TPTP Problem File: ITP113^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP113^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_663__6254006_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Lower_Semicontinuous/prob_663__6254006_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 444 ( 140 unt; 98 typ; 0 def)
% Number of atoms : 1001 ( 251 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2614 ( 129 ~; 25 |; 41 &;1931 @)
% ( 0 <=>; 488 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 180 ( 180 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 73 usr; 20 con; 0-2 aty)
% Number of variables : 963 ( 65 ^; 874 !; 24 ?; 963 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:18.365
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
produc792423270l_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
produc1931999653t_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1350454001_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1691597095_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
set_Pr1928503567a_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
produc957004601l_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
produc581916509al_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
produc845587165t_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
product_prod_a_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (73)
thf(sy_c_Elementary__Topology_Oclosure_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
elemen1695521870a_real: set_Pr1928503567a_real > set_Pr1928503567a_real ).
thf(sy_c_Elementary__Topology_Oclosure_001t__Real__Oreal,type,
elemen1168706977e_real: set_real > set_real ).
thf(sy_c_Elementary__Topology_Oclosure_001tf__a,type,
elementary_closure_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
lower_870348689a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > set_Pr1158285382l_real ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Real__Oreal,type,
lower_1607476196h_real: set_real > ( real > extended_ereal ) > set_Pr147102617l_real ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001tf__a,type,
lower_930854854raph_a: set_a > ( a > extended_ereal ) > set_Pr1928503567a_real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le297355219_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
ord_le1379034467a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
ord_le1950669437t_real: produc845587165t_real > produc845587165t_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
ord_le1686998781al_nat: produc581916509al_nat > produc581916509al_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
ord_le1586073967a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
top_to1567080838real_o: product_prod_a_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
top_to991334818t_real: produc1312517202t_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_Itf__a_J_J,type,
top_to345582004_set_a: produc25772804_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
top_to522338794_set_a: produc2021989178_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
top_to2037809826a_real: produc211508562a_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
top_to95743733t_real: produc1931999653t_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_Itf__a_J_J,type,
top_to1153982881_set_a: produc1350454001_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
top_to724687040_set_a: produc304417648_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
top_to829322572a_real: produc509513372a_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
top_to35724191t_real: produc232195311t_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
top_to1477940855_set_a: produc1691597095_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
top_to2138011583a_real: set_Pr1928503567a_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
produc49576085t_real: nat > real > produc845587165t_real ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_001t__Real__Oreal,type,
produc1214388894l_real: product_prod_a_real > real > produc792423270l_real ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
produc1541812501al_nat: real > nat > produc581916509al_nat ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
produc705216881l_real: real > real > produc957004601l_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc471223946t_real: set_Pr1928503567a_real > set_real > produc1312517202t_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_001t__Set__Oset_Itf__a_J,type,
produc1220567734_set_a: set_Pr1928503567a_real > set_a > produc25772804_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1162518380_set_a: set_real > produc1691597095_set_a > produc2021989178_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
produc150798858a_real: set_real > set_Pr1928503567a_real > produc211508562a_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc119681757t_real: set_real > set_real > produc1931999653t_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_Itf__a_J,type,
produc190993315_set_a: set_real > set_a > produc1350454001_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc272677600_set_a: set_a > produc1691597095_set_a > produc304417648_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
produc383769494a_real: set_a > set_Pr1928503567a_real > produc509513372a_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Real__Oreal_J,type,
produc1266579561t_real: set_a > set_real > produc232195311t_real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Real__Oreal,type,
product_Pair_a_real: a > real > product_prod_a_real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
real_V404783528a_real: product_prod_a_real > product_prod_a_real > real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
real_V1934908667t_real: real > real > real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001tf__a,type,
real_V1514887919dist_a: a > a > real ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
collec1714955950a_real: ( product_prod_a_real > $o ) > set_Pr1928503567a_real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
member565909519l_real: produc792423270l_real > set_Pr1158285382l_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_e____,type,
e: real ).
thf(sy_v_f,type,
f: a > extended_ereal ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_y____,type,
y: real ).
thf(sy_v_z____,type,
z: real ).
% Relevant facts (344)
thf(fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062,axiom,
? [X: product_prod_a_real] :
( ( member1103263856a_real @ X @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
& ( ord_less_real @ ( real_V404783528a_real @ X @ ( product_Pair_a_real @ x @ y ) ) @ e ) ) ).
% \<open>\<exists>ya\<in>Epigraph UNIV f. dist ya (x, y) < e\<close>
thf(fact_1__092_060open_0620_A_060_Ae_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ e ).
% \<open>0 < e\<close>
thf(fact_2_xy,axiom,
( ( member1103263856a_real @ ( product_Pair_a_real @ x @ y ) @ ( elemen1695521870a_real @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) )
& ( ord_less_eq_real @ y @ z ) ) ).
% xy
thf(fact_3_UNIV__I,axiom,
! [X2: product_prod_a_real] : ( member1103263856a_real @ X2 @ top_to2138011583a_real ) ).
% UNIV_I
thf(fact_4_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_5_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_6_iso__tuple__UNIV__I,axiom,
! [X2: product_prod_a_real] : ( member1103263856a_real @ X2 @ top_to2138011583a_real ) ).
% iso_tuple_UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_9_prod_Oinject,axiom,
! [X1: set_a,X22: set_a,Y1: set_a,Y2: set_a] :
( ( ( produc1928581911_set_a @ X1 @ X22 )
= ( produc1928581911_set_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: a,X22: real,Y1: a,Y2: real] :
( ( ( product_Pair_a_real @ X1 @ X22 )
= ( product_Pair_a_real @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
( ( ( produc1928581911_set_a @ A @ B )
= ( produc1928581911_set_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: a,B: real,A2: a,B2: real] :
( ( ( product_Pair_a_real @ A @ B )
= ( product_Pair_a_real @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_dist__commute__lessI,axiom,
! [Y: product_prod_a_real,X2: product_prod_a_real,E: real] :
( ( ord_less_real @ ( real_V404783528a_real @ Y @ X2 ) @ E )
=> ( ord_less_real @ ( real_V404783528a_real @ X2 @ Y ) @ E ) ) ).
% dist_commute_lessI
thf(fact_14_top__prod__def,axiom,
( top_to1477940855_set_a
= ( produc1928581911_set_a @ top_top_set_a @ top_top_set_a ) ) ).
% top_prod_def
thf(fact_15_top__prod__def,axiom,
( top_to35724191t_real
= ( produc1266579561t_real @ top_top_set_a @ top_top_set_real ) ) ).
% top_prod_def
thf(fact_16_top__prod__def,axiom,
( top_to1153982881_set_a
= ( produc190993315_set_a @ top_top_set_real @ top_top_set_a ) ) ).
% top_prod_def
thf(fact_17_top__prod__def,axiom,
( top_to95743733t_real
= ( produc119681757t_real @ top_top_set_real @ top_top_set_real ) ) ).
% top_prod_def
thf(fact_18_top__prod__def,axiom,
( top_to829322572a_real
= ( produc383769494a_real @ top_top_set_a @ top_to2138011583a_real ) ) ).
% top_prod_def
thf(fact_19_top__prod__def,axiom,
( top_to2037809826a_real
= ( produc150798858a_real @ top_top_set_real @ top_to2138011583a_real ) ) ).
% top_prod_def
thf(fact_20_top__prod__def,axiom,
( top_to345582004_set_a
= ( produc1220567734_set_a @ top_to2138011583a_real @ top_top_set_a ) ) ).
% top_prod_def
thf(fact_21_top__prod__def,axiom,
( top_to991334818t_real
= ( produc471223946t_real @ top_to2138011583a_real @ top_top_set_real ) ) ).
% top_prod_def
thf(fact_22_top__prod__def,axiom,
( top_to724687040_set_a
= ( produc272677600_set_a @ top_top_set_a @ top_to1477940855_set_a ) ) ).
% top_prod_def
thf(fact_23_top__prod__def,axiom,
( top_to522338794_set_a
= ( produc1162518380_set_a @ top_top_set_real @ top_to1477940855_set_a ) ) ).
% top_prod_def
thf(fact_24_top_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% top.extremum_strict
thf(fact_25_top_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A ) ).
% top.extremum_strict
thf(fact_26_top_Oextremum__strict,axiom,
! [A: set_Pr1928503567a_real] :
~ ( ord_le1379034467a_real @ top_to2138011583a_real @ A ) ).
% top.extremum_strict
thf(fact_27_top_Oextremum__strict,axiom,
! [A: produc1691597095_set_a] :
~ ( ord_le297355219_set_a @ top_to1477940855_set_a @ A ) ).
% top.extremum_strict
thf(fact_28_top_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != top_top_set_a )
= ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_29_top_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != top_top_set_real )
= ( ord_less_set_real @ A @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_30_top_Onot__eq__extremum,axiom,
! [A: set_Pr1928503567a_real] :
( ( A != top_to2138011583a_real )
= ( ord_le1379034467a_real @ A @ top_to2138011583a_real ) ) ).
% top.not_eq_extremum
thf(fact_31_top_Onot__eq__extremum,axiom,
! [A: produc1691597095_set_a] :
( ( A != top_to1477940855_set_a )
= ( ord_le297355219_set_a @ A @ top_to1477940855_set_a ) ) ).
% top.not_eq_extremum
thf(fact_32_epigraph__mono,axiom,
! [X2: real,Y: real,F: real > extended_ereal,Z: real] :
( ( ( member1068169442l_real @ ( produc705216881l_real @ X2 @ Y ) @ ( lower_1607476196h_real @ top_top_set_real @ F ) )
& ( ord_less_eq_real @ Y @ Z ) )
=> ( member1068169442l_real @ ( produc705216881l_real @ X2 @ Z ) @ ( lower_1607476196h_real @ top_top_set_real @ F ) ) ) ).
% epigraph_mono
thf(fact_33_epigraph__mono,axiom,
! [X2: product_prod_a_real,Y: real,F: product_prod_a_real > extended_ereal,Z: real] :
( ( ( member565909519l_real @ ( produc1214388894l_real @ X2 @ Y ) @ ( lower_870348689a_real @ top_to2138011583a_real @ F ) )
& ( ord_less_eq_real @ Y @ Z ) )
=> ( member565909519l_real @ ( produc1214388894l_real @ X2 @ Z ) @ ( lower_870348689a_real @ top_to2138011583a_real @ F ) ) ) ).
% epigraph_mono
thf(fact_34_epigraph__mono,axiom,
! [X2: a,Y: real,F: a > extended_ereal,Z: real] :
( ( ( member1103263856a_real @ ( product_Pair_a_real @ X2 @ Y ) @ ( lower_930854854raph_a @ top_top_set_a @ F ) )
& ( ord_less_eq_real @ Y @ Z ) )
=> ( member1103263856a_real @ ( product_Pair_a_real @ X2 @ Z ) @ ( lower_930854854raph_a @ top_top_set_a @ F ) ) ) ).
% epigraph_mono
thf(fact_35_metric__eq__thm,axiom,
! [X2: real,S: set_real,Y: real] :
( ( member_real @ X2 @ S )
=> ( ( member_real @ Y @ S )
=> ( ( X2 = Y )
= ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ( real_V1934908667t_real @ X2 @ X3 )
= ( real_V1934908667t_real @ Y @ X3 ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_36_metric__eq__thm,axiom,
! [X2: a,S: set_a,Y: a] :
( ( member_a @ X2 @ S )
=> ( ( member_a @ Y @ S )
=> ( ( X2 = Y )
= ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ( ( real_V1514887919dist_a @ X2 @ X3 )
= ( real_V1514887919dist_a @ Y @ X3 ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_37_metric__eq__thm,axiom,
! [X2: product_prod_a_real,S: set_Pr1928503567a_real,Y: product_prod_a_real] :
( ( member1103263856a_real @ X2 @ S )
=> ( ( member1103263856a_real @ Y @ S )
=> ( ( X2 = Y )
= ( ! [X3: product_prod_a_real] :
( ( member1103263856a_real @ X3 @ S )
=> ( ( real_V404783528a_real @ X2 @ X3 )
= ( real_V404783528a_real @ Y @ X3 ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_38_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_39_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_40_Pair__le,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_41_Pair__le,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
= ( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_real @ B @ D ) ) ) ).
% Pair_le
thf(fact_42_Pair__le,axiom,
! [A: real,B: nat,C: real,D: nat] :
( ( ord_le1686998781al_nat @ ( produc1541812501al_nat @ A @ B ) @ ( produc1541812501al_nat @ C @ D ) )
= ( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_43_Pair__le,axiom,
! [A: nat,B: real,C: nat,D: real] :
( ( ord_le1950669437t_real @ ( produc49576085t_real @ A @ B ) @ ( produc49576085t_real @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_real @ B @ D ) ) ) ).
% Pair_le
thf(fact_44_Pair__le,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_45_dist__eq__0__iff,axiom,
! [X2: product_prod_a_real,Y: product_prod_a_real] :
( ( ( real_V404783528a_real @ X2 @ Y )
= zero_zero_real )
= ( X2 = Y ) ) ).
% dist_eq_0_iff
thf(fact_46_dist__self,axiom,
! [X2: product_prod_a_real] :
( ( real_V404783528a_real @ X2 @ X2 )
= zero_zero_real ) ).
% dist_self
thf(fact_47_dist__le__zero__iff,axiom,
! [X2: product_prod_a_real,Y: product_prod_a_real] :
( ( ord_less_eq_real @ ( real_V404783528a_real @ X2 @ Y ) @ zero_zero_real )
= ( X2 = Y ) ) ).
% dist_le_zero_iff
thf(fact_48_zero__less__dist__iff,axiom,
! [X2: product_prod_a_real,Y: product_prod_a_real] :
( ( ord_less_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) )
= ( X2 != Y ) ) ).
% zero_less_dist_iff
thf(fact_49_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_50_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_51_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_52_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_53_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_54_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_55_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_56_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_57_Pair__mono,axiom,
! [X2: set_a,X4: set_a,Y: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X2 @ X4 )
=> ( ( ord_less_eq_set_a @ Y @ Y4 )
=> ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ ( produc1928581911_set_a @ X4 @ Y4 ) ) ) ) ).
% Pair_mono
thf(fact_58_Pair__mono,axiom,
! [X2: real,X4: real,Y: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ X4 )
=> ( ( ord_less_eq_real @ Y @ Y4 )
=> ( ord_le1342644953l_real @ ( produc705216881l_real @ X2 @ Y ) @ ( produc705216881l_real @ X4 @ Y4 ) ) ) ) ).
% Pair_mono
thf(fact_59_Pair__mono,axiom,
! [X2: real,X4: real,Y: nat,Y4: nat] :
( ( ord_less_eq_real @ X2 @ X4 )
=> ( ( ord_less_eq_nat @ Y @ Y4 )
=> ( ord_le1686998781al_nat @ ( produc1541812501al_nat @ X2 @ Y ) @ ( produc1541812501al_nat @ X4 @ Y4 ) ) ) ) ).
% Pair_mono
thf(fact_60_Pair__mono,axiom,
! [X2: nat,X4: nat,Y: real,Y4: real] :
( ( ord_less_eq_nat @ X2 @ X4 )
=> ( ( ord_less_eq_real @ Y @ Y4 )
=> ( ord_le1950669437t_real @ ( produc49576085t_real @ X2 @ Y ) @ ( produc49576085t_real @ X4 @ Y4 ) ) ) ) ).
% Pair_mono
thf(fact_61_Pair__mono,axiom,
! [X2: nat,X4: nat,Y: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ X4 )
=> ( ( ord_less_eq_nat @ Y @ Y4 )
=> ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ) ).
% Pair_mono
thf(fact_62_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_63_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_64_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_65_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_66_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : Y5 = Z2 )
= ( ^ [X3: real,Y6: real] :
( ( ord_less_eq_real @ X3 @ Y6 )
& ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_71_eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_72_antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_73_antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_74_linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linear
thf(fact_75_linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linear
thf(fact_76_eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% eq_refl
thf(fact_77_eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% eq_refl
thf(fact_78_le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% le_cases
thf(fact_79_le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% le_cases
thf(fact_80_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_81_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_82_le__cases3,axiom,
! [X2: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_83_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_84_antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_85_antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_86_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : Y5 = Z2 )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_87_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_88_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_89_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_90_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_91_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_92_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_93_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_94_order__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_95_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_96_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_97_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_98_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_99_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_100_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_101_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_102_mem__Collect__eq,axiom,
! [A: product_prod_a_real,P: product_prod_a_real > $o] :
( ( member1103263856a_real @ A @ ( collec1714955950a_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A5: set_Pr1928503567a_real] :
( ( collec1714955950a_real
@ ^ [X3: product_prod_a_real] : ( member1103263856a_real @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A5: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_108_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : Y5 = Z2 )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_109_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_110_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_111_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_112_zero__le__dist,axiom,
! [X2: product_prod_a_real,Y: product_prod_a_real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) ) ).
% zero_le_dist
thf(fact_113_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_114_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_115_top__set__def,axiom,
( top_to2138011583a_real
= ( collec1714955950a_real @ top_to1567080838real_o ) ) ).
% top_set_def
thf(fact_116_order_Onot__eq__order__implies__strict,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_117_order_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_118_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_119_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_120_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_121_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_122_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_123_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_124_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_125_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_126_dense__le__bounded,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_127_dense__ge__bounded,axiom,
! [Z: real,X2: real,Y: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_128_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_129_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_130_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_131_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_132_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_133_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_134_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_135_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_136_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_137_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_138_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_139_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_140_not__le__imp__less,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_real @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_141_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_142_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_eq_real @ X3 @ Y6 )
& ~ ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_143_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_144_le__imp__less__or__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_145_le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_146_le__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% le_less_linear
thf(fact_147_le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% le_less_linear
thf(fact_148_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_149_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_150_less__le__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% less_le_trans
thf(fact_151_less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% less_le_trans
thf(fact_152_le__less__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% le_less_trans
thf(fact_153_le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% le_less_trans
thf(fact_154_less__imp__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% less_imp_le
thf(fact_155_less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% less_imp_le
thf(fact_156_antisym__conv2,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_157_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_158_antisym__conv1,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_159_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_160_le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_161_le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_162_not__less,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ord_less_eq_real @ Y @ X2 ) ) ).
% not_less
thf(fact_163_not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% not_less
thf(fact_164_not__le,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
= ( ord_less_real @ Y @ X2 ) ) ).
% not_le
thf(fact_165_not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% not_le
thf(fact_166_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_167_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_168_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_169_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_170_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_171_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_172_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_173_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_174_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_175_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_176_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_177_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_178_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_179_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_180_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_181_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_182_less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_eq_real @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% less_le
thf(fact_183_less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% less_le
thf(fact_184_le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_real @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% le_less
thf(fact_185_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% le_less
thf(fact_186_leI,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% leI
thf(fact_187_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_188_leD,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_real @ X2 @ Y ) ) ).
% leD
thf(fact_189_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_190_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_191_top_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
=> ( A = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_192_top_Oextremum__uniqueI,axiom,
! [A: set_Pr1928503567a_real] :
( ( ord_le1586073967a_real @ top_to2138011583a_real @ A )
=> ( A = top_to2138011583a_real ) ) ).
% top.extremum_uniqueI
thf(fact_193_top_Oextremum__uniqueI,axiom,
! [A: produc1691597095_set_a] :
( ( ord_le486764743_set_a @ top_to1477940855_set_a @ A )
=> ( A = top_to1477940855_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_194_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_195_top_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
= ( A = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_196_top_Oextremum__unique,axiom,
! [A: set_Pr1928503567a_real] :
( ( ord_le1586073967a_real @ top_to2138011583a_real @ A )
= ( A = top_to2138011583a_real ) ) ).
% top.extremum_unique
thf(fact_197_top_Oextremum__unique,axiom,
! [A: produc1691597095_set_a] :
( ( ord_le486764743_set_a @ top_to1477940855_set_a @ A )
= ( A = top_to1477940855_set_a ) ) ).
% top.extremum_unique
thf(fact_198_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_199_top__greatest,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% top_greatest
thf(fact_200_top__greatest,axiom,
! [A: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ A @ top_to2138011583a_real ) ).
% top_greatest
thf(fact_201_top__greatest,axiom,
! [A: produc1691597095_set_a] : ( ord_le486764743_set_a @ A @ top_to1477940855_set_a ) ).
% top_greatest
thf(fact_202_dist__not__less__zero,axiom,
! [X2: product_prod_a_real,Y: product_prod_a_real] :
~ ( ord_less_real @ ( real_V404783528a_real @ X2 @ Y ) @ zero_zero_real ) ).
% dist_not_less_zero
thf(fact_203_dist__pos__lt,axiom,
! [X2: product_prod_a_real,Y: product_prod_a_real] :
( ( X2 != Y )
=> ( ord_less_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) ) ) ).
% dist_pos_lt
thf(fact_204_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_205_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_206_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_207_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_208_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ( ord_less_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_209_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_210_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_211_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_212_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_213_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_214_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_215_less__imp__not__less,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_216_less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_217_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_218_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_219_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_220_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_221_linorder__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_222_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_223_less__imp__triv,axiom,
! [X2: real,Y: real,P: $o] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_224_less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_225_less__imp__not__eq2,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_226_less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_227_antisym__conv3,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_real @ Y @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_228_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_229_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X )
=> ( P @ Y7 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_230_less__not__sym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_231_less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_232_less__imp__not__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_233_less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_234_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_235_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_236_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_237_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_238_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_239_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_240_less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% less_irrefl
thf(fact_241_less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% less_irrefl
thf(fact_242_less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% less_linear
thf(fact_243_less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% less_linear
thf(fact_244_less__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% less_trans
thf(fact_245_less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% less_trans
thf(fact_246_less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% less_asym'
thf(fact_247_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_248_less__asym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% less_asym
thf(fact_249_less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% less_asym
thf(fact_250_less__imp__neq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_251_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_252_dense,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X2 @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_253_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_254_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_255_neq__iff,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
= ( ( ord_less_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ) ).
% neq_iff
thf(fact_256_neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% neq_iff
thf(fact_257_neqE,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% neqE
thf(fact_258_neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% neqE
thf(fact_259_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_260_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_261_lt__ex,axiom,
! [X2: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% lt_ex
thf(fact_262_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_263_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_264_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_265_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_266_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_267_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_268_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_269_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_270_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_271_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_272_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_273_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_274_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_275_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_276_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_277_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_278_old_Oprod_Oinducts,axiom,
! [P: product_prod_a_real > $o,Prod: product_prod_a_real] :
( ! [A4: a,B4: real] : ( P @ ( product_Pair_a_real @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_279_old_Oprod_Oinducts,axiom,
! [P: produc1691597095_set_a > $o,Prod: produc1691597095_set_a] :
( ! [A4: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_280_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_real] :
~ ! [A4: a,B4: real] :
( Y
!= ( product_Pair_a_real @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_281_old_Oprod_Oexhaust,axiom,
! [Y: produc1691597095_set_a] :
~ ! [A4: set_a,B4: set_a] :
( Y
!= ( produc1928581911_set_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_282_Pair__inject,axiom,
! [A: a,B: real,A2: a,B2: real] :
( ( ( product_Pair_a_real @ A @ B )
= ( product_Pair_a_real @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_283_Pair__inject,axiom,
! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
( ( ( produc1928581911_set_a @ A @ B )
= ( produc1928581911_set_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_284_prod__cases,axiom,
! [P: product_prod_a_real > $o,P4: product_prod_a_real] :
( ! [A4: a,B4: real] : ( P @ ( product_Pair_a_real @ A4 @ B4 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_285_prod__cases,axiom,
! [P: produc1691597095_set_a > $o,P4: produc1691597095_set_a] :
( ! [A4: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A4 @ B4 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_286_surj__pair,axiom,
! [P4: product_prod_a_real] :
? [X: a,Y3: real] :
( P4
= ( product_Pair_a_real @ X @ Y3 ) ) ).
% surj_pair
thf(fact_287_surj__pair,axiom,
! [P4: produc1691597095_set_a] :
? [X: set_a,Y3: set_a] :
( P4
= ( produc1928581911_set_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_288_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_289_UNIV__witness,axiom,
? [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_witness
thf(fact_290_UNIV__witness,axiom,
? [X: product_prod_a_real] : ( member1103263856a_real @ X @ top_to2138011583a_real ) ).
% UNIV_witness
thf(fact_291_UNIV__eq__I,axiom,
! [A5: set_a] :
( ! [X: a] : ( member_a @ X @ A5 )
=> ( top_top_set_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_292_UNIV__eq__I,axiom,
! [A5: set_real] :
( ! [X: real] : ( member_real @ X @ A5 )
=> ( top_top_set_real = A5 ) ) ).
% UNIV_eq_I
thf(fact_293_UNIV__eq__I,axiom,
! [A5: set_Pr1928503567a_real] :
( ! [X: product_prod_a_real] : ( member1103263856a_real @ X @ A5 )
=> ( top_to2138011583a_real = A5 ) ) ).
% UNIV_eq_I
thf(fact_294_dist__commute,axiom,
( real_V404783528a_real
= ( ^ [X3: product_prod_a_real,Y6: product_prod_a_real] : ( real_V404783528a_real @ Y6 @ X3 ) ) ) ).
% dist_commute
thf(fact_295_closure__UNIV,axiom,
( ( elementary_closure_a @ top_top_set_a )
= top_top_set_a ) ).
% closure_UNIV
thf(fact_296_closure__UNIV,axiom,
( ( elemen1168706977e_real @ top_top_set_real )
= top_top_set_real ) ).
% closure_UNIV
thf(fact_297_closure__UNIV,axiom,
( ( elemen1695521870a_real @ top_to2138011583a_real )
= top_to2138011583a_real ) ).
% closure_UNIV
thf(fact_298_closure__approachable__le,axiom,
! [X2: real,S2: set_real] :
( ( member_real @ X2 @ ( elemen1168706977e_real @ S2 ) )
= ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [X3: real] :
( ( member_real @ X3 @ S2 )
& ( ord_less_eq_real @ ( real_V1934908667t_real @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% closure_approachable_le
thf(fact_299_closure__approachable__le,axiom,
! [X2: a,S2: set_a] :
( ( member_a @ X2 @ ( elementary_closure_a @ S2 ) )
= ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [X3: a] :
( ( member_a @ X3 @ S2 )
& ( ord_less_eq_real @ ( real_V1514887919dist_a @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% closure_approachable_le
thf(fact_300_closure__approachable__le,axiom,
! [X2: product_prod_a_real,S2: set_Pr1928503567a_real] :
( ( member1103263856a_real @ X2 @ ( elemen1695521870a_real @ S2 ) )
= ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [X3: product_prod_a_real] :
( ( member1103263856a_real @ X3 @ S2 )
& ( ord_less_eq_real @ ( real_V404783528a_real @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% closure_approachable_le
thf(fact_301_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_302_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_303_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D @ N4 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_304_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_305_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_306_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_307_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_308_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_309_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_310_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_311_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_312_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_313_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_314_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_315_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_316_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_317_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_318_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_319_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_320_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_321_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_322_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_323_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_324_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_325_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_326_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_327_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_328_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_329_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_330_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_331_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_332_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_333_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_334_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_335_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_336_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_337_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_338_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_339_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_340_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_341_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_342_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y6: real] :
( ( ord_less_real @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_343_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Z4 ) )
=> ? [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y3 ) )
& ! [Z4: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
% Conjectures (2)
thf(conj_0,hypothesis,
! [A6: a,B5: real] :
( ( ( member1103263856a_real @ ( product_Pair_a_real @ A6 @ B5 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
& ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ A6 @ B5 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------