_eq_nat @ A_43 @ B_38 ) ) ).
thf(fact_423_add__le__imp__le__right,axiom,
! [A_42: nat,C_19: nat,B_37: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_42 @ C_19 ) @ ( plus_plus_nat @ B_37 @ C_19 ) )
=> ( ord_less_eq_nat @ A_42 @ B_37 ) ) ).
thf(fact_424_add__mono,axiom,
! [C_18: nat,D_1: nat,A_41: nat,B_36: nat] :
( ( ord_less_eq_nat @ A_41 @ B_36 )
=> ( ( ord_less_eq_nat @ C_18 @ D_1 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_41 @ C_18 ) @ ( plus_plus_nat @ B_36 @ D_1 ) ) ) ) ).
thf(fact_425_add__left__mono,axiom,
! [C_17: nat,A_40: nat,B_35: nat] :
( ( ord_less_eq_nat @ A_40 @ B_35 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_17 @ A_40 ) @ ( plus_plus_nat @ C_17 @ B_35 ) ) ) ).
thf(fact_426_add__right__mono,axiom,
! [C_16: nat,A_39: nat,B_34: nat] :
( ( ord_less_eq_nat @ A_39 @ B_34 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_39 @ C_16 ) @ ( plus_plus_nat @ B_34 @ C_16 ) ) ) ).
thf(fact_427_add__le__cancel__left,axiom,
! [C_15: nat,A_38: nat,B_33: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_15 @ A_38 ) @ ( plus_plus_nat @ C_15 @ B_33 ) )
<=> ( ord_less_eq_nat @ A_38 @ B_33 ) ) ).
thf(fact_428_add__le__cancel__right,axiom,
! [A_37: nat,C_14: nat,B_32: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_37 @ C_14 ) @ ( plus_plus_nat @ B_32 @ C_14 ) )
<=> ( ord_less_eq_nat @ A_37 @ B_32 ) ) ).
thf(fact_429_add__less__imp__less__left,axiom,
! [C_13: nat,A_36: nat,B_31: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_13 @ A_36 ) @ ( plus_plus_nat @ C_13 @ B_31 ) )
=> ( ord_less_nat @ A_36 @ B_31 ) ) ).
thf(fact_430_add__less__imp__less__right,axiom,
! [A_35: nat,C_12: nat,B_30: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_35 @ C_12 ) @ ( plus_plus_nat @ B_30 @ C_12 ) )
=> ( ord_less_nat @ A_35 @ B_30 ) ) ).
thf(fact_431_add__strict__mono,axiom,
! [C_11: nat,D: nat,A_34: nat,B_29: nat] :
( ( ord_less_nat @ A_34 @ B_29 )
=> ( ( ord_less_nat @ C_11 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A_34 @ C_11 ) @ ( plus_plus_nat @ B_29 @ D ) ) ) ) ).
thf(fact_432_add__strict__left__mono,axiom,
! [C_10: nat,A_33: nat,B_28: nat] :
( ( ord_less_nat @ A_33 @ B_28 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_10 @ A_33 ) @ ( plus_plus_nat @ C_10 @ B_28 ) ) ) ).
thf(fact_433_add__strict__right__mono,axiom,
! [C_9: nat,A_32: nat,B_27: nat] :
( ( ord_less_nat @ A_32 @ B_27 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_32 @ C_9 ) @ ( plus_plus_nat @ B_27 @ C_9 ) ) ) ).
thf(fact_434_add__less__cancel__left,axiom,
! [C_8: nat,A_31: nat,B_26: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_8 @ A_31 ) @ ( plus_plus_nat @ C_8 @ B_26 ) )
<=> ( ord_less_nat @ A_31 @ B_26 ) ) ).
thf(fact_435_add__less__cancel__right,axiom,
! [A_30: nat,C_7: nat,B_25: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_30 @ C_7 ) @ ( plus_plus_nat @ B_25 @ C_7 ) )
<=> ( ord_less_nat @ A_30 @ B_25 ) ) ).
thf(fact_436_le__funI,axiom,
! [F_8: nat > $o,G_3: nat > $o] :
( ! [X_1: nat] : ( ord_less_eq_o @ ( F_8 @ X_1 ) @ ( G_3 @ X_1 ) )
=> ( ord_less_eq_nat_o @ F_8 @ G_3 ) ) ).
thf(fact_437_equalityI,axiom,
! [A_29: nat > $o,B_24: nat > $o] :
( ( ord_less_eq_nat_o @ A_29 @ B_24 )
=> ( ( ord_less_eq_nat_o @ B_24 @ A_29 )
=> ( A_29 = B_24 ) ) ) ).
thf(fact_438_subsetD,axiom,
! [C_6: $o,A_28: $o > $o,B_23: $o > $o] :
( ( ord_less_eq_o_o @ A_28 @ B_23 )
=> ( ( member_o @ C_6 @ A_28 )
=> ( member_o @ C_6 @ B_23 ) ) ) ).
thf(fact_439_subsetD,axiom,
! [C_6: product_unit,A_28: product_unit > $o,B_23: product_unit > $o] :
( ( ord_le1511552390unit_o @ A_28 @ B_23 )
=> ( ( member_Product_unit @ C_6 @ A_28 )
=> ( member_Product_unit @ C_6 @ B_23 ) ) ) ).
thf(fact_440_subsetD,axiom,
! [C_6: arrow_1429601828e_indi,A_28: arrow_1429601828e_indi > $o,B_23: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_28 @ B_23 )
=> ( ( member2052026769e_indi @ C_6 @ A_28 )
=> ( member2052026769e_indi @ C_6 @ B_23 ) ) ) ).
thf(fact_441_subsetD,axiom,
! [C_6: nat,A_28: nat > $o,B_23: nat > $o] :
( ( ord_less_eq_nat_o @ A_28 @ B_23 )
=> ( ( member_nat @ C_6 @ A_28 )
=> ( member_nat @ C_6 @ B_23 ) ) ) ).
thf(fact_442_subsetD,axiom,
! [C_6: produc1501160679le_alt > $o,A_28: ( produc1501160679le_alt > $o ) > $o,B_23: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le1063113995lt_o_o @ A_28 @ B_23 )
=> ( ( member377231867_alt_o @ C_6 @ A_28 )
=> ( member377231867_alt_o @ C_6 @ B_23 ) ) ) ).
thf(fact_443_subsetD,axiom,
! [C_6: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_28: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_23: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le134800455lt_o_o @ A_28 @ B_23 )
=> ( ( member616898751_alt_o @ C_6 @ A_28 )
=> ( member616898751_alt_o @ C_6 @ B_23 ) ) ) ).
thf(fact_444_subsetD,axiom,
! [C_6: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_28: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_23: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o @ A_28 @ B_23 )
=> ( ( member526088951_alt_o @ C_6 @ A_28 )
=> ( member526088951_alt_o @ C_6 @ B_23 ) ) ) ).
thf(fact_445_subsetD,axiom,
! [C_6: produc1501160679le_alt,A_28: produc1501160679le_alt > $o,B_23: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o @ A_28 @ B_23 )
=> ( ( member214075476le_alt @ C_6 @ A_28 )
=> ( member214075476le_alt @ C_6 @ B_23 ) ) ) ).
thf(fact_446_psubset__eq,axiom,
! [A_27: nat > $o,B_22: nat > $o] :
( ( ord_less_nat_o @ A_27 @ B_22 )
<=> ( ( ord_less_eq_nat_o @ A_27 @ B_22 )
& ( A_27 != B_22 ) ) ) ).
thf(fact_447_psubset__eq,axiom,
! [A_27: product_unit > $o,B_22: product_unit > $o] :
( ( ord_le232288914unit_o @ A_27 @ B_22 )
<=> ( ( ord_le1511552390unit_o @ A_27 @ B_22 )
& ( A_27 != B_22 ) ) ) ).
thf(fact_448_psubset__eq,axiom,
! [A_27: arrow_1429601828e_indi > $o,B_22: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_27 @ B_22 )
<=> ( ( ord_le1799070453indi_o @ A_27 @ B_22 )
& ( A_27 != B_22 ) ) ) ).
thf(fact_449_subset__iff__psubset__eq,axiom,
! [A_26: nat > $o,B_21: nat > $o] :
( ( ord_less_eq_nat_o @ A_26 @ B_21 )
<=> ( ( ord_less_nat_o @ A_26 @ B_21 )
| ( A_26 = B_21 ) ) ) ).
thf(fact_450_subset__iff__psubset__eq,axiom,
! [A_26: product_unit > $o,B_21: product_unit > $o] :
( ( ord_le1511552390unit_o @ A_26 @ B_21 )
<=> ( ( ord_le232288914unit_o @ A_26 @ B_21 )
| ( A_26 = B_21 ) ) ) ).
thf(fact_451_subset__iff__psubset__eq,axiom,
! [A_26: arrow_1429601828e_indi > $o,B_21: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_26 @ B_21 )
<=> ( ( ord_le777687553indi_o @ A_26 @ B_21 )
| ( A_26 = B_21 ) ) ) ).
thf(fact_452_psubset__imp__subset,axiom,
! [A_25: nat > $o,B_20: nat > $o] :
( ( ord_less_nat_o @ A_25 @ B_20 )
=> ( ord_less_eq_nat_o @ A_25 @ B_20 ) ) ).
thf(fact_453_psubset__imp__subset,axiom,
! [A_25: product_unit > $o,B_20: product_unit > $o] :
( ( ord_le232288914unit_o @ A_25 @ B_20 )
=> ( ord_le1511552390unit_o @ A_25 @ B_20 ) ) ).
thf(fact_454_psubset__imp__subset,axiom,
! [A_25: arrow_1429601828e_indi > $o,B_20: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_25 @ B_20 )
=> ( ord_le1799070453indi_o @ A_25 @ B_20 ) ) ).
thf(fact_455_psubset__subset__trans,axiom,
! [C_5: nat > $o,A_24: nat > $o,B_19: nat > $o] :
( ( ord_less_nat_o @ A_24 @ B_19 )
=> ( ( ord_less_eq_nat_o @ B_19 @ C_5 )
=> ( ord_less_nat_o @ A_24 @ C_5 ) ) ) ).
thf(fact_456_psubset__subset__trans,axiom,
! [C_5: product_unit > $o,A_24: product_unit > $o,B_19: product_unit > $o] :
( ( ord_le232288914unit_o @ A_24 @ B_19 )
=> ( ( ord_le1511552390unit_o @ B_19 @ C_5 )
=> ( ord_le232288914unit_o @ A_24 @ C_5 ) ) ) ).
thf(fact_457_psubset__subset__trans,axiom,
! [C_5: arrow_1429601828e_indi > $o,A_24: arrow_1429601828e_indi > $o,B_19: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_24 @ B_19 )
=> ( ( ord_le1799070453indi_o @ B_19 @ C_5 )
=> ( ord_le777687553indi_o @ A_24 @ C_5 ) ) ) ).
thf(fact_458_subset__psubset__trans,axiom,
! [C_4: nat > $o,A_23: nat > $o,B_18: nat > $o] :
( ( ord_less_eq_nat_o @ A_23 @ B_18 )
=> ( ( ord_less_nat_o @ B_18 @ C_4 )
=> ( ord_less_nat_o @ A_23 @ C_4 ) ) ) ).
thf(fact_459_subset__psubset__trans,axiom,
! [C_4: product_unit > $o,A_23: product_unit > $o,B_18: product_unit > $o] :
( ( ord_le1511552390unit_o @ A_23 @ B_18 )
=> ( ( ord_le232288914unit_o @ B_18 @ C_4 )
=> ( ord_le232288914unit_o @ A_23 @ C_4 ) ) ) ).
thf(fact_460_subset__psubset__trans,axiom,
! [C_4: arrow_1429601828e_indi > $o,A_23: arrow_1429601828e_indi > $o,B_18: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_23 @ B_18 )
=> ( ( ord_le777687553indi_o @ B_18 @ C_4 )
=> ( ord_le777687553indi_o @ A_23 @ C_4 ) ) ) ).
thf(fact_461_equalityE,axiom,
! [A_22: nat > $o,B_17: nat > $o] :
( ( A_22 = B_17 )
=> ~ ( ( ord_less_eq_nat_o @ A_22 @ B_17 )
=> ~ ( ord_less_eq_nat_o @ B_17 @ A_22 ) ) ) ).
thf(fact_462_subset__trans,axiom,
! [C_3: nat > $o,A_21: nat > $o,B_16: nat > $o] :
( ( ord_less_eq_nat_o @ A_21 @ B_16 )
=> ( ( ord_less_eq_nat_o @ B_16 @ C_3 )
=> ( ord_less_eq_nat_o @ A_21 @ C_3 ) ) ) ).
thf(fact_463_set__mp,axiom,
! [X_7: $o,A_20: $o > $o,B_15: $o > $o] :
( ( ord_less_eq_o_o @ A_20 @ B_15 )
=> ( ( member_o @ X_7 @ A_20 )
=> ( member_o @ X_7 @ B_15 ) ) ) ).
thf(fact_464_set__mp,axiom,
! [X_7: product_unit,A_20: product_unit > $o,B_15: product_unit > $o] :
( ( ord_le1511552390unit_o @ A_20 @ B_15 )
=> ( ( member_Product_unit @ X_7 @ A_20 )
=> ( member_Product_unit @ X_7 @ B_15 ) ) ) ).
thf(fact_465_set__mp,axiom,
! [X_7: arrow_1429601828e_indi,A_20: arrow_1429601828e_indi > $o,B_15: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_20 @ B_15 )
=> ( ( member2052026769e_indi @ X_7 @ A_20 )
=> ( member2052026769e_indi @ X_7 @ B_15 ) ) ) ).
thf(fact_466_set__mp,axiom,
! [X_7: nat,A_20: nat > $o,B_15: nat > $o] :
( ( ord_less_eq_nat_o @ A_20 @ B_15 )
=> ( ( member_nat @ X_7 @ A_20 )
=> ( member_nat @ X_7 @ B_15 ) ) ) ).
thf(fact_467_set__mp,axiom,
! [X_7: produc1501160679le_alt > $o,A_20: ( produc1501160679le_alt > $o ) > $o,B_15: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le1063113995lt_o_o @ A_20 @ B_15 )
=> ( ( member377231867_alt_o @ X_7 @ A_20 )
=> ( member377231867_alt_o @ X_7 @ B_15 ) ) ) ).
thf(fact_468_set__mp,axiom,
! [X_7: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_20: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_15: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le134800455lt_o_o @ A_20 @ B_15 )
=> ( ( member616898751_alt_o @ X_7 @ A_20 )
=> ( member616898751_alt_o @ X_7 @ B_15 ) ) ) ).
thf(fact_469_set__mp,axiom,
! [X_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_20: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_15: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o @ A_20 @ B_15 )
=> ( ( member526088951_alt_o @ X_7 @ A_20 )
=> ( member526088951_alt_o @ X_7 @ B_15 ) ) ) ).
thf(fact_470_set__mp,axiom,
! [X_7: produc1501160679le_alt,A_20: produc1501160679le_alt > $o,B_15: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o @ A_20 @ B_15 )
=> ( ( member214075476le_alt @ X_7 @ A_20 )
=> ( member214075476le_alt @ X_7 @ B_15 ) ) ) ).
thf(fact_471_set__rev__mp,axiom,
! [B_14: $o > $o,X_6: $o,A_19: $o > $o] :
( ( member_o @ X_6 @ A_19 )
=> ( ( ord_less_eq_o_o @ A_19 @ B_14 )
=> ( member_o @ X_6 @ B_14 ) ) ) ).
thf(fact_472_set__rev__mp,axiom,
! [B_14: product_unit > $o,X_6: product_unit,A_19: product_unit > $o] :
( ( member_Product_unit @ X_6 @ A_19 )
=> ( ( ord_le1511552390unit_o @ A_19 @ B_14 )
=> ( member_Product_unit @ X_6 @ B_14 ) ) ) ).
thf(fact_473_set__rev__mp,axiom,
! [B_14: arrow_1429601828e_indi > $o,X_6: arrow_1429601828e_indi,A_19: arrow_1429601828e_indi > $o] :
( ( member2052026769e_indi @ X_6 @ A_19 )
=> ( ( ord_le1799070453indi_o @ A_19 @ B_14 )
=> ( member2052026769e_indi @ X_6 @ B_14 ) ) ) ).
thf(fact_474_set__rev__mp,axiom,
! [B_14: nat > $o,X_6: nat,A_19: nat > $o] :
( ( member_nat @ X_6 @ A_19 )
=> ( ( ord_less_eq_nat_o @ A_19 @ B_14 )
=> ( member_nat @ X_6 @ B_14 ) ) ) ).
thf(fact_475_set__rev__mp,axiom,
! [B_14: ( produc1501160679le_alt > $o ) > $o,X_6: produc1501160679le_alt > $o,A_19: ( produc1501160679le_alt > $o ) > $o] :
( ( member377231867_alt_o @ X_6 @ A_19 )
=> ( ( ord_le1063113995lt_o_o @ A_19 @ B_14 )
=> ( member377231867_alt_o @ X_6 @ B_14 ) ) ) ).
thf(fact_476_set__rev__mp,axiom,
! [B_14: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,X_6: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_19: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ X_6 @ A_19 )
=> ( ( ord_le134800455lt_o_o @ A_19 @ B_14 )
=> ( member616898751_alt_o @ X_6 @ B_14 ) ) ) ).
thf(fact_477_set__rev__mp,axiom,
! [B_14: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,X_6: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_19: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ X_6 @ A_19 )
=> ( ( ord_le1992928527lt_o_o @ A_19 @ B_14 )
=> ( member526088951_alt_o @ X_6 @ B_14 ) ) ) ).
thf(fact_478_set__rev__mp,axiom,
! [B_14: produc1501160679le_alt > $o,X_6: produc1501160679le_alt,A_19: produc1501160679le_alt > $o] :
( ( member214075476le_alt @ X_6 @ A_19 )
=> ( ( ord_le97612146_alt_o @ A_19 @ B_14 )
=> ( member214075476le_alt @ X_6 @ B_14 ) ) ) ).
thf(fact_479_predicate1D,axiom,
! [X_5: nat,P_2: nat > $o,Q_1: nat > $o] :
( ( ord_less_eq_nat_o @ P_2 @ Q_1 )
=> ( ( P_2 @ X_5 )
=> ( Q_1 @ X_5 ) ) ) ).
thf(fact_480_in__mono,axiom,
! [X_4: $o,A_18: $o > $o,B_13: $o > $o] :
( ( ord_less_eq_o_o @ A_18 @ B_13 )
=> ( ( member_o @ X_4 @ A_18 )
=> ( member_o @ X_4 @ B_13 ) ) ) ).
thf(fact_481_in__mono,axiom,
! [X_4: product_unit,A_18: product_unit > $o,B_13: product_unit > $o] :
( ( ord_le1511552390unit_o @ A_18 @ B_13 )
=> ( ( member_Product_unit @ X_4 @ A_18 )
=> ( member_Product_unit @ X_4 @ B_13 ) ) ) ).
thf(fact_482_in__mono,axiom,
! [X_4: arrow_1429601828e_indi,A_18: arrow_1429601828e_indi > $o,B_13: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o @ A_18 @ B_13 )
=> ( ( member2052026769e_indi @ X_4 @ A_18 )
=> ( member2052026769e_indi @ X_4 @ B_13 ) ) ) ).
thf(fact_483_in__mono,axiom,
! [X_4: nat,A_18: nat > $o,B_13: nat > $o] :
( ( ord_less_eq_nat_o @ A_18 @ B_13 )
=> ( ( member_nat @ X_4 @ A_18 )
=> ( member_nat @ X_4 @ B_13 ) ) ) ).
thf(fact_484_in__mono,axiom,
! [X_4: produc1501160679le_alt > $o,A_18: ( produc1501160679le_alt > $o ) > $o,B_13: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le1063113995lt_o_o @ A_18 @ B_13 )
=> ( ( member377231867_alt_o @ X_4 @ A_18 )
=> ( member377231867_alt_o @ X_4 @ B_13 ) ) ) ).
thf(fact_485_in__mono,axiom,
! [X_4: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_18: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_13: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le134800455lt_o_o @ A_18 @ B_13 )
=> ( ( member616898751_alt_o @ X_4 @ A_18 )
=> ( member616898751_alt_o @ X_4 @ B_13 ) ) ) ).
thf(fact_486_in__mono,axiom,
! [X_4: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_18: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_13: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o @ A_18 @ B_13 )
=> ( ( member526088951_alt_o @ X_4 @ A_18 )
=> ( member526088951_alt_o @ X_4 @ B_13 ) ) ) ).
thf(fact_487_in__mono,axiom,
! [X_4: produc1501160679le_alt,A_18: produc1501160679le_alt > $o,B_13: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o @ A_18 @ B_13 )
=> ( ( member214075476le_alt @ X_4 @ A_18 )
=> ( member214075476le_alt @ X_4 @ B_13 ) ) ) ).
thf(fact_488_equalityD2,axiom,
! [A_17: nat > $o,B_12: nat > $o] :
( ( A_17 = B_12 )
=> ( ord_less_eq_nat_o @ B_12 @ A_17 ) ) ).
thf(fact_489_equalityD1,axiom,
! [A_16: nat > $o,B_11: nat > $o] :
( ( A_16 = B_11 )
=> ( ord_less_eq_nat_o @ A_16 @ B_11 ) ) ).
thf(fact_490_rev__predicate1D,axiom,
! [Q: nat > $o,P_1: nat > $o,X_3: nat] :
( ( P_1 @ X_3 )
=> ( ( ord_less_eq_nat_o @ P_1 @ Q )
=> ( Q @ X_3 ) ) ) ).
thf(fact_491_set__eq__subset,axiom,
! [A_15: nat > $o,B_10: nat > $o] :
( ( A_15 = B_10 )
<=> ( ( ord_less_eq_nat_o @ A_15 @ B_10 )
& ( ord_less_eq_nat_o @ B_10 @ A_15 ) ) ) ).
thf(fact_492_subset__refl,axiom,
! [A_14: nat > $o] : ( ord_less_eq_nat_o @ A_14 @ A_14 ) ).
thf(fact_493_pred__subset__eq,axiom,
! [R_1: $o > $o,S_1: $o > $o] :
( ( ord_less_eq_o_o
@ ^ [X_1: $o] : ( member_o @ X_1 @ R_1 )
@ ^ [X_1: $o] : ( member_o @ X_1 @ S_1 ) )
<=> ( ord_less_eq_o_o @ R_1 @ S_1 ) ) ).
thf(fact_494_pred__subset__eq,axiom,
! [R_1: product_unit > $o,S_1: product_unit > $o] :
( ( ord_le1511552390unit_o
@ ^ [X_1: product_unit] : ( member_Product_unit @ X_1 @ R_1 )
@ ^ [X_1: product_unit] : ( member_Product_unit @ X_1 @ S_1 ) )
<=> ( ord_le1511552390unit_o @ R_1 @ S_1 ) ) ).
thf(fact_495_pred__subset__eq,axiom,
! [R_1: arrow_1429601828e_indi > $o,S_1: arrow_1429601828e_indi > $o] :
( ( ord_le1799070453indi_o
@ ^ [X_1: arrow_1429601828e_indi] : ( member2052026769e_indi @ X_1 @ R_1 )
@ ^ [X_1: arrow_1429601828e_indi] : ( member2052026769e_indi @ X_1 @ S_1 ) )
<=> ( ord_le1799070453indi_o @ R_1 @ S_1 ) ) ).
thf(fact_496_pred__subset__eq,axiom,
! [R_1: nat > $o,S_1: nat > $o] :
( ( ord_less_eq_nat_o
@ ^ [X_1: nat] : ( member_nat @ X_1 @ R_1 )
@ ^ [X_1: nat] : ( member_nat @ X_1 @ S_1 ) )
<=> ( ord_less_eq_nat_o @ R_1 @ S_1 ) ) ).
thf(fact_497_pred__subset__eq,axiom,
! [R_1: ( produc1501160679le_alt > $o ) > $o,S_1: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le1063113995lt_o_o
@ ^ [X_1: produc1501160679le_alt > $o] : ( member377231867_alt_o @ X_1 @ R_1 )
@ ^ [X_1: produc1501160679le_alt > $o] : ( member377231867_alt_o @ X_1 @ S_1 ) )
<=> ( ord_le1063113995lt_o_o @ R_1 @ S_1 ) ) ).
thf(fact_498_pred__subset__eq,axiom,
! [R_1: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,S_1: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le134800455lt_o_o
@ ^ [X_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] : ( member616898751_alt_o @ X_1 @ R_1 )
@ ^ [X_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] : ( member616898751_alt_o @ X_1 @ S_1 ) )
<=> ( ord_le134800455lt_o_o @ R_1 @ S_1 ) ) ).
thf(fact_499_pred__subset__eq,axiom,
! [R_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,S_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1992928527lt_o_o
@ ^ [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] : ( member526088951_alt_o @ X_1 @ R_1 )
@ ^ [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] : ( member526088951_alt_o @ X_1 @ S_1 ) )
<=> ( ord_le1992928527lt_o_o @ R_1 @ S_1 ) ) ).
thf(fact_500_pred__subset__eq,axiom,
! [R_1: produc1501160679le_alt > $o,S_1: produc1501160679le_alt > $o] :
( ( ord_le97612146_alt_o
@ ^ [X_1: produc1501160679le_alt] : ( member214075476le_alt @ X_1 @ R_1 )
@ ^ [X_1: produc1501160679le_alt] : ( member214075476le_alt @ X_1 @ S_1 ) )
<=> ( ord_le97612146_alt_o @ R_1 @ S_1 ) ) ).
thf(fact_501_Pi__mono,axiom,
! [B_9: produc1501160679le_alt > $o > $o,C_2: produc1501160679le_alt > $o > $o,A_13: produc1501160679le_alt > $o] :
( ! [X_1: produc1501160679le_alt] :
( ( member214075476le_alt @ X_1 @ A_13 )
=> ( ord_less_eq_o_o @ ( B_9 @ X_1 ) @ ( C_2 @ X_1 ) ) )
=> ( ord_le1063113995lt_o_o @ ( pi_Pro1701359055_alt_o @ A_13 @ B_9 ) @ ( pi_Pro1701359055_alt_o @ A_13 @ C_2 ) ) ) ).
thf(fact_502_Pi__mono,axiom,
! [B_9: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o,C_2: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o,A_13: arrow_1429601828e_indi > $o] :
( ! [X_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_13 )
=> ( ord_le1063113995lt_o_o @ ( B_9 @ X_1 ) @ ( C_2 @ X_1 ) ) )
=> ( ord_le1992928527lt_o_o @ ( pi_Arr1929480907_alt_o @ A_13 @ B_9 ) @ ( pi_Arr1929480907_alt_o @ A_13 @ C_2 ) ) ) ).
thf(fact_503_Pi__mono,axiom,
! [B_9: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o,C_2: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o,A_13: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ A_13 )
=> ( ord_le1063113995lt_o_o @ ( B_9 @ X_1 ) @ ( C_2 @ X_1 ) ) )
=> ( ord_le134800455lt_o_o @ ( pi_Arr1304755663_alt_o @ A_13 @ B_9 ) @ ( pi_Arr1304755663_alt_o @ A_13 @ C_2 ) ) ) ).
thf(fact_504_subsetI,axiom,
! [B_8: $o > $o,A_12: $o > $o] :
( ! [X_1: $o] :
( ( member_o @ X_1 @ A_12 )
=> ( member_o @ X_1 @ B_8 ) )
=> ( ord_less_eq_o_o @ A_12 @ B_8 ) ) ).
thf(fact_505_subsetI,axiom,
! [B_8: product_unit > $o,A_12: product_unit > $o] :
( ! [X_1: product_unit] :
( ( member_Product_unit @ X_1 @ A_12 )
=> ( member_Product_unit @ X_1 @ B_8 ) )
=> ( ord_le1511552390unit_o @ A_12 @ B_8 ) ) ).
thf(fact_506_subsetI,axiom,
! [B_8: arrow_1429601828e_indi > $o,A_12: arrow_1429601828e_indi > $o] :
( ! [X_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_12 )
=> ( member2052026769e_indi @ X_1 @ B_8 ) )
=> ( ord_le1799070453indi_o @ A_12 @ B_8 ) ) ).
thf(fact_507_subsetI,axiom,
! [B_8: nat > $o,A_12: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_12 )
=> ( member_nat @ X_1 @ B_8 ) )
=> ( ord_less_eq_nat_o @ A_12 @ B_8 ) ) ).
thf(fact_508_subsetI,axiom,
! [B_8: ( produc1501160679le_alt > $o ) > $o,A_12: ( produc1501160679le_alt > $o ) > $o] :
( ! [X_1: produc1501160679le_alt > $o] :
( ( member377231867_alt_o @ X_1 @ A_12 )
=> ( member377231867_alt_o @ X_1 @ B_8 ) )
=> ( ord_le1063113995lt_o_o @ A_12 @ B_8 ) ) ).
thf(fact_509_subsetI,axiom,
! [B_8: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,A_12: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ! [X_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o] :
( ( member616898751_alt_o @ X_1 @ A_12 )
=> ( member616898751_alt_o @ X_1 @ B_8 ) )
=> ( ord_le134800455lt_o_o @ A_12 @ B_8 ) ) ).
thf(fact_510_subsetI,axiom,
! [B_8: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,A_12: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ A_12 )
=> ( member526088951_alt_o @ X_1 @ B_8 ) )
=> ( ord_le1992928527lt_o_o @ A_12 @ B_8 ) ) ).
thf(fact_511_subsetI,axiom,
! [B_8: produc1501160679le_alt > $o,A_12: produc1501160679le_alt > $o] :
( ! [X_1: produc1501160679le_alt] :
( ( member214075476le_alt @ X_1 @ A_12 )
=> ( member214075476le_alt @ X_1 @ B_8 ) )
=> ( ord_le97612146_alt_o @ A_12 @ B_8 ) ) ).
thf(fact_512_psubsetD,axiom,
! [C_1: $o,A_11: $o > $o,B_7: $o > $o] :
( ( ord_less_o_o @ A_11 @ B_7 )
=> ( ( member_o @ C_1 @ A_11 )
=> ( member_o @ C_1 @ B_7 ) ) ) ).
thf(fact_513_psubsetD,axiom,
! [C_1: product_unit,A_11: product_unit > $o,B_7: product_unit > $o] :
( ( ord_le232288914unit_o @ A_11 @ B_7 )
=> ( ( member_Product_unit @ C_1 @ A_11 )
=> ( member_Product_unit @ C_1 @ B_7 ) ) ) ).
thf(fact_514_psubsetD,axiom,
! [C_1: arrow_1429601828e_indi,A_11: arrow_1429601828e_indi > $o,B_7: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_11 @ B_7 )
=> ( ( member2052026769e_indi @ C_1 @ A_11 )
=> ( member2052026769e_indi @ C_1 @ B_7 ) ) ) ).
thf(fact_515_psubsetD,axiom,
! [C_1: nat,A_11: nat > $o,B_7: nat > $o] :
( ( ord_less_nat_o @ A_11 @ B_7 )
=> ( ( member_nat @ C_1 @ A_11 )
=> ( member_nat @ C_1 @ B_7 ) ) ) ).
thf(fact_516_psubsetD,axiom,
! [C_1: produc1501160679le_alt > $o,A_11: ( produc1501160679le_alt > $o ) > $o,B_7: ( produc1501160679le_alt > $o ) > $o] :
( ( ord_le910298367lt_o_o @ A_11 @ B_7 )
=> ( ( member377231867_alt_o @ C_1 @ A_11 )
=> ( member377231867_alt_o @ C_1 @ B_7 ) ) ) ).
thf(fact_517_psubsetD,axiom,
! [C_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_11: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o,B_7: ( ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o ) > $o] :
( ( ord_le1859604819lt_o_o @ A_11 @ B_7 )
=> ( ( member616898751_alt_o @ C_1 @ A_11 )
=> ( member616898751_alt_o @ C_1 @ B_7 ) ) ) ).
thf(fact_518_psubsetD,axiom,
! [C_1: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_11: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_7: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( ord_le157835011lt_o_o @ A_11 @ B_7 )
=> ( ( member526088951_alt_o @ C_1 @ A_11 )
=> ( member526088951_alt_o @ C_1 @ B_7 ) ) ) ).
thf(fact_519_psubsetD,axiom,
! [C_1: produc1501160679le_alt,A_11: produc1501160679le_alt > $o,B_7: produc1501160679le_alt > $o] :
( ( ord_le988258430_alt_o @ A_11 @ B_7 )
=> ( ( member214075476le_alt @ C_1 @ A_11 )
=> ( member214075476le_alt @ C_1 @ B_7 ) ) ) ).
thf(fact_520_psubset__trans,axiom,
! [C: nat > $o,A_10: nat > $o,B_6: nat > $o] :
( ( ord_less_nat_o @ A_10 @ B_6 )
=> ( ( ord_less_nat_o @ B_6 @ C )
=> ( ord_less_nat_o @ A_10 @ C ) ) ) ).
thf(fact_521_psubset__trans,axiom,
! [C: product_unit > $o,A_10: product_unit > $o,B_6: product_unit > $o] :
( ( ord_le232288914unit_o @ A_10 @ B_6 )
=> ( ( ord_le232288914unit_o @ B_6 @ C )
=> ( ord_le232288914unit_o @ A_10 @ C ) ) ) ).
thf(fact_522_psubset__trans,axiom,
! [C: arrow_1429601828e_indi > $o,A_10: arrow_1429601828e_indi > $o,B_6: arrow_1429601828e_indi > $o] :
( ( ord_le777687553indi_o @ A_10 @ B_6 )
=> ( ( ord_le777687553indi_o @ B_6 @ C )
=> ( ord_le777687553indi_o @ A_10 @ C ) ) ) ).
thf(fact_523_in__below,axiom,
! [X: arrow_475358991le_alt,Y: arrow_475358991le_alt,L_2: produc1501160679le_alt > $o,A_9: arrow_475358991le_alt,B_5: arrow_475358991le_alt] :
( ( A_9 != B_5 )
=> ( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ ( arrow_2098199487_below @ L_2 @ A_9 @ B_5 ) )
<=> ( ( X != Y )
& ( ( Y = A_9 )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ X @ B_5 ) @ L_2 ) )
& ( ( Y != A_9 )
=> ( ( ( X = A_9 )
=> ( ( Y = B_5 )
| ( member214075476le_alt @ ( produc1347929815le_alt @ B_5 @ Y ) @ L_2 ) ) )
& ( ( X != A_9 )
=> ( member214075476le_alt @ ( produc1347929815le_alt @ X @ Y ) @ L_2 ) ) ) ) ) ) ) ) ).
thf(fact_524_add__leE,axiom,
! [M_2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M_2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M_2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_525_add__leD1,axiom,
! [M_2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M_2 @ K ) @ N )
=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_526_add__leD2,axiom,
! [M_2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M_2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
thf(fact_527_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_528_nat__neq__iff,axiom,
! [M_2: nat,N: nat] :
( ( M_2 != N )
<=> ( ( ord_less_nat @ M_2 @ N )
| ( ord_less_nat @ N @ M_2 ) ) ) ).
thf(fact_529_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_530_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_531_less__not__refl2,axiom,
! [N: nat,M_2: nat] :
( ( ord_less_nat @ N @ M_2 )
=> ( M_2 != N ) ) ).
thf(fact_532_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_533_nat__less__cases,axiom,
! [P: nat > nat > $o,M_2: nat,N: nat] :
( ( ( ord_less_nat @ M_2 @ N )
=> ( P @ N @ M_2 ) )
=> ( ( ( M_2 = N )
=> ( P @ N @ M_2 ) )
=> ( ( ( ord_less_nat @ N @ M_2 )
=> ( P @ N @ M_2 ) )
=> ( P @ N @ M_2 ) ) ) ) ).
thf(fact_534_nat__add__commute,axiom,
! [M_2: nat,N: nat] :
( ( plus_plus_nat @ M_2 @ N )
= ( plus_plus_nat @ N @ M_2 ) ) ).
thf(fact_535_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z ) ) ) ).
thf(fact_536_nat__add__assoc,axiom,
! [M_2: nat,N: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M_2 @ N ) @ K )
= ( plus_plus_nat @ M_2 @ ( plus_plus_nat @ N @ K ) ) ) ).
thf(fact_537_nat__add__left__cancel,axiom,
! [K: nat,M_2: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M_2 )
= ( plus_plus_nat @ K @ N ) )
<=> ( M_2 = N ) ) ).
thf(fact_538_nat__add__right__cancel,axiom,
! [M_2: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M_2 @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M_2 = N ) ) ).
thf(fact_539_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_540_nat__le__linear,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
| ( ord_less_eq_nat @ N @ M_2 ) ) ).
thf(fact_541_eq__imp__le,axiom,
! [M_2: nat,N: nat] :
( ( M_2 = N )
=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_542_le__trans,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( ord_less_eq_nat @ J_1 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
thf(fact_543_le__antisym,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( ( ord_less_eq_nat @ N @ M_2 )
=> ( M_2 = N ) ) ) ).
thf(fact_544_below__Lin,axiom,
! [L_2: produc1501160679le_alt > $o,X: arrow_475358991le_alt,Y: arrow_475358991le_alt] :
( ( X != Y )
=> ( ( member377231867_alt_o @ L_2 @ arrow_823908191le_Lin )
=> ( member377231867_alt_o @ ( arrow_2098199487_below @ L_2 @ X @ Y ) @ arrow_823908191le_Lin ) ) ) ).
thf(fact_545_inj__on__add__nat,axiom,
! [K: nat,N_2: nat > $o] :
( inj_on_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K )
@ N_2 ) ).
thf(fact_546_not__add__less1,axiom,
! [I: nat,J_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J_1 ) @ I ) ).
thf(fact_547_not__add__less2,axiom,
! [J_1: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J_1 @ I ) @ I ) ).
thf(fact_548_nat__add__left__cancel__less,axiom,
! [K: nat,M_2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M_2 ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_nat @ M_2 @ N ) ) ).
thf(fact_549_trans__less__add1,axiom,
! [M_2: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J_1 @ M_2 ) ) ) ).
thf(fact_550_trans__less__add2,axiom,
! [M_2: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M_2 @ J_1 ) ) ) ).
thf(fact_551_add__less__mono1,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_552_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_553_less__add__eq__less,axiom,
! [M_2: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M_2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M_2 @ N ) ) ) ).
thf(fact_554_add__lessD1,axiom,
! [I: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J_1 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
thf(fact_555_nat__less__le,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
<=> ( ( ord_less_eq_nat @ M_2 @ N )
& ( M_2 != N ) ) ) ).
thf(fact_556_le__eq__less__or__eq,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
<=> ( ( ord_less_nat @ M_2 @ N )
| ( M_2 = N ) ) ) ).
thf(fact_557_less__imp__le__nat,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_558_le__neq__implies__less,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( ( M_2 != N )
=> ( ord_less_nat @ M_2 @ N ) ) ) ).
thf(fact_559_less__or__eq__imp__le,axiom,
! [M_2: nat,N: nat] :
( ( ( ord_less_nat @ M_2 @ N )
| ( M_2 = N ) )
=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_560_le__add2,axiom,
! [N: nat,M_2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M_2 @ N ) ) ).
thf(fact_561_le__add1,axiom,
! [N: nat,M_2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M_2 ) ) ).
thf(fact_562_le__iff__add,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
<=> ? [K_1: nat] :
( N
= ( plus_plus_nat @ M_2 @ K_1 ) ) ) ).
thf(fact_563_nat__add__left__cancel__le,axiom,
! [K: nat,M_2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M_2 ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_564_trans__le__add1,axiom,
! [M_2: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J_1 @ M_2 ) ) ) ).
thf(fact_565_trans__le__add2,axiom,
! [M_2: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M_2 @ J_1 ) ) ) ).
thf(fact_566_add__le__mono1,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_567_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_568_mono__nat__linear__lb,axiom,
! [M_2: nat,K: nat,F: nat > nat] :
( ! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N_1 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M_2 ) @ K ) @ ( F @ ( plus_plus_nat @ M_2 @ K ) ) ) ) ).
thf(fact_569_less__mono__imp__le__mono,axiom,
! [I: nat,J_1: nat,F: nat > nat] :
( ! [I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ ( F @ I_1 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J_1 ) ) ) ) ).
thf(fact_570_termination__basic__simps_I3_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_571_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_572_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_573_termination__basic__simps_I1_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_574_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_575_in__rel__def,axiom,
! [R: produc1501160679le_alt > $o,X_2: arrow_475358991le_alt,Y_2: arrow_475358991le_alt] :
( ( in_rel1252994498le_alt @ R @ X_2 @ Y_2 )
<=> ( member214075476le_alt @ ( produc1347929815le_alt @ X_2 @ Y_2 ) @ R ) ) ).
thf(fact_576_funcsetI,axiom,
! [F_7: produc1501160679le_alt > $o,B_4: $o > $o,A_8: produc1501160679le_alt > $o] :
( ! [X_1: produc1501160679le_alt] :
( ( member214075476le_alt @ X_1 @ A_8 )
=> ( member_o @ ( F_7 @ X_1 ) @ B_4 ) )
=> ( member377231867_alt_o @ F_7
@ ( pi_Pro1701359055_alt_o @ A_8
@ ^ [Uu: produc1501160679le_alt] : B_4 ) ) ) ).
thf(fact_577_funcsetI,axiom,
! [F_7: arrow_1429601828e_indi > produc1501160679le_alt > $o,B_4: ( produc1501160679le_alt > $o ) > $o,A_8: arrow_1429601828e_indi > $o] :
( ! [X_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_8 )
=> ( member377231867_alt_o @ ( F_7 @ X_1 ) @ B_4 ) )
=> ( member526088951_alt_o @ F_7
@ ( pi_Arr1929480907_alt_o @ A_8
@ ^ [Uu: arrow_1429601828e_indi] : B_4 ) ) ) ).
thf(fact_578_funcsetI,axiom,
! [F_7: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,B_4: ( produc1501160679le_alt > $o ) > $o,A_8: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ A_8 )
=> ( member377231867_alt_o @ ( F_7 @ X_1 ) @ B_4 ) )
=> ( member616898751_alt_o @ F_7
@ ( pi_Arr1304755663_alt_o @ A_8
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : B_4 ) ) ) ).
thf(fact_579_linear__alt,axiom,
? [L_1: produc1501160679le_alt > $o] : ( member377231867_alt_o @ L_1 @ arrow_823908191le_Lin ) ).
thf(fact_580_Pi__cong,axiom,
! [B_3: produc1501160679le_alt > $o > $o,G_2: produc1501160679le_alt > $o,F_6: produc1501160679le_alt > $o,A_7: produc1501160679le_alt > $o] :
( ! [W: produc1501160679le_alt] :
( ( member214075476le_alt @ W @ A_7 )
=> ( ( F_6 @ W )
<=> ( G_2 @ W ) ) )
=> ( ( member377231867_alt_o @ F_6 @ ( pi_Pro1701359055_alt_o @ A_7 @ B_3 ) )
<=> ( member377231867_alt_o @ G_2 @ ( pi_Pro1701359055_alt_o @ A_7 @ B_3 ) ) ) ) ).
thf(fact_581_Pi__cong,axiom,
! [B_3: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o,F_6: arrow_1429601828e_indi > produc1501160679le_alt > $o,G_2: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_7: arrow_1429601828e_indi > $o] :
( ! [W: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ W @ A_7 )
=> ( ( F_6 @ W )
= ( G_2 @ W ) ) )
=> ( ( member526088951_alt_o @ F_6 @ ( pi_Arr1929480907_alt_o @ A_7 @ B_3 ) )
<=> ( member526088951_alt_o @ G_2 @ ( pi_Arr1929480907_alt_o @ A_7 @ B_3 ) ) ) ) ).
thf(fact_582_Pi__cong,axiom,
! [B_3: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o,F_6: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,G_2: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_7: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ! [W: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ W @ A_7 )
=> ( ( F_6 @ W )
= ( G_2 @ W ) ) )
=> ( ( member616898751_alt_o @ F_6 @ ( pi_Arr1304755663_alt_o @ A_7 @ B_3 ) )
<=> ( member616898751_alt_o @ G_2 @ ( pi_Arr1304755663_alt_o @ A_7 @ B_3 ) ) ) ) ).
thf(fact_583_Pi__I_H,axiom,
! [F_5: produc1501160679le_alt > $o,B_2: produc1501160679le_alt > $o > $o,A_6: produc1501160679le_alt > $o] :
( ! [X_1: produc1501160679le_alt] :
( ( member214075476le_alt @ X_1 @ A_6 )
=> ( member_o @ ( F_5 @ X_1 ) @ ( B_2 @ X_1 ) ) )
=> ( member377231867_alt_o @ F_5 @ ( pi_Pro1701359055_alt_o @ A_6 @ B_2 ) ) ) ).
thf(fact_584_Pi__I_H,axiom,
! [F_5: arrow_1429601828e_indi > produc1501160679le_alt > $o,B_2: arrow_1429601828e_indi > ( produc1501160679le_alt > $o ) > $o,A_6: arrow_1429601828e_indi > $o] :
( ! [X_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_6 )
=> ( member377231867_alt_o @ ( F_5 @ X_1 ) @ ( B_2 @ X_1 ) ) )
=> ( member526088951_alt_o @ F_5 @ ( pi_Arr1929480907_alt_o @ A_6 @ B_2 ) ) ) ).
thf(fact_585_Pi__I_H,axiom,
! [F_5: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,B_2: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > ( produc1501160679le_alt > $o ) > $o,A_6: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ! [X_1: arrow_1429601828e_indi > produc1501160679le_alt > $o] :
( ( member526088951_alt_o @ X_1 @ A_6 )
=> ( member377231867_alt_o @ ( F_5 @ X_1 ) @ ( B_2 @ X_1 ) ) )
=> ( member616898751_alt_o @ F_5 @ ( pi_Arr1304755663_alt_o @ A_6 @ B_2 ) ) ) ).
thf(fact_586_card__inj,axiom,
! [F_4: nat > nat,A_5: nat > $o,B_1: nat > $o] :
( ( member_nat_nat @ F_4
@ ( pi_nat_nat @ A_5
@ ^ [Uu: nat] : B_1 ) )
=> ( ( inj_on_nat_nat @ F_4 @ A_5 )
=> ( ( finite_finite_nat @ B_1 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_5 ) @ ( finite_card_nat @ B_1 ) ) ) ) ) ).
thf(fact_587_card__inj,axiom,
! [F_4: arrow_1429601828e_indi > nat,A_5: arrow_1429601828e_indi > $o,B_1: nat > $o] :
( ( member1315464153di_nat @ F_4
@ ( pi_Arr251692973di_nat @ A_5
@ ^ [Uu: arrow_1429601828e_indi] : B_1 ) )
=> ( ( inj_on978774663di_nat @ F_4 @ A_5 )
=> ( ( finite_finite_nat @ B_1 )
=> ( ord_less_eq_nat @ ( finite97476818e_indi @ A_5 ) @ ( finite_card_nat @ B_1 ) ) ) ) ) ).
thf(fact_588_card__inj,axiom,
! [F_4: produc1501160679le_alt > $o,A_5: produc1501160679le_alt > $o,B_1: $o > $o] :
( ( member377231867_alt_o @ F_4
@ ( pi_Pro1701359055_alt_o @ A_5
@ ^ [Uu: produc1501160679le_alt] : B_1 ) )
=> ( ( inj_on1911943593_alt_o @ F_4 @ A_5 )
=> ( ( finite_finite_o @ B_1 )
=> ( ord_less_eq_nat @ ( finite537683861le_alt @ A_5 ) @ ( finite_card_o @ B_1 ) ) ) ) ) ).
thf(fact_589_card__inj,axiom,
! [F_4: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_5: arrow_1429601828e_indi > $o,B_1: ( produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ F_4
@ ( pi_Arr1929480907_alt_o @ A_5
@ ^ [Uu: arrow_1429601828e_indi] : B_1 ) )
=> ( ( inj_on1190919077_alt_o @ F_4 @ A_5 )
=> ( ( finite2112685307_alt_o @ B_1 )
=> ( ord_less_eq_nat @ ( finite97476818e_indi @ A_5 ) @ ( finite28306938_alt_o @ B_1 ) ) ) ) ) ).
thf(fact_590_card__inj,axiom,
! [F_4: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_5: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B_1: ( produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ F_4
@ ( pi_Arr1304755663_alt_o @ A_5
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : B_1 ) )
=> ( ( inj_on1284293749_alt_o @ F_4 @ A_5 )
=> ( ( finite2112685307_alt_o @ B_1 )
=> ( ord_less_eq_nat @ ( finite120663670_alt_o @ A_5 ) @ ( finite28306938_alt_o @ B_1 ) ) ) ) ) ).
thf(fact_591_inj__onI,axiom,
! [F_3: nat > nat,A_4: nat > $o] :
( ! [X_1: nat,Y_1: nat] :
( ( member_nat @ X_1 @ A_4 )
=> ( ( member_nat @ Y_1 @ A_4 )
=> ( ( ( F_3 @ X_1 )
= ( F_3 @ Y_1 ) )
=> ( X_1 = Y_1 ) ) ) )
=> ( inj_on_nat_nat @ F_3 @ A_4 ) ) ).
thf(fact_592_inj__onI,axiom,
! [F_3: arrow_1429601828e_indi > nat,A_4: arrow_1429601828e_indi > $o] :
( ! [X_1: arrow_1429601828e_indi,Y_1: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ X_1 @ A_4 )
=> ( ( member2052026769e_indi @ Y_1 @ A_4 )
=> ( ( ( F_3 @ X_1 )
= ( F_3 @ Y_1 ) )
=> ( X_1 = Y_1 ) ) ) )
=> ( inj_on978774663di_nat @ F_3 @ A_4 ) ) ).
thf(fact_593_inj__on__cong,axiom,
! [F_2: nat > nat,G_1: nat > nat,A_2: nat > $o] :
( ! [A_3: nat] :
( ( member_nat @ A_3 @ A_2 )
=> ( ( F_2 @ A_3 )
= ( G_1 @ A_3 ) ) )
=> ( ( inj_on_nat_nat @ F_2 @ A_2 )
<=> ( inj_on_nat_nat @ G_1 @ A_2 ) ) ) ).
thf(fact_594_inj__on__cong,axiom,
! [F_2: arrow_1429601828e_indi > nat,G_1: arrow_1429601828e_indi > nat,A_2: arrow_1429601828e_indi > $o] :
( ! [A_3: arrow_1429601828e_indi] :
( ( member2052026769e_indi @ A_3 @ A_2 )
=> ( ( F_2 @ A_3 )
= ( G_1 @ A_3 ) ) )
=> ( ( inj_on978774663di_nat @ F_2 @ A_2 )
<=> ( inj_on978774663di_nat @ G_1 @ A_2 ) ) ) ).
thf(fact_595_finite__indi,axiom,
finite664979089e_indi @ top_to988227749indi_o ).
thf(fact_596_card__bij,axiom,
! [G: arrow_1429601828e_indi > produc1501160679le_alt > $o,F_1: ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi,A_1: ( produc1501160679le_alt > $o ) > $o,B: arrow_1429601828e_indi > $o] :
( ( member304866663e_indi @ F_1
@ ( pi_Pro468373057e_indi @ A_1
@ ^ [Uu: produc1501160679le_alt > $o] : B ) )
=> ( ( inj_on1877294875e_indi @ F_1 @ A_1 )
=> ( ( member526088951_alt_o @ G
@ ( pi_Arr1929480907_alt_o @ B
@ ^ [Uu: arrow_1429601828e_indi] : A_1 ) )
=> ( ( inj_on1190919077_alt_o @ G @ B )
=> ( ( finite2112685307_alt_o @ A_1 )
=> ( ( finite664979089e_indi @ B )
=> ( ( finite28306938_alt_o @ A_1 )
= ( finite97476818e_indi @ B ) ) ) ) ) ) ) ) ).
thf(fact_597_card__bij,axiom,
! [G: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,F_1: ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi > produc1501160679le_alt > $o,A_1: ( produc1501160679le_alt > $o ) > $o,B: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o] :
( ( member530241719_alt_o @ F_1
@ ( pi_Pro763888199_alt_o @ A_1
@ ^ [Uu: produc1501160679le_alt > $o] : B ) )
=> ( ( inj_on743426285_alt_o @ F_1 @ A_1 )
=> ( ( member616898751_alt_o @ G
@ ( pi_Arr1304755663_alt_o @ B
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : A_1 ) )
=> ( ( inj_on1284293749_alt_o @ G @ B )
=> ( ( finite2112685307_alt_o @ A_1 )
=> ( ( finite1956767223_alt_o @ B )
=> ( ( finite28306938_alt_o @ A_1 )
= ( finite120663670_alt_o @ B ) ) ) ) ) ) ) ) ).
thf(fact_598_card__bij,axiom,
! [G: $o > produc1501160679le_alt,F_1: produc1501160679le_alt > $o,A_1: produc1501160679le_alt > $o,B: $o > $o] :
( ( member377231867_alt_o @ F_1
@ ( pi_Pro1701359055_alt_o @ A_1
@ ^ [Uu: produc1501160679le_alt] : B ) )
=> ( ( inj_on1911943593_alt_o @ F_1 @ A_1 )
=> ( ( member492167345le_alt @ G
@ ( pi_o_P657324555le_alt @ B
@ ^ [Uu: $o] : A_1 ) )
=> ( ( inj_on867909093le_alt @ G @ B )
=> ( ( finite449174868le_alt @ A_1 )
=> ( ( finite_finite_o @ B )
=> ( ( finite537683861le_alt @ A_1 )
= ( finite_card_o @ B ) ) ) ) ) ) ) ) ).
thf(fact_599_card__bij,axiom,
! [G: ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi,F_1: arrow_1429601828e_indi > produc1501160679le_alt > $o,A_1: arrow_1429601828e_indi > $o,B: ( produc1501160679le_alt > $o ) > $o] :
( ( member526088951_alt_o @ F_1
@ ( pi_Arr1929480907_alt_o @ A_1
@ ^ [Uu: arrow_1429601828e_indi] : B ) )
=> ( ( inj_on1190919077_alt_o @ F_1 @ A_1 )
=> ( ( member304866663e_indi @ G
@ ( pi_Pro468373057e_indi @ B
@ ^ [Uu: produc1501160679le_alt > $o] : A_1 ) )
=> ( ( inj_on1877294875e_indi @ G @ B )
=> ( ( finite664979089e_indi @ A_1 )
=> ( ( finite2112685307_alt_o @ B )
=> ( ( finite97476818e_indi @ A_1 )
= ( finite28306938_alt_o @ B ) ) ) ) ) ) ) ) ).
thf(fact_600_card__bij,axiom,
! [G: ( produc1501160679le_alt > $o ) > arrow_1429601828e_indi > produc1501160679le_alt > $o,F_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > produc1501160679le_alt > $o,A_1: ( arrow_1429601828e_indi > produc1501160679le_alt > $o ) > $o,B: ( produc1501160679le_alt > $o ) > $o] :
( ( member616898751_alt_o @ F_1
@ ( pi_Arr1304755663_alt_o @ A_1
@ ^ [Uu: arrow_1429601828e_indi > produc1501160679le_alt > $o] : B ) )
=> ( ( inj_on1284293749_alt_o @ F_1 @ A_1 )
=> ( ( member530241719_alt_o @ G
@ ( pi_Pro763888199_alt_o @ B
@ ^ [Uu: produc1501160679le_alt > $o] : A_1 ) )
=> ( ( inj_on743426285_alt_o @ G @ B )
=> ( ( finite1956767223_alt_o @ A_1 )
=> ( ( finite2112685307_alt_o @ B )
=> ( ( finite120663670_alt_o @ A_1 )
= ( finite28306938_alt_o @ B ) ) ) ) ) ) ) ) ).
thf(fact_601_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K ) ) ) ).
thf(fact_602_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
thf(fact_603_card__UNIV__unit,axiom,
( ( finite1949902593t_unit @ top_to1984820022unit_o )
= one_one_nat ) ).
thf(fact_604_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_nat @ I_1 @ N ) ) )
= N ) ).
thf(fact_605_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_nat_o ) ).
thf(fact_606_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I ) ) ) ) ).
thf(fact_607_finite__nat__set__iff__bounded__le,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_eq_nat @ X_1 @ M ) ) ) ).
thf(fact_608_finite__nat__set__iff__bounded,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_nat @ X_1 @ M ) ) ) ).
thf(fact_609_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
thf(fact_610_bounded__nat__set__is__finite,axiom,
! [N: nat,N_2: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_nat @ X_1 @ N ) )
=> ( finite_finite_nat @ N_2 ) ) ).
thf(fact_611_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_612_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_613_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_614_gr__implies__not0,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_615_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_616_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_617_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_618_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_619_Nat_Oadd__0__right,axiom,
! [M_2: nat] :
( ( plus_plus_nat @ M_2 @ zero_zero_nat )
= M_2 ) ).
thf(fact_620_add__is__0,axiom,
! [M_2: nat,N: nat] :
( ( ( plus_plus_nat @ M_2 @ N )
= zero_zero_nat )
<=> ( ( M_2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_621_add__eq__self__zero,axiom,
! [M_2: nat,N: nat] :
( ( ( plus_plus_nat @ M_2 @ N )
= M_2 )
=> ( N = zero_zero_nat ) ) ).
thf(fact_622_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_623_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_624_add__gr__0,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M_2 @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M_2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_625_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N )
& ! [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K_1 )
=> ~ ( P @ I_1 ) )
& ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_626_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
& ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ K_1 )
=> ~ ( P @ I_1 ) )
& ( P @ K_1 ) ) ) ) ).
thf(fact_627_less__imp__add__positive,axiom,
! [I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I @ K_1 )
= J_1 ) ) ) ).
thf(fact_628_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_629_Suc__mono,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( ord_less_nat @ ( suc @ M_2 ) @ ( suc @ N ) ) ) ).
thf(fact_630_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_631_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_632_not__less__eq__eq,axiom,
! [M_2: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M_2 @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M_2 ) ) ).
thf(fact_633_le__Suc__eq,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ ( suc @ N ) )
<=> ( ( ord_less_eq_nat @ M_2 @ N )
| ( M_2
= ( suc @ N ) ) ) ) ).
thf(fact_634_Suc__le__mono,axiom,
! [N: nat,M_2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M_2 ) )
<=> ( ord_less_eq_nat @ N @ M_2 ) ) ).
thf(fact_635_le__SucI,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( ord_less_eq_nat @ M_2 @ ( suc @ N ) ) ) ).
thf(fact_636_le__SucE,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M_2 @ N )
=> ( M_2
= ( suc @ N ) ) ) ) ).
thf(fact_637_Suc__leD,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M_2 ) @ N )
=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_638_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_639_nat_Oinject,axiom,
! [Nat_3: nat,Nat_1: nat] :
( ( ( suc @ Nat_3 )
= ( suc @ Nat_1 ) )
<=> ( Nat_3 = Nat_1 ) ) ).
thf(fact_640_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_641_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_642_add__Suc__shift,axiom,
! [M_2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M_2 ) @ N )
= ( plus_plus_nat @ M_2 @ ( suc @ N ) ) ) ).
thf(fact_643_add__Suc,axiom,
! [M_2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M_2 ) @ N )
= ( suc @ ( plus_plus_nat @ M_2 @ N ) ) ) ).
thf(fact_644_add__Suc__right,axiom,
! [M_2: nat,N: nat] :
( ( plus_plus_nat @ M_2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M_2 @ N ) ) ) ).
thf(fact_645_not__less__eq,axiom,
! [M_2: nat,N: nat] :
( ~ ( ord_less_nat @ M_2 @ N )
<=> ( ord_less_nat @ N @ ( suc @ M_2 ) ) ) ).
thf(fact_646_less__Suc__eq,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M_2 @ N )
| ( M_2 = N ) ) ) ).
thf(fact_647_Suc__less__eq,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M_2 ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M_2 @ N ) ) ).
thf(fact_648_not__less__less__Suc__eq,axiom,
! [N: nat,M_2: nat] :
( ~ ( ord_less_nat @ N @ M_2 )
=> ( ( ord_less_nat @ N @ ( suc @ M_2 ) )
<=> ( N = M_2 ) ) ) ).
thf(fact_649_less__antisym,axiom,
! [N: nat,M_2: nat] :
( ~ ( ord_less_nat @ N @ M_2 )
=> ( ( ord_less_nat @ N @ ( suc @ M_2 ) )
=> ( M_2 = N ) ) ) ).
thf(fact_650_less__SucI,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( ord_less_nat @ M_2 @ ( suc @ N ) ) ) ).
thf(fact_651_Suc__lessI,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( ( ( suc @ M_2 )
!= N )
=> ( ord_less_nat @ ( suc @ M_2 ) @ N ) ) ) ).
thf(fact_652_less__trans__Suc,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ( ord_less_nat @ J_1 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
thf(fact_653_less__SucE,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M_2 @ N )
=> ( M_2 = N ) ) ) ).
thf(fact_654_Suc__lessD,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M_2 ) @ N )
=> ( ord_less_nat @ M_2 @ N ) ) ).
thf(fact_655_Suc__less__SucD,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M_2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M_2 @ N ) ) ).
thf(fact_656_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_657_Suc__le__lessD,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M_2 ) @ N )
=> ( ord_less_nat @ M_2 @ N ) ) ).
thf(fact_658_le__less__Suc__eq,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M_2 ) )
<=> ( N = M_2 ) ) ) ).
thf(fact_659_Suc__leI,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M_2 ) @ N ) ) ).
thf(fact_660_le__imp__less__Suc,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( ord_less_nat @ M_2 @ ( suc @ N ) ) ) ).
thf(fact_661_Suc__le__eq,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M_2 ) @ N )
<=> ( ord_less_nat @ M_2 @ N ) ) ).
thf(fact_662_less__Suc__eq__le,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M_2 @ N ) ) ).
thf(fact_663_less__eq__Suc__le,axiom,
! [N: nat,M_2: nat] :
( ( ord_less_nat @ N @ M_2 )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M_2 ) ) ).
thf(fact_664_less__iff__Suc__add,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
<=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M_2 @ K_1 ) ) ) ) ).
thf(fact_665_less__add__Suc2,axiom,
! [I: nat,M_2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M_2 @ I ) ) ) ).
thf(fact_666_less__add__Suc1,axiom,
! [I: nat,M_2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M_2 ) ) ) ).
thf(fact_667_one__is__add,axiom,
! [M_2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M_2 @ N ) )
<=> ( ( ( M_2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M_2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_668_add__is__1,axiom,
! [M_2: nat,N: nat] :
( ( ( plus_plus_nat @ M_2 @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M_2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M_2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_669_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
thf(fact_670_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_671_less__Suc__eq__0__disj,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ ( suc @ N ) )
<=> ( ( M_2 = zero_zero_nat )
| ? [J: nat] :
( ( M_2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
thf(fact_672_inj__Suc,axiom,
! [N_2: nat > $o] : ( inj_on_nat_nat @ suc @ N_2 ) ).
thf(fact_673_Suc__neq__Zero,axiom,
! [M_2: nat] :
( ( suc @ M_2 )
!= zero_zero_nat ) ).
thf(fact_674_Zero__neq__Suc,axiom,
! [M_2: nat] :
( zero_zero_nat
!= ( suc @ M_2 ) ) ).
thf(fact_675_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] :
( ( suc @ Nat_2 )
!= zero_zero_nat ) ).
thf(fact_676_Suc__not__Zero,axiom,
! [M_2: nat] :
( ( suc @ M_2 )
!= zero_zero_nat ) ).
thf(fact_677_nat_Osimps_I2_J,axiom,
! [Nat_1: nat] :
( zero_zero_nat
!= ( suc @ Nat_1 ) ) ).
thf(fact_678_Zero__not__Suc,axiom,
! [M_2: nat] :
( zero_zero_nat
!= ( suc @ M_2 ) ) ).
thf(fact_679_less__eq__Suc__le__raw,axiom,
! [X_1: nat] :
( ( ord_less_nat @ X_1 )
= ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
thf(fact_680_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_eq_nat @ I_1 @ N ) ) )
= ( suc @ N ) ) ).
thf(fact_681_card__less__Suc,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_3 ) @ ( ord_less_nat @ K_1 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_3 ) @ ( ord_less_nat @ K_1 @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_682_card__less,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_3 ) @ ( ord_less_nat @ K_1 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
thf(fact_683_card__less__Suc2,axiom,
! [I: nat,M_3: nat > $o] :
( ~ ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_3 ) @ ( ord_less_nat @ K_1 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_3 ) @ ( ord_less_nat @ K_1 @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_684_inc__induct,axiom,
! [P: nat > $o,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( P @ J_1 )
=> ( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( P @ ( suc @ I_1 ) )
=> ( P @ I_1 ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_685_less__imp__Suc__add,axiom,
! [M_2: nat,N: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M_2 @ K_1 ) ) ) ) ).
thf(fact_686_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
thf(fact_687_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_688_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_689_Suc__le__D,axiom,
! [N: nat,M_1: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M_1 )
=> ? [M: nat] :
( M_1
= ( suc @ M ) ) ) ).
thf(fact_690_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
thf(fact_691_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
thf(fact_692_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
thf(fact_693_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
thf(fact_694_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_695_nat__induct,axiom,
! [N: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N_1: nat] :
( ( P @ N_1 )
=> ( P @ ( suc @ N_1 ) ) )
=> ( P @ N ) ) ) ).
thf(fact_696_surjh,axiom,
( ( image_484224243di_nat @ h @ top_to988227749indi_o )
= ( ord_at4362885an_nat @ zero_zero_nat @ ( finite97476818e_indi @ top_to988227749indi_o ) ) ) ).
thf(fact_697_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at4362885an_nat @ L @ U ) ) ).
thf(fact_698__096_B_Bthesis_O_A_I_B_Bh_O_A_091_124_Ainj_Ah_059_Arange_Ah_A_061_A_123,axiom,
~ ! [H: arrow_1429601828e_indi > nat] :
( ( inj_on978774663di_nat @ H @ top_to988227749indi_o )
=> ( ( image_484224243di_nat @ H @ top_to988227749indi_o )
!= ( ord_at4362885an_nat @ zero_zero_nat @ ( finite97476818e_indi @ top_to988227749indi_o ) ) ) ) ).
thf(fact_699_subset__card__intvl__is__intvl,axiom,
! [A: nat > $o,K: nat] :
( ( ord_less_eq_nat_o @ A @ ( ord_at4362885an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) )
=> ( A
= ( ord_at4362885an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) ) ) ).
%----Helper facts (3)
thf(help_If_1_1_If_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv_,axiom,
! [X: produc1501160679le_alt > $o,Y: produc1501160679le_alt > $o] :
( ( if_Pro1561232536_alt_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv_,axiom,
! [X: produc1501160679le_alt > $o,Y: produc1501160679le_alt > $o] :
( ( if_Pro1561232536_alt_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000_062_Itc__prod_Itc__Arrow____Order____Mirabelle____lcilvlkkzv_,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member214075476le_alt @ ( produc1347929815le_alt @ c @ d ) @ ( f @ p ) ).
%------------------------------------------------------------------------------