%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP113^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:39 AM UTC 2026
% Result : Theorem 2.70s 1.62s
% Output : Refutation 2.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP113^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n015.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:46 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP113^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer Lower_Semicontinuous problem prob_663__6254006_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : Lower_Semicontinuous/prob_663__6254006_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.00 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 368 ( 96 unt; 48 typ; 0 def)
% 0.19/0.46 % Number of atoms : 951 ( 174 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.19/0.46 % Number of connectives : 3688 ( 109 ~; 17 |; 82 &;2973 @)
% 0.19/0.46 % ( 0 <=>; 507 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 28 ( 8 avg)
% 0.19/0.46 % Number of types : 5 ( 4 usr)
% 0.19/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 45 ( 44 usr; 5 con; 0-5 aty)
% 0.19/0.46 % Number of variables : 1118 ( 61 ^; 978 !; 39 ?;1118 :)
% 0.19/0.46 % ( 40 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.46 % SPC : TH1_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:16.540
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (6)
% 0.19/0.46 thf(ty_t_Extended__Real_Oereal,type,
% 0.19/0.46 extended_ereal: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.19/0.46 product_prod: $tType > $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Real_Oreal,type,
% 0.19/0.46 real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Set_Oset,type,
% 0.19/0.46 set: $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Nat_Onat,type,
% 0.19/0.46 nat: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_tf_a,type,
% 0.19/0.46 a: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (42)
% 0.19/0.46 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 0.19/0.46 euclid925273238_space:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_HOL_Otype,type,
% 0.19/0.46 type:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.19/0.46 zero:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.46 ord:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.19/0.46 top:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.46 order:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.19/0.46 no_bot:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.19/0.46 no_top:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.46 linorder:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.46 preorder:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.19/0.46 order_top:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.19/0.46 wellorder:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.19/0.46 dense_order:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.19/0.46 linordered_field:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.19/0.46 dense_linorder:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 0.19/0.46 real_V2090557954_space:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 0.19/0.46 topolo890362671_space:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
% 0.19/0.46 real_V1759440057_space:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.19/0.46 linord1659791738miring:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 0.19/0.46 topolo503727757_space:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.46 canoni770627133id_add:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 0.19/0.46 real_V55928688vector:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.19/0.46 condit1656338222tinuum:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.19/0.46 condit1037483654norder:
% 0.19/0.46 !>[A: $tType] : $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Elementary__Topology_Oclosure,type,
% 0.19/0.46 elementary_closure:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.46 zero_zero:
% 0.19/0.46 !>[A: $tType] : A ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph,type,
% 0.19/0.46 lower_261390618igraph:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > ( set @ ( product_prod @ A @ real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.19/0.46 ord_less:
% 0.19/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.46 ord_less_eq:
% 0.19/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.19/0.46 top_top:
% 0.19/0.46 !>[A: $tType] : A ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.19/0.46 product_Pair:
% 0.19/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.19/0.46 product_rec_prod:
% 0.19/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
% 0.19/0.46 real_V2000881966t_dist:
% 0.19/0.46 !>[A: $tType] : ( A > A > real ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_OCollect,type,
% 0.19/0.46 collect:
% 0.19/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Uniform__Limit_Ouniformly__Cauchy__on,type,
% 0.19/0.46 unifor760693226chy_on:
% 0.19/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( nat > A > B ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member,type,
% 0.19/0.46 member:
% 0.19/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_e____,type,
% 0.19/0.46 e: real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_f,type,
% 0.19/0.46 f: a > extended_ereal ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_thesis____,type,
% 0.19/0.46 thesis: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_x____,type,
% 0.19/0.46 x: a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_y____,type,
% 0.19/0.46 y: real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_z____,type,
% 0.19/0.46 z: real ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (254)
% 0.19/0.46 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,
% 0.19/0.46 ? [X: product_prod @ a @ real] :
% 0.19/0.46 ( ( member @ ( product_prod @ a @ real ) @ X @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
% 0.19/0.46 & ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ X @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<exists>ya\<in>Epigraph UNIV f. dist ya (x, y) < e\<close>
% 0.19/0.46 thf(fact_1__092_060open_0620_A_060_Ae_092_060close_062,axiom,
% 0.19/0.46 ord_less @ real @ ( zero_zero @ real ) @ e ).
% 0.19/0.46
% 0.19/0.46 % \<open>0 < e\<close>
% 0.19/0.46 thf(fact_2_xy,axiom,
% 0.19/0.46 ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ x @ y ) @ ( elementary_closure @ ( product_prod @ a @ real ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) ) )
% 0.19/0.46 & ( ord_less_eq @ real @ y @ z ) ) ).
% 0.19/0.46
% 0.19/0.46 % xy
% 0.19/0.46 thf(fact_3_UNIV__I,axiom,
% 0.19/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % UNIV_I
% 0.19/0.46 thf(fact_4_iso__tuple__UNIV__I,axiom,
% 0.19/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % iso_tuple_UNIV_I
% 0.19/0.46 thf(fact_5_prod_Oinject,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.19/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.19/0.46 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.19/0.46 = ( ( X1 = Y1 )
% 0.19/0.46 & ( X22 = Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % prod.inject
% 0.19/0.46 thf(fact_6_old_Oprod_Oinject,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.19/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.19/0.46 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.19/0.46 = ( ( A2 = A3 )
% 0.19/0.46 & ( B2 = B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % old.prod.inject
% 0.19/0.46 thf(fact_7_top__apply,axiom,
% 0.19/0.46 ! [C: $tType,D: $tType] :
% 0.19/0.46 ( ( top @ C )
% 0.19/0.46 => ( ( top_top @ ( D > C ) )
% 0.19/0.46 = ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % top_apply
% 0.19/0.46 thf(fact_8_dist__commute__lessI,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [Y: A,X2: A,E: real] :
% 0.19/0.46 ( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X2 ) @ E )
% 0.19/0.46 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ E ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dist_commute_lessI
% 0.19/0.46 thf(fact_9_top__prod__def,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( top @ B )
% 0.19/0.46 & ( top @ A ) )
% 0.19/0.46 => ( ( top_top @ ( product_prod @ A @ B ) )
% 0.19/0.46 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % top_prod_def
% 0.19/0.46 thf(fact_10_top_Oextremum__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order_top @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % top.extremum_strict
% 0.19/0.46 thf(fact_11_top_Onot__eq__extremum,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order_top @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 != ( top_top @ A ) )
% 0.19/0.46 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % top.not_eq_extremum
% 0.19/0.46 thf(fact_12_epigraph__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [X2: A,Y: real,F: A > extended_ereal,Z: real] :
% 0.19/0.46 ( ( ( member @ ( product_prod @ A @ real ) @ ( product_Pair @ A @ real @ X2 @ Y ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) )
% 0.19/0.46 & ( ord_less_eq @ real @ Y @ Z ) )
% 0.19/0.46 => ( member @ ( product_prod @ A @ real ) @ ( product_Pair @ A @ real @ X2 @ Z ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % epigraph_mono
% 0.19/0.46 thf(fact_13_metric__eq__thm,axiom,
% 0.19/0.46 ! [A4: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A4 )
% 0.19/0.46 => ! [X4: A4,S: set @ A4,Y3: A4] :
% 0.19/0.46 ( ( member @ A4 @ X4 @ S )
% 0.19/0.46 => ( ( member @ A4 @ Y3 @ S )
% 0.19/0.46 => ( ( X4 = Y3 )
% 0.19/0.46 = ( ! [X3: A4] :
% 0.19/0.46 ( ( member @ A4 @ X3 @ S )
% 0.19/0.46 => ( ( real_V2000881966t_dist @ A4 @ X4 @ X3 )
% 0.19/0.46 = ( real_V2000881966t_dist @ A4 @ Y3 @ X3 ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % metric_eq_thm
% 0.19/0.46 thf(fact_14_order__refl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_15_Pair__le,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,B2: B,C2: A,D2: B] :
% 0.19/0.46 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D2 ) )
% 0.19/0.46 = ( ( ord_less_eq @ A @ A2 @ C2 )
% 0.19/0.46 & ( ord_less_eq @ B @ B2 @ D2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_le
% 0.19/0.46 thf(fact_16_dist__eq__0__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ( real_V2000881966t_dist @ A @ X2 @ Y )
% 0.19/0.46 = ( zero_zero @ real ) )
% 0.19/0.46 = ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dist_eq_0_iff
% 0.19/0.46 thf(fact_17_dist__self,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ( ( real_V2000881966t_dist @ A @ X2 @ X2 )
% 0.19/0.46 = ( zero_zero @ real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dist_self
% 0.19/0.46 thf(fact_18_dist__le__zero__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ ( zero_zero @ real ) )
% 0.19/0.46 = ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dist_le_zero_iff
% 0.19/0.46 thf(fact_19_zero__less__dist__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) )
% 0.19/0.46 = ( X2 != Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_less_dist_iff
% 0.19/0.46 thf(fact_20_le__funD,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funD
% 0.19/0.46 thf(fact_21_le__funE,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funE
% 0.19/0.46 thf(fact_22_le__funI,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ! [F: A > B,G: A > B] :
% 0.19/0.46 ( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
% 0.19/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funI
% 0.19/0.46 thf(fact_23_le__fun__def,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.19/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.19/0.46 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_fun_def
% 0.19/0.46 thf(fact_24_order__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( order @ B )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.19/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.19/0.46 => ( ! [X: B,Y4: B] :
% 0.19/0.46 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_25_order__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C: $tType] :
% 0.19/0.46 ( ( ( order @ C )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.19/0.46 => ( ! [X: A,Y4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_26_Pair__mono,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [X2: A,X5: A,Y: B,Y5: B] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ X5 )
% 0.19/0.46 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 0.19/0.46 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_mono
% 0.19/0.46 thf(fact_27_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( F @ B2 ) )
% 0.19/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.19/0.46 => ( ! [X: B,Y4: B] :
% 0.19/0.46 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_28_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( ( F @ B2 )
% 0.19/0.46 = C2 )
% 0.19/0.46 => ( ! [X: A,Y4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_29_eq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.19/0.46 = ( ^ [X3: A,Y7: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 0.19/0.46 & ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_30_antisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_31_linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_32_eq__refl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_33_le__cases,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_34_order_Otrans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_35_le__cases3,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_36_antisym__conv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [Y: A,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 = ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_37_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.19/0.46 = ( ^ [A5: A,B4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.19/0.46 & ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_38_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.46 ( ( A2 = B2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_39_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( B2 = C2 )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_40_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_41_order__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_42_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_43_linorder__wlog,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.19/0.46 ( ! [A6: A,B5: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.19/0.46 => ( P @ A6 @ B5 ) )
% 0.19/0.46 => ( ! [A6: A,B5: A] :
% 0.19/0.46 ( ( P @ B5 @ A6 )
% 0.19/0.46 => ( P @ A6 @ B5 ) )
% 0.19/0.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_44_dual__order_Otrans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B2: A,A2: A,C2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.19/0.46 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.19/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.19/0.46 = ( P @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A: $tType,A7: set @ A] :
% 0.19/0.46 ( ( collect @ A
% 0.19/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A7 ) )
% 0.19/0.46 = A7 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_47_Collect__cong,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.46 ( ! [X: A] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 = ( Q @ X ) )
% 0.19/0.46 => ( ( collect @ A @ P )
% 0.19/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_48_ext,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.19/0.46 ( ! [X: A] :
% 0.19/0.46 ( ( F @ X )
% 0.19/0.46 = ( G @ X ) )
% 0.19/0.46 => ( F = G ) ) ).
% 0.19/0.46
% 0.19/0.46 % ext
% 0.19/0.46 thf(fact_49_dual__order_Oeq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.19/0.46 = ( ^ [A5: A,B4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.19/0.46 & ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.eq_iff
% 0.19/0.46 thf(fact_50_dual__order_Oantisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B2: A,A2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.antisym
% 0.19/0.46 thf(fact_51_zero__le__dist,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( real_V2090557954_space @ A )
% 0.19/0.46 => ! [X2: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_le_dist
% 0.19/0.46 thf(fact_52_top__set__def,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( top_top @ ( set @ A ) )
% 0.19/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % top_set_def
% 0.19/0.46 thf(fact_53_order_Onot__eq__order__implies__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B2: A] :
% 0.19/0.46 ( ( A2 != B2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.not_eq_order_implies_strict
% 0.19/0.46 thf(fact_54_dual__order_Ostrict__implies__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B2: A,A2: A] :
% 0.19/0.46 ( ( ord_less @ A @ B2 @ A2 )
% 0.19/0.46 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_implies_order
% 0.19/0.46 thf(fact_55_dual__order_Ostrict__iff__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [B4: A,A5: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.19/0.46 & ( A5 != B4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_iff_order
% 0.19/0.46 thf(fact_56_dual__order_Oorder__iff__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [B4: A,A5: A] :
% 0.19/0.46 ( ( ord_less @ A @ B4 @ A5 )
% 0.19/0.46 | ( A5 = B4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.order_iff_strict
% 0.19/0.46 thf(fact_57_order_Ostrict__implies__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B2: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_implies_order
% 0.19/0.46 thf(fact_58_dense__le__bounded,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ! [W: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ W )
% 0.19/0.46 => ( ( ord_less @ A @ W @ Y )
% 0.19/0.46 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_le_bounded
% 0.19/0.46 thf(fact_59_dense__ge__bounded,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [Z: A,X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z @ X2 )
% 0.19/0.46 => ( ! [W: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z @ W )
% 0.19/0.46 => ( ( ord_less @ A @ W @ X2 )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_ge_bounded
% 0.19/0.46 thf(fact_60_dual__order_Ostrict__trans2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B2: A,A2: A,C2: A] :
% 0.19/0.46 ( ( ord_less @ A @ B2 @ A2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.19/0.46 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans2
% 0.19/0.46 thf(fact_61_dual__order_Ostrict__trans1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B2: A,A2: A,C2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.46 => ( ( ord_less @ A @ C2 @ B2 )
% 0.19/0.46 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans1
% 0.19/0.46 thf(fact_62_order_Ostrict__iff__order,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [A5: A,B4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.19/0.46 & ( A5 != B4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_iff_order
% 0.19/0.46 thf(fact_63_order_Oorder__iff__strict,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [A5: A,B4: A] :
% 0.19/0.46 ( ( ord_less @ A @ A5 @ B4 )
% 0.19/0.46 | ( A5 = B4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.order_iff_strict
% 0.19/0.46 thf(fact_64_order_Ostrict__trans2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans2
% 0.19/0.46 thf(fact_65_order_Ostrict__trans1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.46 => ( ( ord_less @ A @ B2 @ C2 )
% 0.19/0.46 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans1
% 0.19/0.46 thf(fact_66_not__le__imp__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Y: A,X2: A] :
% 0.19/0.46 ( ~ ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_le_imp_less
% 0.19/0.46 thf(fact_67_less__le__not__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ( ( ord_less @ A )
% 0.19/0.46 = ( ^ [X3: A,Y7: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 0.19/0.46 & ~ ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_le_not_le
% 0.19/0.46 thf(fact_68_le__imp__less__or__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 | ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_less_or_eq
% 0.19/0.46 thf(fact_69_le__less__linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less_linear
% 0.19/0.46 thf(fact_70_dense__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [Y: A,Z: A] :
% 0.19/0.46 ( ! [X: A] :
% 0.19/0.46 ( ( ord_less @ A @ X @ Y )
% 0.19/0.46 => ( ord_less_eq @ A @ X @ Z ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_le
% 0.19/0.46 thf(fact_71_dense__ge,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_linorder @ A )
% 0.19/0.46 => ! [Z: A,Y: A] :
% 0.19/0.46 ( ! [X: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z @ X )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ X ) )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense_ge
% 0.19/0.46 thf(fact_72_less__le__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_le_trans
% 0.19/0.46 thf(fact_73_le__less__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less @ A @ Y @ Z )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less_trans
% 0.19/0.46 thf(fact_74_less__imp__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_le
% 0.19/0.46 thf(fact_75_antisym__conv2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.19/0.47 = ( X2 = Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv2
% 0.19/0.47 thf(fact_76_antisym__conv1,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.47 = ( X2 = Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv1
% 0.19/0.47 thf(fact_77_le__neq__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( A2 != B2 )
% 0.19/0.47 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_neq_trans
% 0.19/0.47 thf(fact_78_not__less,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.19/0.47 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_less
% 0.19/0.47 thf(fact_79_not__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
% 0.19/0.47 = ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_le
% 0.19/0.47 thf(fact_80_order__less__le__subst2,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( order @ C )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.19/0.47 => ( ! [X: A,Y4: A] :
% 0.19/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.19/0.47 => ( ord_less @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_le_subst2
% 0.19/0.47 thf(fact_81_order__less__le__subst1,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( order @ B )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.19/0.47 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.19/0.47 => ( ! [X: B,Y4: B] :
% 0.19/0.47 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_le_subst1
% 0.19/0.47 thf(fact_82_order__le__less__subst2,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( order @ C )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.19/0.47 => ( ! [X: A,Y4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.19/0.47 => ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_le_less_subst2
% 0.19/0.47 thf(fact_83_order__le__less__subst1,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( order @ B )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.19/0.47 => ( ( ord_less @ B @ B2 @ C2 )
% 0.19/0.47 => ( ! [X: B,Y4: B] :
% 0.19/0.47 ( ( ord_less @ B @ X @ Y4 )
% 0.19/0.47 => ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_le_less_subst1
% 0.19/0.47 thf(fact_84_less__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ( ( ord_less @ A )
% 0.19/0.47 = ( ^ [X3: A,Y7: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 0.19/0.47 & ( X3 != Y7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_le
% 0.19/0.47 thf(fact_85_le__less,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ( ( ord_less_eq @ A )
% 0.19/0.47 = ( ^ [X3: A,Y7: A] :
% 0.19/0.47 ( ( ord_less @ A @ X3 @ Y7 )
% 0.19/0.47 | ( X3 = Y7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_less
% 0.19/0.47 thf(fact_86_leI,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % leI
% 0.19/0.47 thf(fact_87_leD,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [Y: A,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.47 => ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % leD
% 0.19/0.47 thf(fact_88_top_Oextremum__uniqueI,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_top @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.19/0.47 => ( A2
% 0.19/0.47 = ( top_top @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top.extremum_uniqueI
% 0.19/0.47 thf(fact_89_top_Oextremum__unique,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_top @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( top_top @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top.extremum_unique
% 0.19/0.47 thf(fact_90_top__greatest,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_top @ A )
% 0.19/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top_greatest
% 0.19/0.47 thf(fact_91_dist__not__less__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ~ ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ ( zero_zero @ real ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dist_not_less_zero
% 0.19/0.47 thf(fact_92_dist__pos__lt,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( X2 != Y )
% 0.19/0.47 => ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dist_pos_lt
% 0.19/0.47 thf(fact_93_dual__order_Ostrict__implies__not__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [B2: A,A2: A] :
% 0.19/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.19/0.47 => ( A2 != B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_implies_not_eq
% 0.19/0.47 thf(fact_94_order_Ostrict__implies__not__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ( A2 != B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_implies_not_eq
% 0.19/0.47 thf(fact_95_not__less__iff__gr__or__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.19/0.47 = ( ( ord_less @ A @ Y @ X2 )
% 0.19/0.47 | ( X2 = Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_less_iff_gr_or_eq
% 0.19/0.47 thf(fact_96_dual__order_Ostrict__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.19/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.19/0.47 => ( ( ord_less @ A @ C2 @ B2 )
% 0.19/0.47 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_trans
% 0.19/0.47 thf(fact_97_linorder__less__wlog,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.19/0.47 ( ! [A6: A,B5: A] :
% 0.19/0.47 ( ( ord_less @ A @ A6 @ B5 )
% 0.19/0.47 => ( P @ A6 @ B5 ) )
% 0.19/0.47 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 0.19/0.47 => ( ! [A6: A,B5: A] :
% 0.19/0.47 ( ( P @ B5 @ A6 )
% 0.19/0.47 => ( P @ A6 @ B5 ) )
% 0.19/0.47 => ( P @ A2 @ B2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_less_wlog
% 0.19/0.47 thf(fact_98_exists__least__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( wellorder @ A )
% 0.19/0.47 => ( ( ^ [P2: A > $o] :
% 0.19/0.47 ? [X6: A] : ( P2 @ X6 ) )
% 0.19/0.47 = ( ^ [P3: A > $o] :
% 0.19/0.47 ? [N: A] :
% 0.19/0.47 ( ( P3 @ N )
% 0.19/0.47 & ! [M: A] :
% 0.19/0.47 ( ( ord_less @ A @ M @ N )
% 0.19/0.47 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % exists_least_iff
% 0.19/0.47 thf(fact_99_less__imp__not__less,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_not_less
% 0.19/0.47 thf(fact_100_order_Ostrict__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less @ A @ B2 @ C2 )
% 0.19/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_trans
% 0.19/0.47 thf(fact_101_dual__order_Oirrefl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.irrefl
% 0.19/0.47 thf(fact_102_linorder__cases,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( ( X2 != Y )
% 0.19/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_cases
% 0.19/0.47 thf(fact_103_less__imp__triv,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A,P: $o] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( ( ord_less @ A @ Y @ X2 )
% 0.19/0.47 => P ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_triv
% 0.19/0.47 thf(fact_104_less__imp__not__eq2,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( Y != X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_not_eq2
% 0.19/0.47 thf(fact_105_antisym__conv3,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [Y: A,X2: A] :
% 0.19/0.47 ( ~ ( ord_less @ A @ Y @ X2 )
% 0.19/0.47 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.19/0.47 = ( X2 = Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv3
% 0.19/0.47 thf(fact_106_less__induct,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( wellorder @ A )
% 0.19/0.47 => ! [P: A > $o,A2: A] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ! [Y8: A] :
% 0.19/0.47 ( ( ord_less @ A @ Y8 @ X )
% 0.19/0.47 => ( P @ Y8 ) )
% 0.19/0.47 => ( P @ X ) )
% 0.19/0.47 => ( P @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_induct
% 0.19/0.47 thf(fact_107_less__not__sym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_sym
% 0.19/0.47 thf(fact_108_less__imp__not__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( X2 != Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_not_eq
% 0.19/0.47 thf(fact_109_dual__order_Oasym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [B2: A,A2: A] :
% 0.19/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.19/0.47 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.asym
% 0.19/0.47 thf(fact_110_ord__less__eq__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A )
% 0.19/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( B2 = C2 )
% 0.19/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_less_eq_trans
% 0.19/0.47 thf(fact_111_ord__eq__less__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A )
% 0.19/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.47 ( ( A2 = B2 )
% 0.19/0.47 => ( ( ord_less @ A @ B2 @ C2 )
% 0.19/0.47 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_less_trans
% 0.19/0.47 thf(fact_112_less__irrefl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A] :
% 0.19/0.47 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_irrefl
% 0.19/0.47 thf(fact_113_less__linear,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 | ( X2 = Y )
% 0.19/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_linear
% 0.19/0.47 thf(fact_114_less__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A,Z: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.19/0.47 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_trans
% 0.19/0.47 thf(fact_115_less__asym_H,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_asym'
% 0.19/0.47 thf(fact_116_less__asym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_asym
% 0.19/0.47 thf(fact_117_less__imp__neq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( X2 != Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_neq
% 0.19/0.47 thf(fact_118_dense,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( dense_order @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ? [Z3: A] :
% 0.19/0.47 ( ( ord_less @ A @ X2 @ Z3 )
% 0.19/0.47 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dense
% 0.19/0.47 thf(fact_119_order_Oasym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.asym
% 0.19/0.47 thf(fact_120_neq__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( X2 != Y )
% 0.19/0.47 = ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neq_iff
% 0.19/0.47 thf(fact_121_neqE,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y: A] :
% 0.19/0.47 ( ( X2 != Y )
% 0.19/0.47 => ( ~ ( ord_less @ A @ X2 @ Y )
% 0.19/0.47 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neqE
% 0.19/0.47 thf(fact_122_gt__ex,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( no_top @ A )
% 0.19/0.47 => ! [X2: A] :
% 0.19/0.47 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 0.19/0.47
% 0.19/0.47 % gt_ex
% 0.19/0.47 thf(fact_123_lt__ex,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( no_bot @ A )
% 0.19/0.47 => ! [X2: A] :
% 0.19/0.47 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % lt_ex
% 0.19/0.47 thf(fact_124_order__less__subst2,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( order @ C )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.19/0.47 => ( ! [X: A,Y4: A] :
% 0.19/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.19/0.47 => ( ord_less @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_subst2
% 0.19/0.47 thf(fact_125_order__less__subst1,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( order @ B )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.19/0.47 => ( ( ord_less @ B @ B2 @ C2 )
% 0.19/0.47 => ( ! [X: B,Y4: B] :
% 0.19/0.47 ( ( ord_less @ B @ X @ Y4 )
% 0.19/0.47 => ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_subst1
% 0.19/0.47 thf(fact_126_ord__less__eq__subst,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B )
% 0.19/0.47 & ( ord @ A ) )
% 0.19/0.47 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.19/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.19/0.47 => ( ( ( F @ B2 )
% 0.19/0.47 = C2 )
% 0.19/0.47 => ( ! [X: A,Y4: A] :
% 0.19/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.19/0.47 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_less_eq_subst
% 0.19/0.47 thf(fact_127_ord__eq__less__subst,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B )
% 0.19/0.47 & ( ord @ A ) )
% 0.19/0.47 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( F @ B2 ) )
% 0.19/0.47 => ( ( ord_less @ B @ B2 @ C2 )
% 0.19/0.47 => ( ! [X: B,Y4: B] :
% 0.19/0.47 ( ( ord_less @ B @ X @ Y4 )
% 0.19/0.47 => ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.47 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_less_subst
% 0.19/0.47 thf(fact_128_old_Oprod_Oinducts,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.19/0.47 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 0.19/0.47 => ( P @ Prod ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inducts
% 0.19/0.47 thf(fact_129_old_Oprod_Oexhaust,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.19/0.47 ~ ! [A6: A,B5: B] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.exhaust
% 0.19/0.47 thf(fact_130_prod__induct7,axiom,
% 0.19/0.47 ! [G3: $tType,F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.19/0.47 ( ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct7
% 0.19/0.47 thf(fact_131_prod__induct6,axiom,
% 0.19/0.47 ! [F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 0.19/0.47 ( ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D3 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct6
% 0.19/0.47 thf(fact_132_prod__induct5,axiom,
% 0.19/0.47 ! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 0.19/0.47 ( ! [A6: A,B5: B,C3: C,D3: D,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct5
% 0.19/0.47 thf(fact_133_prod__induct4,axiom,
% 0.19/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.19/0.47 ( ! [A6: A,B5: B,C3: C,D3: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct4
% 0.19/0.47 thf(fact_134_prod__induct3,axiom,
% 0.19/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.47 ( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct3
% 0.19/0.47 thf(fact_135_prod__cases7,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.19/0.47 ~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3,G4: G3] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases7
% 0.19/0.47 thf(fact_136_prod__cases6,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 0.19/0.47 ~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D3 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases6
% 0.19/0.47 thf(fact_137_prod__cases5,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 0.19/0.47 ~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases5
% 0.19/0.47 thf(fact_138_prod__cases4,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.19/0.47 ~ ! [A6: A,B5: B,C3: C,D3: D] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases4
% 0.19/0.47 thf(fact_139_prod__cases3,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.47 ~ ! [A6: A,B5: B,C3: C] :
% 0.19/0.47 ( Y
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases3
% 0.19/0.47 thf(fact_140_Pair__inject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.19/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.19/0.47 => ~ ( ( A2 = A3 )
% 0.19/0.47 => ( B2 != B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_inject
% 0.19/0.47 thf(fact_141_prod__cases,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
% 0.19/0.47 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 0.19/0.47 => ( P @ P4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases
% 0.19/0.47 thf(fact_142_surj__pair,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
% 0.19/0.47 ? [X: A,Y4: B] :
% 0.19/0.47 ( P4
% 0.19/0.47 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % surj_pair
% 0.19/0.47 thf(fact_143_UNIV__witness,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % UNIV_witness
% 0.19/0.47 thf(fact_144_UNIV__eq__I,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A] :
% 0.19/0.47 ( ! [X: A] : ( member @ A @ X @ A7 )
% 0.19/0.47 => ( ( top_top @ ( set @ A ) )
% 0.19/0.47 = A7 ) ) ).
% 0.19/0.47
% 0.19/0.47 % UNIV_eq_I
% 0.19/0.47 thf(fact_145_dist__commute,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ A )
% 0.19/0.47 => ( ( real_V2000881966t_dist @ A )
% 0.19/0.47 = ( ^ [X3: A,Y7: A] : ( real_V2000881966t_dist @ A @ Y7 @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dist_commute
% 0.19/0.47 thf(fact_146_closure__UNIV,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( topolo503727757_space @ A )
% 0.19/0.47 => ( ( elementary_closure @ A @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_UNIV
% 0.19/0.47 thf(fact_147_closure__approachable__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ A )
% 0.19/0.47 => ! [X2: A,S2: set @ A] :
% 0.19/0.47 ( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
% 0.19/0.47 = ( ! [E4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ S2 )
% 0.19/0.47 & ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X3 @ X2 ) @ E4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_approachable_le
% 0.19/0.47 thf(fact_148_not__gr__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N2: A] :
% 0.19/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
% 0.19/0.47 = ( N2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_gr_zero
% 0.19/0.47 thf(fact_149_le__zero__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
% 0.19/0.47 = ( N2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_zero_eq
% 0.19/0.47 thf(fact_150_closure__approachableD,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ A )
% 0.19/0.47 => ! [X2: A,S2: set @ A,E: real] :
% 0.19/0.47 ( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
% 0.19/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
% 0.19/0.47 => ? [X: A] :
% 0.19/0.47 ( ( member @ A @ X @ S2 )
% 0.19/0.47 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ X ) @ E ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_approachableD
% 0.19/0.47 thf(fact_151_closure__approachable,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ A )
% 0.19/0.47 => ! [X2: A,S2: set @ A] :
% 0.19/0.47 ( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
% 0.19/0.47 = ( ! [E4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ S2 )
% 0.19/0.47 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X3 @ X2 ) @ E4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_approachable
% 0.19/0.47 thf(fact_152_old_Oprod_Orec,axiom,
% 0.19/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.19/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.19/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.rec
% 0.19/0.47 thf(fact_153_perfect__choose__dist,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ( real_V2090557954_space @ A )
% 0.19/0.47 & ( topolo890362671_space @ A ) )
% 0.19/0.47 => ! [R: real,X2: A] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 0.19/0.47 => ? [A6: A] :
% 0.19/0.47 ( ( A6 != X2 )
% 0.19/0.47 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ A6 @ X2 ) @ R ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % perfect_choose_dist
% 0.19/0.47 thf(fact_154_vector__choose__dist,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ( real_V55928688vector @ A )
% 0.19/0.47 & ( topolo890362671_space @ A ) )
% 0.19/0.47 => ! [C2: real,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
% 0.19/0.47 => ~ ! [Y4: A] :
% 0.19/0.47 ( ( real_V2000881966t_dist @ A @ X2 @ Y4 )
% 0.19/0.47 != C2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vector_choose_dist
% 0.19/0.47 thf(fact_155_approachable__lt__le2,axiom,
% 0.19/0.47 ! [A: $tType,Q: A > $o,F: A > real,P: A > $o] :
% 0.19/0.47 ( ( ? [D4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 0.19/0.47 & ! [X3: A] :
% 0.19/0.47 ( ( Q @ X3 )
% 0.19/0.47 => ( ( ord_less @ real @ ( F @ X3 ) @ D4 )
% 0.19/0.47 => ( P @ X3 ) ) ) ) )
% 0.19/0.47 = ( ? [D4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 0.19/0.47 & ! [X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ real @ ( F @ X3 ) @ D4 )
% 0.19/0.47 => ( ( Q @ X3 )
% 0.19/0.47 => ( P @ X3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % approachable_lt_le2
% 0.19/0.47 thf(fact_156_closure__closure,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( topolo503727757_space @ A )
% 0.19/0.47 => ! [S2: set @ A] :
% 0.19/0.47 ( ( elementary_closure @ A @ ( elementary_closure @ A @ S2 ) )
% 0.19/0.47 = ( elementary_closure @ A @ S2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_closure
% 0.19/0.47 thf(fact_157_top1I,axiom,
% 0.19/0.47 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.19/0.47
% 0.19/0.47 % top1I
% 0.19/0.47 thf(fact_158_less__fun__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B )
% 0.19/0.47 => ( ( ord_less @ ( A > B ) )
% 0.19/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.19/0.47 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.19/0.47 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_fun_def
% 0.19/0.47 thf(fact_159_less__prod__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ( ord @ A )
% 0.19/0.47 & ( ord @ B ) )
% 0.19/0.47 => ( ( ord_less @ ( product_prod @ A @ B ) )
% 0.19/0.47 = ( ^ [X3: product_prod @ A @ B,Y7: product_prod @ A @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X3 @ Y7 )
% 0.19/0.47 & ~ ( ord_less_eq @ ( product_prod @ A @ B ) @ Y7 @ X3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_prod_def
% 0.19/0.47 thf(fact_160_closure__subset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( topolo503727757_space @ A )
% 0.19/0.47 => ! [S2: set @ A] : ( ord_less_eq @ ( set @ A ) @ S2 @ ( elementary_closure @ A @ S2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_subset
% 0.19/0.47 thf(fact_161_closure__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( topolo503727757_space @ A )
% 0.19/0.47 => ! [S2: set @ A,T2: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T2 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( elementary_closure @ A @ S2 ) @ ( elementary_closure @ A @ T2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % closure_mono
% 0.19/0.47 thf(fact_162_subset__UNIV,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_UNIV
% 0.19/0.47 thf(fact_163_zero__reorient,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( zero @ A )
% 0.19/0.47 => ! [X2: A] :
% 0.19/0.47 ( ( ( zero_zero @ A )
% 0.19/0.47 = X2 )
% 0.19/0.47 = ( X2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_reorient
% 0.19/0.47 thf(fact_164_zero__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_le
% 0.19/0.47 thf(fact_165_gr__zeroI,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N2: A] :
% 0.19/0.47 ( ( N2
% 0.19/0.47 != ( zero_zero @ A ) )
% 0.19/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % gr_zeroI
% 0.19/0.47 thf(fact_166_not__less__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N2: A] :
% 0.19/0.47 ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_less_zero
% 0.19/0.47 thf(fact_167_gr__implies__not__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [M2: A,N2: A] :
% 0.19/0.47 ( ( ord_less @ A @ M2 @ N2 )
% 0.19/0.47 => ( N2
% 0.19/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % gr_implies_not_zero
% 0.19/0.47 thf(fact_168_zero__less__iff__neq__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N2: A] :
% 0.19/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
% 0.19/0.47 = ( N2
% 0.19/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_less_iff_neq_zero
% 0.19/0.47 thf(fact_169_uniformly__convergent__eq__cauchy,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( real_V1759440057_space @ A )
% 0.19/0.47 => ! [P: B > $o,S3: nat > B > A] :
% 0.19/0.47 ( ( ? [L: B > A] :
% 0.19/0.47 ! [E4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 0.19/0.47 => ? [N3: nat] :
% 0.19/0.47 ! [N: nat,X3: B] :
% 0.19/0.47 ( ( ( ord_less_eq @ nat @ N3 @ N )
% 0.19/0.47 & ( P @ X3 ) )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ ( S3 @ N @ X3 ) @ ( L @ X3 ) ) @ E4 ) ) ) )
% 0.19/0.47 = ( ! [E4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 0.19/0.47 => ? [N3: nat] :
% 0.19/0.47 ! [M: nat,N: nat,X3: B] :
% 0.19/0.47 ( ( ( ord_less_eq @ nat @ N3 @ M )
% 0.19/0.47 & ( ord_less_eq @ nat @ N3 @ N )
% 0.19/0.47 & ( P @ X3 ) )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ ( S3 @ M @ X3 ) @ ( S3 @ N @ X3 ) ) @ E4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uniformly_convergent_eq_cauchy
% 0.19/0.47 thf(fact_170_uniformly__cauchy__imp__uniformly__convergent,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( real_V1759440057_space @ B )
% 0.19/0.47 => ! [P: A > $o,S3: nat > A > B,L2: A > B] :
% 0.19/0.47 ( ! [E3: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 0.19/0.47 => ? [N4: nat] :
% 0.19/0.47 ! [M3: nat,N5: nat,X: A] :
% 0.19/0.47 ( ( ( ord_less_eq @ nat @ N4 @ M3 )
% 0.19/0.47 & ( ord_less_eq @ nat @ N4 @ N5 )
% 0.19/0.47 & ( P @ X ) )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ M3 @ X ) @ ( S3 @ N5 @ X ) ) @ E3 ) ) )
% 0.19/0.47 => ( ! [X: A] :
% 0.19/0.47 ( ( P @ X )
% 0.19/0.47 => ! [E3: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 0.19/0.47 => ? [N4: nat] :
% 0.19/0.47 ! [N5: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ N4 @ N5 )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ N5 @ X ) @ ( L2 @ X ) ) @ E3 ) ) ) )
% 0.19/0.47 => ! [E5: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E5 )
% 0.19/0.47 => ? [N6: nat] :
% 0.19/0.47 ! [N7: nat,X7: A] :
% 0.19/0.47 ( ( ( ord_less_eq @ nat @ N6 @ N7 )
% 0.19/0.47 & ( P @ X7 ) )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ N7 @ X7 ) @ ( L2 @ X7 ) ) @ E5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uniformly_cauchy_imp_uniformly_convergent
% 0.19/0.47 thf(fact_171_top__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( top_top @ ( A > $o ) )
% 0.19/0.47 = ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top_empty_eq
% 0.19/0.47 thf(fact_172_subsetI,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ( member @ A @ X @ A7 )
% 0.19/0.47 => ( member @ A @ X @ B6 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_173_psubsetI,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( A7 != B6 )
% 0.19/0.47 => ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubsetI
% 0.19/0.47 thf(fact_174_subset__antisym,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A7 )
% 0.19/0.47 => ( A7 = B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_antisym
% 0.19/0.47 thf(fact_175_in__mono,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( member @ A @ X2 @ A7 )
% 0.19/0.47 => ( member @ A @ X2 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % in_mono
% 0.19/0.47 thf(fact_176_subsetD,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( member @ A @ C2 @ A7 )
% 0.19/0.47 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetD
% 0.19/0.47 thf(fact_177_psubsetE,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubsetE
% 0.19/0.47 thf(fact_178_equalityE,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( A7 = B6 )
% 0.19/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityE
% 0.19/0.47 thf(fact_179_subset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.19/0.47 ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A8 )
% 0.19/0.47 => ( member @ A @ X3 @ B7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_eq
% 0.19/0.47 thf(fact_180_equalityD1,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( A7 = B6 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD1
% 0.19/0.47 thf(fact_181_equalityD2,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( A7 = B6 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD2
% 0.19/0.47 thf(fact_182_psubset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 0.19/0.47 & ( A8 != B7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubset_eq
% 0.19/0.47 thf(fact_183_subset__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.19/0.47 ! [T3: A] :
% 0.19/0.47 ( ( member @ A @ T3 @ A8 )
% 0.19/0.47 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_184_subset__refl,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ A7 ) ).
% 0.19/0.47
% 0.19/0.47 % subset_refl
% 0.19/0.47 thf(fact_185_Collect__mono,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ( P @ X )
% 0.19/0.47 => ( Q @ X ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono
% 0.19/0.47 thf(fact_186_subset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_trans
% 0.19/0.47 thf(fact_187_set__eq__subset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ^ [Y6: set @ A,Z2: set @ A] : ( Y6 = Z2 ) )
% 0.19/0.47 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % set_eq_subset
% 0.19/0.47 thf(fact_188_Collect__mono__iff,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.19/0.47 = ( ! [X3: A] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono_iff
% 0.19/0.47 thf(fact_189_psubset__imp__subset,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubset_imp_subset
% 0.19/0.47 thf(fact_190_psubset__subset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.19/0.47 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubset_subset_trans
% 0.19/0.47 thf(fact_191_subset__not__subset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 0.19/0.47 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_not_subset_eq
% 0.19/0.47 thf(fact_192_subset__psubset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 0.19/0.47 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_psubset_trans
% 0.19/0.47 thf(fact_193_subset__iff__psubset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A8: set @ A,B7: set @ A] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) @ A8 @ B7 )
% 0.19/0.47 | ( A8 = B7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff_psubset_eq
% 0.19/0.47 thf(fact_194_subrelI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.19/0.47 ( ! [X: A,Y4: B] :
% 0.19/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
% 0.19/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S3 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subrelI
% 0.19/0.47 thf(fact_195_seq__mono__lemma,axiom,
% 0.19/0.47 ! [M2: nat,D2: nat > real,E: nat > real] :
% 0.19/0.47 ( ! [N5: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M2 @ N5 )
% 0.19/0.47 => ( ord_less @ real @ ( D2 @ N5 ) @ ( E @ N5 ) ) )
% 0.19/0.47 => ( ! [N5: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M2 @ N5 )
% 0.19/0.47 => ( ord_less_eq @ real @ ( E @ N5 ) @ ( E @ M2 ) ) )
% 0.19/0.47 => ! [N7: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M2 @ N7 )
% 0.19/0.47 => ( ord_less @ real @ ( D2 @ N7 ) @ ( E @ M2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % seq_mono_lemma
% 0.19/0.47 thf(fact_196_le0,axiom,
% 0.19/0.47 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 0.19/0.47
% 0.19/0.47 % le0
% 0.19/0.47 thf(fact_197_bot__nat__0_Oextremum,axiom,
% 0.19/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum
% 0.19/0.47 thf(fact_198_neq0__conv,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ( ( N2
% 0.19/0.47 != ( zero_zero @ nat ) )
% 0.19/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % neq0_conv
% 0.19/0.47 thf(fact_199_less__nat__zero__code,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_nat_zero_code
% 0.19/0.47 thf(fact_200_bot__nat__0_Onot__eq__extremum,axiom,
% 0.19/0.47 ! [A2: nat] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 != ( zero_zero @ nat ) )
% 0.19/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.not_eq_extremum
% 0.19/0.47 thf(fact_201_gr0I,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ( ( N2
% 0.19/0.47 != ( zero_zero @ nat ) )
% 0.19/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % gr0I
% 0.19/0.47 thf(fact_202_not__gr0,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
% 0.19/0.47 = ( N2
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_gr0
% 0.19/0.47 thf(fact_203_not__less0,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_less0
% 0.19/0.47 thf(fact_204_less__zeroE,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_zeroE
% 0.19/0.47 thf(fact_205_gr__implies__not0,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.19/0.47 => ( N2
% 0.19/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % gr_implies_not0
% 0.19/0.47 thf(fact_206_infinite__descent0,axiom,
% 0.19/0.47 ! [P: nat > $o,N2: nat] :
% 0.19/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.47 => ( ! [N5: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
% 0.19/0.47 => ( ~ ( P @ N5 )
% 0.19/0.47 => ? [M4: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ M4 @ N5 )
% 0.19/0.47 & ~ ( P @ M4 ) ) ) )
% 0.19/0.47 => ( P @ N2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_descent0
% 0.19/0.47 thf(fact_207_bot__nat__0_Oextremum__strict,axiom,
% 0.19/0.47 ! [A2: nat] :
% 0.19/0.47 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum_strict
% 0.19/0.47 thf(fact_208_infinite__descent0__measure,axiom,
% 0.19/0.47 ! [A: $tType,V: A > nat,P: A > $o,X2: A] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ( ( V @ X )
% 0.19/0.47 = ( zero_zero @ nat ) )
% 0.19/0.47 => ( P @ X ) )
% 0.19/0.47 => ( ! [X: A] :
% 0.19/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
% 0.19/0.47 => ( ~ ( P @ X )
% 0.19/0.47 => ? [Y8: A] :
% 0.19/0.47 ( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X ) )
% 0.19/0.47 & ~ ( P @ Y8 ) ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_descent0_measure
% 0.19/0.47 thf(fact_209_psubset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 0.19/0.47 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubset_trans
% 0.19/0.47 thf(fact_210_psubsetD,axiom,
% 0.19/0.47 ! [A: $tType,A7: set @ A,B6: set @ A,C2: A] :
% 0.19/0.47 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.19/0.47 => ( ( member @ A @ C2 @ A7 )
% 0.19/0.47 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubsetD
% 0.19/0.47 thf(fact_211_measure__induct,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( wellorder @ B )
% 0.19/0.47 => ! [F: A > B,P: A > $o,A2: A] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ! [Y8: A] :
% 0.19/0.47 ( ( ord_less @ B @ ( F @ Y8 ) @ ( F @ X ) )
% 0.19/0.47 => ( P @ Y8 ) )
% 0.19/0.47 => ( P @ X ) )
% 0.19/0.47 => ( P @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % measure_induct
% 0.19/0.47 thf(fact_212_measure__induct__rule,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( wellorder @ B )
% 0.19/0.47 => ! [F: A > B,P: A > $o,A2: A] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ! [Y8: A] :
% 0.19/0.47 ( ( ord_less @ B @ ( F @ Y8 ) @ ( F @ X ) )
% 0.19/0.47 => ( P @ Y8 ) )
% 0.19/0.47 => ( P @ X ) )
% 0.19/0.47 => ( P @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % measure_induct_rule
% 0.19/0.47 thf(fact_213_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.19/0.47 ! [A2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.19/0.47 => ( A2
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum_uniqueI
% 0.19/0.47 thf(fact_214_bot__nat__0_Oextremum__unique,axiom,
% 0.19/0.47 ! [A2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum_unique
% 0.19/0.47 thf(fact_215_less__mono__imp__le__mono,axiom,
% 0.19/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.19/0.47 ( ! [I2: nat,J2: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.19/0.47 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_mono_imp_le_mono
% 0.19/0.47 thf(fact_216_le__neq__implies__less,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.19/0.47 => ( ( M2 != N2 )
% 0.19/0.47 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_neq_implies_less
% 0.19/0.47 thf(fact_217_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.47 ! [P: nat > $o,K: nat,B2: nat] :
% 0.19/0.47 ( ( P @ K )
% 0.19/0.47 => ( ! [Y4: nat] :
% 0.19/0.47 ( ( P @ Y4 )
% 0.19/0.47 => ( ord_less_eq @ nat @ Y4 @ B2 ) )
% 0.19/0.47 => ? [X: nat] :
% 0.19/0.47 ( ( P @ X )
% 0.19/0.47 & ! [Y8: nat] :
% 0.19/0.47 ( ( P @ Y8 )
% 0.19/0.47 => ( ord_less_eq @ nat @ Y8 @ X ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Nat.ex_has_greatest_nat
% 0.19/0.47 thf(fact_218_less__or__eq__imp__le,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( ( ord_less @ nat @ M2 @ N2 )
% 0.19/0.47 | ( M2 = N2 ) )
% 0.19/0.47 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_or_eq_imp_le
% 0.19/0.47 thf(fact_219_le__eq__less__or__eq,axiom,
% 0.19/0.47 ( ( ord_less_eq @ nat )
% 0.19/0.47 = ( ^ [M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ M @ N )
% 0.19/0.47 | ( M = N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_eq_less_or_eq
% 0.19/0.47 thf(fact_220_less__imp__le__nat,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.19/0.47 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_le_nat
% 0.19/0.47 thf(fact_221_ex__least__nat__le,axiom,
% 0.19/0.47 ! [P: nat > $o,N2: nat] :
% 0.19/0.47 ( ( P @ N2 )
% 0.19/0.47 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.19/0.47 => ? [K2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ K2 @ N2 )
% 0.19/0.47 & ! [I3: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ I3 @ K2 )
% 0.19/0.47 => ~ ( P @ I3 ) )
% 0.19/0.47 & ( P @ K2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_least_nat_le
% 0.19/0.47 thf(fact_222_nat__le__linear,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.19/0.47 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_le_linear
% 0.19/0.47 thf(fact_223_nat__less__le,axiom,
% 0.19/0.47 ( ( ord_less @ nat )
% 0.19/0.47 = ( ^ [M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.47 & ( M != N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_less_le
% 0.19/0.47 thf(fact_224_le__antisym,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 0.19/0.47 => ( M2 = N2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_antisym
% 0.19/0.47 thf(fact_225_eq__imp__le,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( M2 = N2 )
% 0.19/0.47 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_imp_le
% 0.19/0.47 thf(fact_226_le__trans,axiom,
% 0.19/0.47 ! [I: nat,J: nat,K: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.19/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_trans
% 0.19/0.47 thf(fact_227_le__refl,axiom,
% 0.19/0.47 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.19/0.47
% 0.19/0.47 % le_refl
% 0.19/0.47 thf(fact_228_le__0__eq,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
% 0.19/0.47 = ( N2
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_0_eq
% 0.19/0.47 thf(fact_229_less__eq__nat_Osimps_I1_J,axiom,
% 0.19/0.47 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 0.19/0.47
% 0.19/0.47 % less_eq_nat.simps(1)
% 0.19/0.47 thf(fact_230_uniformly__Cauchy__onI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ B )
% 0.19/0.47 => ! [X8: set @ A,F: nat > A > B] :
% 0.19/0.47 ( ! [E3: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 0.19/0.47 => ? [M5: nat] :
% 0.19/0.47 ! [X: A] :
% 0.19/0.47 ( ( member @ A @ X @ X8 )
% 0.19/0.47 => ! [M3: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M5 @ M3 )
% 0.19/0.47 => ! [N5: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M5 @ N5 )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F @ M3 @ X ) @ ( F @ N5 @ X ) ) @ E3 ) ) ) ) )
% 0.19/0.47 => ( unifor760693226chy_on @ A @ B @ X8 @ F ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uniformly_Cauchy_onI
% 0.19/0.47 thf(fact_231_uniformly__Cauchy__on__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ B )
% 0.19/0.47 => ( ( unifor760693226chy_on @ A @ B )
% 0.19/0.47 = ( ^ [X9: set @ A,F2: nat > A > B] :
% 0.19/0.47 ! [E4: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 0.19/0.47 => ? [M6: nat] :
% 0.19/0.47 ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ X9 )
% 0.19/0.47 => ! [M: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M6 @ M )
% 0.19/0.47 => ! [N: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M6 @ N )
% 0.19/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F2 @ M @ X3 ) @ ( F2 @ N @ X3 ) ) @ E4 ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uniformly_Cauchy_on_def
% 0.19/0.47 thf(fact_232_infinite__descent__measure,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 0.19/0.47 ( ! [X: A] :
% 0.19/0.47 ( ~ ( P @ X )
% 0.19/0.47 => ? [Y8: A] :
% 0.19/0.47 ( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X ) )
% 0.19/0.47 & ~ ( P @ Y8 ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_descent_measure
% 0.19/0.47 thf(fact_233_linorder__neqE__nat,axiom,
% 0.19/0.47 ! [X2: nat,Y: nat] :
% 0.19/0.47 ( ( X2 != Y )
% 0.19/0.47 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 0.19/0.47 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_neqE_nat
% 0.19/0.47 thf(fact_234_infinite__descent,axiom,
% 0.19/0.47 ! [P: nat > $o,N2: nat] :
% 0.19/0.47 ( ! [N5: nat] :
% 0.19/0.47 ( ~ ( P @ N5 )
% 0.19/0.47 => ? [M4: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ M4 @ N5 )
% 0.19/0.47 & ~ ( P @ M4 ) ) )
% 0.19/0.47 => ( P @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_descent
% 0.19/0.47 thf(fact_235_nat__less__induct,axiom,
% 0.19/0.47 ! [P: nat > $o,N2: nat] :
% 0.19/0.47 ( ! [N5: nat] :
% 0.19/0.47 ( ! [M4: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ M4 @ N5 )
% 0.19/0.47 => ( P @ M4 ) )
% 0.19/0.47 => ( P @ N5 ) )
% 0.19/0.47 => ( P @ N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_less_induct
% 0.19/0.47 thf(fact_236_less__irrefl__nat,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.19/0.47
% 0.19/0.47 % less_irrefl_nat
% 0.19/0.47 thf(fact_237_less__not__refl3,axiom,
% 0.19/0.47 ! [S3: nat,T4: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ S3 @ T4 )
% 0.19/0.47 => ( S3 != T4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_refl3
% 0.19/0.47 thf(fact_238_less__not__refl2,axiom,
% 0.19/0.47 ! [N2: nat,M2: nat] :
% 0.19/0.47 ( ( ord_less @ nat @ N2 @ M2 )
% 0.19/0.47 => ( M2 != N2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_refl2
% 0.19/0.47 thf(fact_239_less__not__refl,axiom,
% 0.19/0.47 ! [N2: nat] :
% 0.19/0.47 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_refl
% 0.19/0.47 thf(fact_240_nat__neq__iff,axiom,
% 0.19/0.47 ! [M2: nat,N2: nat] :
% 0.19/0.47 ( ( M2 != N2 )
% 0.19/0.47 = ( ( ord_less @ nat @ M2 @ N2 )
% 0.19/0.47 | ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_neq_iff
% 0.19/0.47 thf(fact_241_uniformly__Cauchy__onI_H,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( real_V2090557954_space @ B )
% 0.19/0.47 => ! [X8: set @ A,F: nat > A > B] :
% 0.19/0.47 ( ! [E3: real] :
% 0.19/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 0.19/0.47 => ? [M5: nat] :
% 0.31/0.47 ! [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ X8 )
% 0.31/0.47 => ! [M3: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M5 @ M3 )
% 0.31/0.47 => ! [N5: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M3 @ N5 )
% 0.31/0.47 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F @ M3 @ X ) @ ( F @ N5 @ X ) ) @ E3 ) ) ) ) )
% 0.31/0.47 => ( unifor760693226chy_on @ A @ B @ X8 @ F ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % uniformly_Cauchy_onI'
% 0.31/0.47 thf(fact_242_nat__descend__induct,axiom,
% 0.31/0.47 ! [N2: nat,P: nat > $o,M2: nat] :
% 0.31/0.47 ( ! [K2: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ N2 @ K2 )
% 0.31/0.47 => ( P @ K2 ) )
% 0.31/0.47 => ( ! [K2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ K2 @ N2 )
% 0.31/0.47 => ( ! [I3: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ K2 @ I3 )
% 0.31/0.47 => ( P @ I3 ) )
% 0.31/0.47 => ( P @ K2 ) ) )
% 0.31/0.47 => ( P @ M2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_descend_induct
% 0.31/0.47 thf(fact_243_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 0.31/0.47 ( ( P @ K )
% 0.31/0.47 => ( ! [Y4: A] :
% 0.31/0.47 ( ( P @ Y4 )
% 0.31/0.47 => ( ord_less @ nat @ ( F @ Y4 ) @ B2 ) )
% 0.31/0.47 => ? [X: A] :
% 0.31/0.47 ( ( P @ X )
% 0.31/0.47 & ! [Y8: A] :
% 0.31/0.47 ( ( P @ Y8 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( F @ Y8 ) @ ( F @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Lattices_Big.ex_has_greatest_nat
% 0.31/0.47 thf(fact_244_ex__has__least__nat,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.31/0.47 ( ( P @ K )
% 0.31/0.47 => ? [X: A] :
% 0.31/0.47 ( ( P @ X )
% 0.31/0.47 & ! [Y8: A] :
% 0.31/0.47 ( ( P @ Y8 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ Y8 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ex_has_least_nat
% 0.31/0.47 thf(fact_245_less__eq__real__def,axiom,
% 0.31/0.47 ( ( ord_less_eq @ real )
% 0.31/0.47 = ( ^ [X3: real,Y7: real] :
% 0.31/0.47 ( ( ord_less @ real @ X3 @ Y7 )
% 0.31/0.47 | ( X3 = Y7 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_eq_real_def
% 0.31/0.47 thf(fact_246_zero__prod__def,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( zero @ B )
% 0.31/0.47 & ( zero @ A ) )
% 0.31/0.47 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 0.31/0.47 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_prod_def
% 0.31/0.47 thf(fact_247_complete__real,axiom,
% 0.31/0.47 ! [S2: set @ real] :
% 0.31/0.47 ( ? [X7: real] : ( member @ real @ X7 @ S2 )
% 0.31/0.47 => ( ? [Z4: real] :
% 0.31/0.47 ! [X: real] :
% 0.31/0.47 ( ( member @ real @ X @ S2 )
% 0.31/0.47 => ( ord_less_eq @ real @ X @ Z4 ) )
% 0.31/0.47 => ? [Y4: real] :
% 0.31/0.47 ( ! [X7: real] :
% 0.31/0.47 ( ( member @ real @ X7 @ S2 )
% 0.31/0.47 => ( ord_less_eq @ real @ X7 @ Y4 ) )
% 0.31/0.47 & ! [Z4: real] :
% 0.31/0.47 ( ! [X: real] :
% 0.31/0.47 ( ( member @ real @ X @ S2 )
% 0.31/0.47 => ( ord_less_eq @ real @ X @ Z4 ) )
% 0.31/0.47 => ( ord_less_eq @ real @ Y4 @ Z4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % complete_real
% 0.31/0.47 thf(fact_248_field__lbound__gt__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A )
% 0.31/0.47 => ! [D1: A,D22: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.31/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
% 0.31/0.47 => ? [E3: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
% 0.31/0.47 & ( ord_less @ A @ E3 @ D1 )
% 0.31/0.47 & ( ord_less @ A @ E3 @ D22 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % field_lbound_gt_zero
% 0.31/0.47 thf(fact_249_less__numeral__extra_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(3)
% 0.31/0.47 thf(fact_250_complete__interval,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( condit1037483654norder @ A )
% 0.31/0.47 => ! [A2: A,B2: A,P: A > $o] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( P @ A2 )
% 0.31/0.47 => ( ~ ( P @ B2 )
% 0.31/0.47 => ? [C3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.31/0.47 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.31/0.47 & ! [X7: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ A2 @ X7 )
% 0.31/0.47 & ( ord_less @ A @ X7 @ C3 ) )
% 0.31/0.47 => ( P @ X7 ) )
% 0.31/0.47 & ! [D5: A] :
% 0.31/0.47 ( ! [X: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ A2 @ X )
% 0.31/0.47 & ( ord_less @ A @ X @ D5 ) )
% 0.31/0.47 => ( P @ X ) )
% 0.31/0.47 => ( ord_less_eq @ A @ D5 @ C3 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % complete_interval
% 0.31/0.47 thf(fact_251_ex__gt__or__lt,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( condit1656338222tinuum @ A )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ? [B5: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B5 )
% 0.31/0.47 | ( ord_less @ A @ B5 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ex_gt_or_lt
% 0.31/0.47 thf(fact_252_le__numeral__extra_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(3)
% 0.31/0.47 thf(fact_253_minf_I8_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [T4: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X7: A] :
% 0.31/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ T4 @ X7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(8)
% 0.31/0.47
% 0.31/0.47 % Subclasses (2)
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( type @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( real_V2090557954_space @ A )
% 0.31/0.47 => ( topolo503727757_space @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % Type constructors (61)
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( euclid925273238_space @ A4 )
% 0.31/0.47 & ( euclid925273238_space @ A9 ) )
% 0.31/0.47 => ( euclid925273238_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space_1,axiom,
% 0.31/0.47 euclid925273238_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( topolo503727757_space @ A9 )
% 0.31/0.47 => ( topolo503727757_space @ ( A4 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( order_top @ A9 )
% 0.31/0.47 => ( order_top @ ( A4 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( preorder @ A9 )
% 0.31/0.47 => ( preorder @ ( A4 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( order @ A9 )
% 0.31/0.47 => ( order @ ( A4 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( top @ A9 )
% 0.31/0.47 => ( top @ ( A4 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ord @ A9 )
% 0.31/0.47 => ( ord @ ( A4 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.31/0.47 condit1037483654norder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.31/0.47 canoni770627133id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_2,axiom,
% 0.31/0.47 topolo503727757_space @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.31/0.47 linord1659791738miring @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.31/0.47 wellorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
% 0.31/0.47 preorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.31/0.47 linorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.31/0.47 no_top @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
% 0.31/0.47 order @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
% 0.31/0.47 ord @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.31/0.47 zero @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_6,axiom,
% 0.31/0.47 ! [A4: $tType] : ( order_top @ ( set @ A4 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.31/0.47 ! [A4: $tType] : ( preorder @ ( set @ A4 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 0.31/0.47 ! [A4: $tType] : ( order @ ( set @ A4 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Otop_9,axiom,
% 0.31/0.47 ! [A4: $tType] : ( top @ ( set @ A4 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.31/0.47 ! [A4: $tType] : ( ord @ ( set @ A4 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Groups_Ozero_11,axiom,
% 0.31/0.47 ! [A4: $tType] :
% 0.31/0.47 ( ( zero @ A4 )
% 0.31/0.47 => ( zero @ ( set @ A4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_12,axiom,
% 0.31/0.47 topolo503727757_space @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
% 0.31/0.47 order_top @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 0.31/0.47 preorder @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
% 0.31/0.47 linorder @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.31/0.47 order @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
% 0.31/0.47 top @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 0.31/0.47 ord @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
% 0.31/0.47 condit1037483654norder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
% 0.31/0.47 condit1656338222tinuum @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 0.31/0.47 real_V55928688vector @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_20,axiom,
% 0.31/0.47 topolo503727757_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_21,axiom,
% 0.31/0.47 linord1659791738miring @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
% 0.31/0.47 real_V1759440057_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
% 0.31/0.47 topolo890362671_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 0.31/0.47 real_V2090557954_space @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 0.31/0.47 dense_linorder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.31/0.47 linordered_field @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 0.31/0.47 dense_order @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Opreorder_22,axiom,
% 0.31/0.47 preorder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_23,axiom,
% 0.31/0.47 linorder @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Ono__top_24,axiom,
% 0.31/0.47 no_top @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 0.31/0.47 no_bot @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_25,axiom,
% 0.31/0.47 order @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Orderings_Oord_26,axiom,
% 0.31/0.47 ord @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
% 0.31/0.47 zero @ real ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_28,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( real_V55928688vector @ A4 )
% 0.31/0.47 & ( real_V55928688vector @ A9 ) )
% 0.31/0.47 => ( real_V55928688vector @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_29,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( topolo503727757_space @ A4 )
% 0.31/0.47 & ( topolo503727757_space @ A9 ) )
% 0.31/0.47 => ( topolo503727757_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ocomplete__space_30,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( real_V1759440057_space @ A4 )
% 0.31/0.47 & ( real_V1759440057_space @ A9 ) )
% 0.31/0.47 => ( real_V1759440057_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Topological__Spaces_Operfect__space_31,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( euclid925273238_space @ A4 )
% 0.31/0.47 & ( euclid925273238_space @ A9 ) )
% 0.31/0.47 => ( topolo890362671_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_32,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( real_V2090557954_space @ A4 )
% 0.31/0.47 & ( real_V2090557954_space @ A9 ) )
% 0.31/0.47 => ( real_V2090557954_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_33,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( order_top @ A4 )
% 0.31/0.47 & ( order_top @ A9 ) )
% 0.31/0.47 => ( order_top @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_34,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( preorder @ A4 )
% 0.31/0.47 & ( preorder @ A9 ) )
% 0.31/0.47 => ( preorder @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oorder_35,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( order @ A4 )
% 0.31/0.47 & ( order @ A9 ) )
% 0.31/0.47 => ( order @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Orderings_Otop_36,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( top @ A4 )
% 0.31/0.47 & ( top @ A9 ) )
% 0.31/0.47 => ( top @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Orderings_Oord_37,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( ord @ A4 )
% 0.31/0.47 & ( ord @ A9 ) )
% 0.31/0.47 => ( ord @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Groups_Ozero_38,axiom,
% 0.31/0.47 ! [A4: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( zero @ A4 )
% 0.31/0.47 & ( zero @ A9 ) )
% 0.31/0.47 => ( zero @ ( product_prod @ A4 @ A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Free types (1)
% 0.31/0.47 thf(tfree_0,hypothesis,
% 0.31/0.47 real_V2090557954_space @ a ).
% 0.31/0.47
% 0.31/0.47 % Conjectures (2)
% 0.31/0.47 thf(conj_0,hypothesis,
% 0.31/0.47 ! [A10: a,B8: real] :
% 0.31/0.47 ( ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A10 @ B8 ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
% 0.31/0.47 & ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A10 @ B8 ) @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) )
% 0.31/0.47 => thesis ) ).
% 0.31/0.47
% 0.31/0.47 thf(conj_1,conjecture,
% 0.31/0.47 thesis ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.37/1.49 ---- Embedded in THF ---
% 2.37/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.37/1.49 %%% Generated on Tue Feb 24 08:26:47 EST 2026
% 2.37/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.37/1.49 %%% Logic specification used:
% 2.37/1.49 %%% thf(spec, logic, $$dhol).
% 2.37/1.49
% 2.37/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.65X5awu5pu/DTF2DTF27033.p
% 2.37/1.49 thf(ty_t_Extended__Real_Oereal, type, extended_ereal: $tType).
% 2.37/1.49 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.37/1.49 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.37/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.37/1.49 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.37/1.49 thf(ty_tf_a, type, a: $tType).
% 2.37/1.49 thf(sy_cl_Euclidean__Space_Oeuclidean__space, type, euclid925273238_space: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V2090557954_space: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Topological__Spaces_Operfect__space, type, topolo890362671_space: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Real__Vector__Spaces_Ocomplete__space, type, real_V1759440057_space: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo503727757_space: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector, type, real_V55928688vector: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.37/1.49 thf(sy_c_Elementary__Topology_Oclosure, type, elementary_closure: (!> [A:$tType]: (((set @ A) > (set @ A))))).
% 2.37/1.49 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.37/1.49 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph, type, lower_261390618igraph: (!> [A:$tType]: (((set @ A) > ((A > extended_ereal) > (set @ ((product_prod @ A) @ real))))))).
% 2.37/1.49 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.37/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.37/1.49 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.37/1.49 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.37/1.49 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.37/1.49 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist, type, real_V2000881966t_dist: (!> [A:$tType]: ((A > (A > real))))).
% 2.37/1.49 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.37/1.49 thf(sy_c_Uniform__Limit_Ouniformly__Cauchy__on, type, unifor760693226chy_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((nat > (A > B)) > $o))))).
% 2.37/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.37/1.49 thf(sy_v_e____, type, e: real).
% 2.37/1.49 thf(sy_v_f, type, f: (a > extended_ereal)).
% 2.37/1.49 thf(sy_v_thesis____, type, thesis: $o).
% 2.37/1.49 thf(sy_v_x____, type, x: a).
% 2.37/1.49 thf(sy_v_y____, type, y: real).
% 2.37/1.49 thf(sy_v_z____, type, z: real).
% 2.37/1.49 thf(fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062, axiom, (? [X:((product_prod @ a) @ real)]: (((((member @ ((product_prod @ a) @ real)) @ X) @ (((lower_261390618igraph @ a) @ (top_top @ (set @ a))) @ f)) & (((ord_less @ real) @ (((real_V2000881966t_dist @ ((product_prod @ a) @ real)) @ X) @ ((((product_Pair @ a) @ real) @ x) @ y))) @ e))))).
% 2.37/1.49 thf(fact_1__092_060open_0620_A_060_Ae_092_060close_062, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ e)).
% 2.37/1.49 thf(fact_2_xy, axiom, ((((member @ ((product_prod @ a) @ real)) @ ((((product_Pair @ a) @ real) @ x) @ y)) @ ((elementary_closure @ ((product_prod @ a) @ real)) @ (((lower_261390618igraph @ a) @ (top_top @ (set @ a))) @ f))) & (((ord_less_eq @ real) @ y) @ z))).
% 2.37/1.49 thf(fact_3_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.37/1.49 thf(fact_4_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.37/1.49 thf(fact_5_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 2.37/1.49 thf(fact_6_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.37/1.49 thf(fact_7_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X3:D]: ((top_top @ C)))))))).
% 2.37/1.49 thf(fact_8_dist__commute__lessI, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [Y:A,X2:A,E:real]: (((((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ Y) @ X2)) @ E) => (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ X2) @ Y)) @ E)))))))).
% 2.37/1.49 thf(fact_9_top__prod__def, axiom, (! [A:$tType,B:$tType]: ((((top @ B) & (top @ A)) => ((top_top @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (top_top @ A)) @ (top_top @ B))))))).
% 2.37/1.49 thf(fact_10_top_Oextremum__strict, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A2))))))))).
% 2.37/1.49 thf(fact_11_top_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((A2 != (top_top @ A)) = (((ord_less @ A) @ A2) @ (top_top @ A))))))))).
% 2.37/1.49 thf(fact_12_epigraph__mono, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:real,F:(A > extended_ereal),Z:real]: ((((((member @ ((product_prod @ A) @ real)) @ ((((product_Pair @ A) @ real) @ X2) @ Y)) @ (((lower_261390618igraph @ A) @ (top_top @ (set @ A))) @ F)) & (((ord_less_eq @ real) @ Y) @ Z)) => (((member @ ((product_prod @ A) @ real)) @ ((((product_Pair @ A) @ real) @ X2) @ Z)) @ (((lower_261390618igraph @ A) @ (top_top @ (set @ A))) @ F))))))))).
% 2.37/1.49 thf(fact_13_metric__eq__thm, axiom, (! [A4:$tType]: (((real_V2090557954_space @ A4) => (! [X4:A4,S:(set @ A4),Y3:A4]: (((((member @ A4) @ X4) @ S) => ((((member @ A4) @ Y3) @ S) => ((X4 = Y3) = (! [X3:A4]: (((((member @ A4) @ X3) @ S) => ((((real_V2000881966t_dist @ A4) @ X4) @ X3) = (((real_V2000881966t_dist @ A4) @ Y3) @ X3)))))))))))))).
% 2.37/1.49 thf(fact_14_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.37/1.49 thf(fact_15_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:B,C2:A,D2:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D2)) = ((((ord_less_eq @ A) @ A2) @ C2) & (((ord_less_eq @ B) @ B2) @ D2))))))))).
% 2.37/1.49 thf(fact_16_dist__eq__0__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:A]: ((((((real_V2000881966t_dist @ A) @ X2) @ Y) = (zero_zero @ real)) = (X2 = Y)))))))).
% 2.37/1.49 thf(fact_17_dist__self, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A]: (((((real_V2000881966t_dist @ A) @ X2) @ X2) = (zero_zero @ real)))))))).
% 2.37/1.49 thf(fact_18_dist__le__zero__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ real) @ (((real_V2000881966t_dist @ A) @ X2) @ Y)) @ (zero_zero @ real)) = (X2 = Y)))))))).
% 2.37/1.49 thf(fact_19_zero__less__dist__iff, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((real_V2000881966t_dist @ A) @ X2) @ Y)) = (X2 != Y)))))))).
% 2.37/1.49 thf(fact_20_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.37/1.49 thf(fact_21_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.37/1.49 thf(fact_22_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X:A]: ((((ord_less_eq @ B) @ (F @ X)) @ (G @ X)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.37/1.49 thf(fact_23_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.37/1.49 thf(fact_24_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.37/1.49 thf(fact_25_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ C) @ (F @ X)) @ (F @ Y4))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.37/1.49 thf(fact_26_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X2:A,X5:A,Y:B,Y5:B]: (((((ord_less_eq @ A) @ X2) @ X5) => ((((ord_less_eq @ B) @ Y) @ Y5) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ ((((product_Pair @ A) @ B) @ X5) @ Y5)))))))))).
% 2.37/1.49 thf(fact_27_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.37/1.49 thf(fact_28_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.37/1.49 thf(fact_29_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y6:A,Z2:A]: ((Y6 = Z2))) = (^ [X3:A,Y7:A]: (((((ord_less_eq @ A) @ X3) @ Y7) & (((ord_less_eq @ A) @ Y7) @ X3))))))))).
% 2.37/1.49 thf(fact_30_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.37/1.49 thf(fact_31_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.37/1.49 thf(fact_32_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.37/1.49 thf(fact_33_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.37/1.49 thf(fact_34_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_35_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z)))) => (((((ord_less_eq @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.37/1.49 thf(fact_36_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.37/1.49 thf(fact_37_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y6:A,Z2:A]: ((Y6 = Z2))) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) & (((ord_less_eq @ A) @ B4) @ A5))))))))).
% 2.37/1.49 thf(fact_38_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_39_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_40_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.37/1.49 thf(fact_41_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X2) @ Z))))))))).
% 2.37/1.49 thf(fact_42_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.37/1.49 thf(fact_43_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) => ((P @ A6) @ B5)))) => ((! [A6:A,B5:A]: ((((P @ B5) @ A6) => ((P @ A6) @ B5)))) => ((P @ A2) @ B2))))))))).
% 2.37/1.49 thf(fact_44_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.37/1.49 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.37/1.49 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A7:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A7)))) = A7)))).
% 2.37/1.49 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.37/1.49 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 2.37/1.49 thf(fact_49_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y6:A,Z2:A]: ((Y6 = Z2))) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A5) & (((ord_less_eq @ A) @ A5) @ B4))))))))).
% 2.37/1.49 thf(fact_50_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.37/1.49 thf(fact_51_zero__le__dist, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (((real_V2000881966t_dist @ A) @ X2) @ Y)))))))).
% 2.37/1.49 thf(fact_52_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.37/1.49 thf(fact_53_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.37/1.49 thf(fact_54_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.37/1.49 thf(fact_55_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B4:A,A5:A]: (((((ord_less_eq @ A) @ B4) @ A5) & (A5 != B4))))))))).
% 2.37/1.49 thf(fact_56_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B4:A,A5:A]: (((((ord_less @ A) @ B4) @ A5) | (A5 = B4))))))))).
% 2.37/1.49 thf(fact_57_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.37/1.49 thf(fact_58_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ X2) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X2) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.37/1.49 thf(fact_59_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,X2:A,Y:A]: (((((ord_less @ A) @ Z) @ X2) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X2) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.37/1.49 thf(fact_60_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.37/1.49 thf(fact_61_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.37/1.49 thf(fact_62_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) & (A5 != B4))))))))).
% 2.37/1.49 thf(fact_63_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) | (A5 = B4))))))))).
% 2.37/1.49 thf(fact_64_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_65_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_66_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X2:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X2))) => (((ord_less @ A) @ X2) @ Y)))))))).
% 2.37/1.49 thf(fact_67_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X3:A,Y7:A]: (((((ord_less_eq @ A) @ X3) @ Y7) & (~ ((((ord_less_eq @ A) @ Y7) @ X3))))))))))).
% 2.37/1.49 thf(fact_68_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less @ A) @ X2) @ Y) | (X2 = Y))))))))).
% 2.37/1.49 thf(fact_69_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less @ A) @ Y) @ X2)))))))).
% 2.37/1.49 thf(fact_70_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z:A]: (((! [X:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.37/1.49 thf(fact_71_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z:A,Y:A]: (((! [X:A]: (((((ord_less @ A) @ Z) @ X) => (((ord_less_eq @ A) @ Y) @ X)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.37/1.49 thf(fact_72_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X2) @ Z))))))))).
% 2.37/1.49 thf(fact_73_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X2) @ Z))))))))).
% 2.37/1.49 thf(fact_74_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.37/1.49 thf(fact_75_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) = (X2 = Y))))))))).
% 2.37/1.49 thf(fact_76_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.37/1.49 thf(fact_77_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.37/1.49 thf(fact_78_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.37/1.49 thf(fact_79_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) = (((ord_less @ A) @ Y) @ X2)))))))).
% 2.37/1.49 thf(fact_80_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less @ C) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 2.37/1.49 thf(fact_81_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.37/1.49 thf(fact_82_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ C) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 2.37/1.49 thf(fact_83_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X:B,Y4:B]: (((((ord_less @ B) @ X) @ Y4) => (((ord_less @ A) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.37/1.49 thf(fact_84_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X3:A,Y7:A]: (((((ord_less_eq @ A) @ X3) @ Y7) & (X3 != Y7))))))))).
% 2.37/1.49 thf(fact_85_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y7:A]: (((((ord_less @ A) @ X3) @ Y7) | (X3 = Y7))))))))).
% 2.37/1.49 thf(fact_86_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.37/1.49 thf(fact_87_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less @ A) @ X2) @ Y)))))))))).
% 2.37/1.49 thf(fact_88_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.37/1.49 thf(fact_89_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.37/1.49 thf(fact_90_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.37/1.49 thf(fact_91_dist__not__less__zero, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:A]: ((~ ((((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ X2) @ Y)) @ (zero_zero @ real)))))))))).
% 2.37/1.49 thf(fact_92_dist__pos__lt, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,Y:A]: (((X2 != Y) => (((ord_less @ real) @ (zero_zero @ real)) @ (((real_V2000881966t_dist @ A) @ X2) @ Y))))))))).
% 2.37/1.49 thf(fact_93_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.37/1.49 thf(fact_94_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.37/1.49 thf(fact_95_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = ((((ord_less @ A) @ Y) @ X2) | (X2 = Y))))))))).
% 2.37/1.49 thf(fact_96_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))).
% 2.37/1.49 thf(fact_97_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B5:A]: (((((ord_less @ A) @ A6) @ B5) => ((P @ A6) @ B5)))) => ((! [A6:A]: (((P @ A6) @ A6))) => ((! [A6:A,B5:A]: ((((P @ B5) @ A6) => ((P @ A6) @ B5)))) => ((P @ A2) @ B2)))))))))).
% 2.37/1.49 thf(fact_98_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X6:A]: ((P2 @ X6))))) = (^ [P3:(A > $o)]: ((? [N:A]: (((P3 @ N) & (! [M:A]: (((((ord_less @ A) @ M) @ N) => (~ ((P3 @ M)))))))))))))))).
% 2.37/1.49 thf(fact_99_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.37/1.49 thf(fact_100_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_101_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.37/1.49 thf(fact_102_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => ((X2 != Y) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.37/1.49 thf(fact_103_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,P:$o]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ X2) => P)))))))).
% 2.37/1.49 thf(fact_104_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (Y != X2)))))))).
% 2.37/1.49 thf(fact_105_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X2:A]: (((~ ((((ord_less @ A) @ Y) @ X2))) => ((~ ((((ord_less @ A) @ X2) @ Y))) = (X2 = Y))))))))).
% 2.37/1.49 thf(fact_106_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X:A]: (((! [Y8:A]: (((((ord_less @ A) @ Y8) @ X) => (P @ Y8)))) => (P @ X)))) => (P @ A2)))))))).
% 2.37/1.49 thf(fact_107_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.37/1.49 thf(fact_108_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.37/1.49 thf(fact_109_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.37/1.49 thf(fact_110_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_111_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))).
% 2.37/1.49 thf(fact_112_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((~ ((((ord_less @ A) @ X2) @ X2))))))))).
% 2.37/1.49 thf(fact_113_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) | ((X2 = Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.37/1.49 thf(fact_114_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X2) @ Z))))))))).
% 2.37/1.49 thf(fact_115_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.37/1.49 thf(fact_116_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.37/1.49 thf(fact_117_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.37/1.49 thf(fact_118_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X2) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 2.37/1.49 thf(fact_119_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.37/1.49 thf(fact_120_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) = ((((ord_less @ A) @ X2) @ Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.37/1.49 thf(fact_121_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.37/1.49 thf(fact_122_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))).
% 2.37/1.49 thf(fact_123_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X2:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X2))))))))).
% 2.37/1.49 thf(fact_124_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less @ C) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A2)) @ C2)))))))))).
% 2.37/1.49 thf(fact_125_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X:B,Y4:B]: (((((ord_less @ B) @ X) @ Y4) => (((ord_less @ A) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.37/1.49 thf(fact_126_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less @ B) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A2)) @ C2)))))))))).
% 2.37/1.49 thf(fact_127_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X:B,Y4:B]: (((((ord_less @ B) @ X) @ Y4) => (((ord_less @ A) @ (F @ X)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C2))))))))))).
% 2.37/1.49 thf(fact_128_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B5)))) => (P @ Prod))))).
% 2.37/1.49 thf(fact_129_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A6) @ B5))))))))).
% 2.37/1.49 thf(fact_130_prod__induct7, axiom, (! [G3:$tType,F3:$tType,E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))))]: (((! [A6:A,B5:B,C3:C,D3:D,E3:E2,F4:F3,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F3) @ G3)) @ E3) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4))))))))) => (P @ X2))))).
% 2.37/1.49 thf(fact_131_prod__induct6, axiom, (! [F3:$tType,E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))))]: (((! [A6:A,B5:B,C3:C,D3:D,E3:E2,F4:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ F3)) @ D3) @ ((((product_Pair @ E2) @ F3) @ E3) @ F4)))))))) => (P @ X2))))).
% 2.37/1.49 thf(fact_132_prod__induct5, axiom, (! [E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))))]: (((! [A6:A,B5:B,C3:C,D3:D,E3:E2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E2)) @ C3) @ ((((product_Pair @ D) @ E2) @ D3) @ E3))))))) => (P @ X2))))).
% 2.37/1.49 thf(fact_133_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A6:A,B5:B,C3:C,D3:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D3)))))) => (P @ X2))))).
% 2.37/1.49 thf(fact_134_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A6:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X2))))).
% 2.37/1.49 thf(fact_135_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,F3:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))))]: ((~ ((! [A6:A,B5:B,C3:C,D3:D,E3:E2,F4:F3,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F3) @ G3)) @ E3) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4)))))))))))))).
% 2.37/1.49 thf(fact_136_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,F3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))))]: ((~ ((! [A6:A,B5:B,C3:C,D3:D,E3:E2,F4:F3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ F3)) @ D3) @ ((((product_Pair @ E2) @ F3) @ E3) @ F4))))))))))))).
% 2.37/1.49 thf(fact_137_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))))]: ((~ ((! [A6:A,B5:B,C3:C,D3:D,E3:E2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E2)) @ C3) @ ((((product_Pair @ D) @ E2) @ D3) @ E3)))))))))))).
% 2.37/1.49 thf(fact_138_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A6:A,B5:B,C3:C,D3:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D3))))))))))).
% 2.37/1.49 thf(fact_139_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A6:A,B5:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))).
% 2.37/1.49 thf(fact_140_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.37/1.49 thf(fact_141_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P4:((product_prod @ A) @ B)]: (((! [A6:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B5)))) => (P @ P4))))).
% 2.37/1.49 thf(fact_142_surj__pair, axiom, (! [A:$tType,B:$tType,P4:((product_prod @ A) @ B)]: ((? [X:A,Y4:B]: ((P4 = ((((product_Pair @ A) @ B) @ X) @ Y4))))))).
% 2.37/1.49 thf(fact_143_UNIV__witness, axiom, (! [A:$tType]: ((? [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))))).
% 2.37/1.49 thf(fact_144_UNIV__eq__I, axiom, (! [A:$tType,A7:(set @ A)]: (((! [X:A]: ((((member @ A) @ X) @ A7))) => ((top_top @ (set @ A)) = A7))))).
% 2.37/1.49 thf(fact_145_dist__commute, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => ((real_V2000881966t_dist @ A) = (^ [X3:A,Y7:A]: ((((real_V2000881966t_dist @ A) @ Y7) @ X3)))))))).
% 2.37/1.49 thf(fact_146_closure__UNIV, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (((elementary_closure @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))).
% 2.37/1.49 thf(fact_147_closure__approachable__le, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,S2:(set @ A)]: (((((member @ A) @ X2) @ ((elementary_closure @ A) @ S2)) = (! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [X3:A]: (((((member @ A) @ X3) @ S2) & (((ord_less_eq @ real) @ (((real_V2000881966t_dist @ A) @ X3) @ X2)) @ E4)))))))))))))).
% 2.37/1.49 thf(fact_148_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N2))) = (N2 = (zero_zero @ A))))))))).
% 2.37/1.49 thf(fact_149_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: (((((ord_less_eq @ A) @ N2) @ (zero_zero @ A)) = (N2 = (zero_zero @ A))))))))).
% 2.37/1.49 thf(fact_150_closure__approachableD, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,S2:(set @ A),E:real]: (((((member @ A) @ X2) @ ((elementary_closure @ A) @ S2)) => ((((ord_less @ real) @ (zero_zero @ real)) @ E) => (? [X:A]: (((((member @ A) @ X) @ S2) & (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ X2) @ X)) @ E)))))))))))).
% 2.37/1.49 thf(fact_151_closure__approachable, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (! [X2:A,S2:(set @ A)]: (((((member @ A) @ X2) @ ((elementary_closure @ A) @ S2)) = (! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [X3:A]: (((((member @ A) @ X3) @ S2) & (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ X3) @ X2)) @ E4)))))))))))))).
% 2.37/1.49 thf(fact_152_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.37/1.49 thf(fact_153_perfect__choose__dist, axiom, (! [A:$tType]: ((((real_V2090557954_space @ A) & (topolo890362671_space @ A)) => (! [R:real,X2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ R) => (? [A6:A]: (((A6 != X2) & (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ A6) @ X2)) @ R))))))))))).
% 2.37/1.49 thf(fact_154_vector__choose__dist, axiom, (! [A:$tType]: ((((real_V55928688vector @ A) & (topolo890362671_space @ A)) => (! [C2:real,X2:A]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ C2) => (~ ((! [Y4:A]: (((((real_V2000881966t_dist @ A) @ X2) @ Y4) != C2)))))))))))).
% 2.37/1.49 thf(fact_155_approachable__lt__le2, axiom, (! [A:$tType,Q:(A > $o),F:(A > real),P:(A > $o)]: (((? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [X3:A]: (((Q @ X3) => ((((ord_less @ real) @ (F @ X3)) @ D4) => (P @ X3)))))))) = (? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [X3:A]: (((((ord_less_eq @ real) @ (F @ X3)) @ D4) => ((Q @ X3) => (P @ X3)))))))))))).
% 2.37/1.49 thf(fact_156_closure__closure, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S2:(set @ A)]: ((((elementary_closure @ A) @ ((elementary_closure @ A) @ S2)) = ((elementary_closure @ A) @ S2)))))))).
% 2.37/1.49 thf(fact_157_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 2.37/1.49 thf(fact_158_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.37/1.49 thf(fact_159_less__prod__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => ((ord_less @ ((product_prod @ A) @ B)) = (^ [X3:((product_prod @ A) @ B),Y7:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X3) @ Y7) & (~ ((((ord_less_eq @ ((product_prod @ A) @ B)) @ Y7) @ X3))))))))))).
% 2.37/1.49 thf(fact_160_closure__subset, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ S2) @ ((elementary_closure @ A) @ S2)))))))).
% 2.37/1.49 thf(fact_161_closure__mono, axiom, (! [A:$tType]: (((topolo503727757_space @ A) => (! [S2:(set @ A),T2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ T2) => (((ord_less_eq @ (set @ A)) @ ((elementary_closure @ A) @ S2)) @ ((elementary_closure @ A) @ T2))))))))).
% 2.37/1.49 thf(fact_162_subset__UNIV, axiom, (! [A:$tType,A7:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A7) @ (top_top @ (set @ A)))))).
% 2.37/1.49 thf(fact_163_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 2.37/1.49 thf(fact_164_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X2))))))).
% 2.37/1.49 thf(fact_165_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: (((N2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N2)))))))).
% 2.37/1.49 thf(fact_166_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: ((~ ((((ord_less @ A) @ N2) @ (zero_zero @ A)))))))))).
% 2.37/1.49 thf(fact_167_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M2:A,N2:A]: (((((ord_less @ A) @ M2) @ N2) => (N2 != (zero_zero @ A))))))))).
% 2.37/1.49 thf(fact_168_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N2) = (N2 != (zero_zero @ A))))))))).
% 2.37/1.49 thf(fact_169_uniformly__convergent__eq__cauchy, axiom, (! [A:$tType,B:$tType]: (((real_V1759440057_space @ A) => (! [P:(B > $o),S3:(nat > (B > A))]: (((? [L:(B > A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [N3:nat]: ((! [N:nat,X3:B]: ((((((ord_less_eq @ nat) @ N3) @ N) & (P @ X3)) => (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ ((S3 @ N) @ X3)) @ (L @ X3))) @ E4))))))))))) = (! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [N3:nat]: ((! [M:nat,N:nat,X3:B]: ((((((ord_less_eq @ nat) @ N3) @ M) & ((((ord_less_eq @ nat) @ N3) @ N) & (P @ X3))) => (((ord_less @ real) @ (((real_V2000881966t_dist @ A) @ ((S3 @ M) @ X3)) @ ((S3 @ N) @ X3))) @ E4)))))))))))))))).
% 2.37/1.49 thf(fact_170_uniformly__cauchy__imp__uniformly__convergent, axiom, (! [A:$tType,B:$tType]: (((real_V1759440057_space @ B) => (! [P:(A > $o),S3:(nat > (A > B)),L2:(A > B)]: (((! [E3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [N4:nat]: ((! [M3:nat,N5:nat,X:A]: ((((((ord_less_eq @ nat) @ N4) @ M3) & ((((ord_less_eq @ nat) @ N4) @ N5) & (P @ X))) => (((ord_less @ real) @ (((real_V2000881966t_dist @ B) @ ((S3 @ M3) @ X)) @ ((S3 @ N5) @ X))) @ E3))))))))) => ((! [X:A]: (((P @ X) => (! [E3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [N4:nat]: ((! [N5:nat]: (((((ord_less_eq @ nat) @ N4) @ N5) => (((ord_less @ real) @ (((real_V2000881966t_dist @ B) @ ((S3 @ N5) @ X)) @ (L2 @ X))) @ E3)))))))))))) => (! [E5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E5) => (? [N6:nat]: ((! [N7:nat,X7:A]: ((((((ord_less_eq @ nat) @ N6) @ N7) & (P @ X7)) => (((ord_less @ real) @ (((real_V2000881966t_dist @ B) @ ((S3 @ N7) @ X7)) @ (L2 @ X7))) @ E5))))))))))))))))).
% 2.37/1.49 thf(fact_171_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A))))))))).
% 2.37/1.49 thf(fact_172_subsetI, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ A7) => (((member @ A) @ X) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A7) @ B6))))).
% 2.37/1.49 thf(fact_173_psubsetI, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => ((A7 != B6) => (((ord_less @ (set @ A)) @ A7) @ B6)))))).
% 2.37/1.49 thf(fact_174_subset__antisym, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A7) => (A7 = B6)))))).
% 2.37/1.49 thf(fact_175_in__mono, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => ((((member @ A) @ X2) @ A7) => (((member @ A) @ X2) @ B6)))))).
% 2.37/1.49 thf(fact_176_subsetD, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => ((((member @ A) @ C2) @ A7) => (((member @ A) @ C2) @ B6)))))).
% 2.37/1.49 thf(fact_177_psubsetE, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) => (~ (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))).
% 2.37/1.49 thf(fact_178_equalityE, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((A7 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))))).
% 2.37/1.49 thf(fact_179_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A8) => (((member @ A) @ X3) @ B7)))))))))).
% 2.37/1.49 thf(fact_180_equalityD1, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((A7 = B6) => (((ord_less_eq @ (set @ A)) @ A7) @ B6))))).
% 2.37/1.49 thf(fact_181_equalityD2, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((A7 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A7))))).
% 2.37/1.49 thf(fact_182_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B7) & (A8 != B7)))))))).
% 2.37/1.49 thf(fact_183_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A8) => (((member @ A) @ T3) @ B7)))))))))).
% 2.37/1.49 thf(fact_184_subset__refl, axiom, (! [A:$tType,A7:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A7) @ A7)))).
% 2.37/1.49 thf(fact_185_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) => (Q @ X)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.37/1.49 thf(fact_186_subset__trans, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ A7) @ C4)))))).
% 2.37/1.49 thf(fact_187_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y6:(set @ A),Z2:(set @ A)]: ((Y6 = Z2))) = (^ [A8:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A8)))))))).
% 2.37/1.49 thf(fact_188_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.37/1.49 thf(fact_189_psubset__imp__subset, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) => (((ord_less_eq @ (set @ A)) @ A7) @ B6))))).
% 2.37/1.49 thf(fact_190_psubset__subset__trans, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A7) @ C4)))))).
% 2.37/1.49 thf(fact_191_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B7) & (~ ((((ord_less_eq @ (set @ A)) @ B7) @ A8)))))))))).
% 2.37/1.49 thf(fact_192_subset__psubset__trans, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A7) @ C4)))))).
% 2.37/1.49 thf(fact_193_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A8) @ B7) | (A8 = B7)))))))).
% 2.37/1.49 thf(fact_194_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((! [X:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ S3)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S3))))).
% 2.37/1.49 thf(fact_195_seq__mono__lemma, axiom, (! [M2:nat,D2:(nat > real),E:(nat > real)]: (((! [N5:nat]: (((((ord_less_eq @ nat) @ M2) @ N5) => (((ord_less @ real) @ (D2 @ N5)) @ (E @ N5))))) => ((! [N5:nat]: (((((ord_less_eq @ nat) @ M2) @ N5) => (((ord_less_eq @ real) @ (E @ N5)) @ (E @ M2))))) => (! [N7:nat]: (((((ord_less_eq @ nat) @ M2) @ N7) => (((ord_less @ real) @ (D2 @ N7)) @ (E @ M2)))))))))).
% 2.37/1.49 thf(fact_196_le0, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))).
% 2.37/1.49 thf(fact_197_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.37/1.49 thf(fact_198_neq0__conv, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))).
% 2.37/1.49 thf(fact_199_less__nat__zero__code, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 2.37/1.49 thf(fact_200_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.37/1.49 thf(fact_201_gr0I, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))).
% 2.37/1.49 thf(fact_202_not__gr0, axiom, (! [N2:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2))) = (N2 = (zero_zero @ nat)))))).
% 2.37/1.49 thf(fact_203_not__less0, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 2.37/1.49 thf(fact_204_less__zeroE, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 2.37/1.49 thf(fact_205_gr__implies__not0, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (N2 != (zero_zero @ nat)))))).
% 2.37/1.49 thf(fact_206_infinite__descent0, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((! [N5:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N5) => ((~ ((P @ N5))) => (? [M4:nat]: (((((ord_less @ nat) @ M4) @ N5) & (~ ((P @ M4)))))))))) => (P @ N2)))))).
% 2.37/1.49 thf(fact_207_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.37/1.49 thf(fact_208_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X2:A]: (((! [X:A]: ((((V @ X) = (zero_zero @ nat)) => (P @ X)))) => ((! [X:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X)) => ((~ ((P @ X))) => (? [Y8:A]: (((((ord_less @ nat) @ (V @ Y8)) @ (V @ X)) & (~ ((P @ Y8)))))))))) => (P @ X2)))))).
% 2.37/1.49 thf(fact_209_psubset__trans, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A7) @ C4)))))).
% 2.37/1.49 thf(fact_210_psubsetD, axiom, (! [A:$tType,A7:(set @ A),B6:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A7) @ B6) => ((((member @ A) @ C2) @ A7) => (((member @ A) @ C2) @ B6)))))).
% 2.37/1.49 thf(fact_211_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X:A]: (((! [Y8:A]: (((((ord_less @ B) @ (F @ Y8)) @ (F @ X)) => (P @ Y8)))) => (P @ X)))) => (P @ A2)))))))).
% 2.37/1.49 thf(fact_212_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X:A]: (((! [Y8:A]: (((((ord_less @ B) @ (F @ Y8)) @ (F @ X)) => (P @ Y8)))) => (P @ X)))) => (P @ A2)))))))).
% 2.37/1.49 thf(fact_213_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.37/1.49 thf(fact_214_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.37/1.49 thf(fact_215_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))))))).
% 2.37/1.49 thf(fact_216_le__neq__implies__less, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((M2 != N2) => (((ord_less @ nat) @ M2) @ N2)))))).
% 2.37/1.49 thf(fact_217_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (? [X:nat]: (((P @ X) & (! [Y8:nat]: (((P @ Y8) => (((ord_less_eq @ nat) @ Y8) @ X)))))))))))).
% 2.37/1.49 thf(fact_218_less__or__eq__imp__le, axiom, (! [M2:nat,N2:nat]: ((((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.37/1.49 thf(fact_219_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 2.37/1.49 thf(fact_220_less__imp__le__nat, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.37/1.49 thf(fact_221_ex__least__nat__le, axiom, (! [P:(nat > $o),N2:nat]: (((P @ N2) => ((~ ((P @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N2) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P @ I3)))))) & (P @ K2)))))))))).
% 2.37/1.49 thf(fact_222_nat__le__linear, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) | (((ord_less_eq @ nat) @ N2) @ M2))))).
% 2.37/1.49 thf(fact_223_nat__less__le, axiom, ((ord_less @ nat) = (^ [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) & (M != N)))))).
% 2.37/1.49 thf(fact_224_le__antisym, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M2) => (M2 = N2)))))).
% 2.37/1.49 thf(fact_225_eq__imp__le, axiom, (! [M2:nat,N2:nat]: (((M2 = N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.37/1.49 thf(fact_226_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.37/1.49 thf(fact_227_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 2.37/1.49 thf(fact_228_le__0__eq, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ N2) @ (zero_zero @ nat)) = (N2 = (zero_zero @ nat)))))).
% 2.37/1.49 thf(fact_229_less__eq__nat_Osimps_I1_J, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))).
% 2.37/1.49 thf(fact_230_uniformly__Cauchy__onI, axiom, (! [B:$tType,A:$tType]: (((real_V2090557954_space @ B) => (! [X8:(set @ A),F:(nat > (A > B))]: (((! [E3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [M5:nat]: ((! [X:A]: (((((member @ A) @ X) @ X8) => (! [M3:nat]: (((((ord_less_eq @ nat) @ M5) @ M3) => (! [N5:nat]: (((((ord_less_eq @ nat) @ M5) @ N5) => (((ord_less @ real) @ (((real_V2000881966t_dist @ B) @ ((F @ M3) @ X)) @ ((F @ N5) @ X))) @ E3))))))))))))))) => ((((unifor760693226chy_on @ A) @ B) @ X8) @ F)))))))).
% 2.37/1.49 thf(fact_231_uniformly__Cauchy__on__def, axiom, (! [B:$tType,A:$tType]: (((real_V2090557954_space @ B) => (((unifor760693226chy_on @ A) @ B) = (^ [X9:(set @ A),F2:(nat > (A > B))]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [M6:nat]: ((! [X3:A]: (((((member @ A) @ X3) @ X9) => (! [M:nat]: (((((ord_less_eq @ nat) @ M6) @ M) => (! [N:nat]: (((((ord_less_eq @ nat) @ M6) @ N) => (((ord_less @ real) @ (((real_V2000881966t_dist @ B) @ ((F2 @ M) @ X3)) @ ((F2 @ N) @ X3))) @ E4)))))))))))))))))))))).
% 2.37/1.49 thf(fact_232_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X2:A]: (((! [X:A]: (((~ ((P @ X))) => (? [Y8:A]: (((((ord_less @ nat) @ (V @ Y8)) @ (V @ X)) & (~ ((P @ Y8))))))))) => (P @ X2))))).
% 2.37/1.49 thf(fact_233_linorder__neqE__nat, axiom, (! [X2:nat,Y:nat]: (((X2 != Y) => ((~ ((((ord_less @ nat) @ X2) @ Y))) => (((ord_less @ nat) @ Y) @ X2)))))).
% 2.37/1.49 thf(fact_234_infinite__descent, axiom, (! [P:(nat > $o),N2:nat]: (((! [N5:nat]: (((~ ((P @ N5))) => (? [M4:nat]: (((((ord_less @ nat) @ M4) @ N5) & (~ ((P @ M4))))))))) => (P @ N2))))).
% 2.37/1.49 thf(fact_235_nat__less__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N5:nat]: (((! [M4:nat]: (((((ord_less @ nat) @ M4) @ N5) => (P @ M4)))) => (P @ N5)))) => (P @ N2))))).
% 2.37/1.49 thf(fact_236_less__irrefl__nat, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 2.37/1.49 thf(fact_237_less__not__refl3, axiom, (! [S3:nat,T4:nat]: (((((ord_less @ nat) @ S3) @ T4) => (S3 != T4))))).
% 2.37/1.49 thf(fact_238_less__not__refl2, axiom, (! [N2:nat,M2:nat]: (((((ord_less @ nat) @ N2) @ M2) => (M2 != N2))))).
% 2.37/1.49 thf(fact_239_less__not__refl, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 2.37/1.49 thf(fact_240_nat__neq__iff, axiom, (! [M2:nat,N2:nat]: (((M2 != N2) = ((((ord_less @ nat) @ M2) @ N2) | (((ord_less @ nat) @ N2) @ M2)))))).
% 2.37/1.49 thf(fact_241_uniformly__Cauchy__onI_H, axiom, (! [B:$tType,A:$tType]: (((real_V2090557954_space @ B) => (! [X8:(set @ A),F:(nat > (A > B))]: (((! [E3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [M5:nat]: ((! [X:A]: (((((member @ A) @ X) @ X8) => (! [M3:nat]: (((((ord_less_eq @ nat) @ M5) @ M3) => (! [N5:nat]: (((((ord_less @ nat) @ M3) @ N5) => (((ord_less @ real) @ (((real_V2000881966t_dist @ B) @ ((F @ M3) @ X)) @ ((F @ N5) @ X))) @ E3))))))))))))))) => ((((unifor760693226chy_on @ A) @ B) @ X8) @ F)))))))).
% 2.37/1.49 thf(fact_242_nat__descend__induct, axiom, (! [N2:nat,P:(nat > $o),M2:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N2) @ K2) => (P @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N2) => ((! [I3:nat]: (((((ord_less @ nat) @ K2) @ I3) => (P @ I3)))) => (P @ K2))))) => (P @ M2)))))).
% 2.37/1.49 thf(fact_243_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B2:nat]: (((P @ K) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (F @ Y4)) @ B2)))) => (? [X:A]: (((P @ X) & (! [Y8:A]: (((P @ Y8) => (((ord_less_eq @ nat) @ (F @ Y8)) @ (F @ X))))))))))))).
% 2.37/1.49 thf(fact_244_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (? [X:A]: (((P @ X) & (! [Y8:A]: (((P @ Y8) => (((ord_less_eq @ nat) @ (M2 @ X)) @ (M2 @ Y8)))))))))))).
% 2.37/1.50 thf(fact_245_less__eq__real__def, axiom, ((ord_less_eq @ real) = (^ [X3:real,Y7:real]: (((((ord_less @ real) @ X3) @ Y7) | (X3 = Y7)))))).
% 2.37/1.50 thf(fact_246_zero__prod__def, axiom, (! [A:$tType,B:$tType]: ((((zero @ B) & (zero @ A)) => ((zero_zero @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (zero_zero @ A)) @ (zero_zero @ B))))))).
% 2.37/1.50 thf(fact_247_complete__real, axiom, (! [S2:(set @ real)]: (((? [X7:real]: ((((member @ real) @ X7) @ S2))) => ((? [Z4:real]: ((! [X:real]: (((((member @ real) @ X) @ S2) => (((ord_less_eq @ real) @ X) @ Z4)))))) => (? [Y4:real]: (((! [X7:real]: (((((member @ real) @ X7) @ S2) => (((ord_less_eq @ real) @ X7) @ Y4)))) & (! [Z4:real]: (((! [X:real]: (((((member @ real) @ X) @ S2) => (((ord_less_eq @ real) @ X) @ Z4)))) => (((ord_less_eq @ real) @ Y4) @ Z4)))))))))))).
% 2.37/1.50 thf(fact_248_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D22:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D22) => (? [E3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E3) & ((((ord_less @ A) @ E3) @ D1) & (((ord_less @ A) @ E3) @ D22))))))))))))).
% 2.37/1.50 thf(fact_249_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.37/1.50 thf(fact_250_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X7:A]: ((((((ord_less_eq @ A) @ A2) @ X7) & (((ord_less @ A) @ X7) @ C3)) => (P @ X7)))) & (! [D5:A]: (((! [X:A]: ((((((ord_less_eq @ A) @ A2) @ X) & (((ord_less @ A) @ X) @ D5)) => (P @ X)))) => (((ord_less_eq @ A) @ D5) @ C3)))))))))))))))))).
% 2.37/1.50 thf(fact_251_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A2:A]: ((? [B5:A]: (((((ord_less @ A) @ A2) @ B5) | (((ord_less @ A) @ B5) @ A2)))))))))).
% 2.37/1.50 thf(fact_252_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.37/1.50 thf(fact_253_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T4:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (~ ((((ord_less_eq @ A) @ T4) @ X7)))))))))))))).
% 2.37/1.50 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (type @ A))))).
% 2.37/1.50 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space, axiom, (! [A:$tType]: (((real_V2090557954_space @ A) => (topolo503727757_space @ A))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space, axiom, (! [A4:$tType,A9:$tType]: ((((euclid925273238_space @ A4) & (euclid925273238_space @ A9)) => (euclid925273238_space @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space_1, axiom, (euclid925273238_space @ real)).
% 2.37/1.50 thf(tcon_fun___Topological__Spaces_Otopological__space, axiom, (! [A4:$tType,A9:$tType]: (((topolo503727757_space @ A9) => (topolo503727757_space @ (A4 > A9)))))).
% 2.37/1.50 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A4:$tType,A9:$tType]: (((order_top @ A9) => (order_top @ (A4 > A9)))))).
% 2.37/1.50 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A4:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A4 > A9)))))).
% 2.37/1.50 thf(tcon_fun___Orderings_Oorder, axiom, (! [A4:$tType,A9:$tType]: (((order @ A9) => (order @ (A4 > A9)))))).
% 2.37/1.50 thf(tcon_fun___Orderings_Otop, axiom, (! [A4:$tType,A9:$tType]: (((top @ A9) => (top @ (A4 > A9)))))).
% 2.37/1.50 thf(tcon_fun___Orderings_Oord, axiom, (! [A4:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A4 > A9)))))).
% 2.37/1.50 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_2, axiom, (topolo503727757_space @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Orderings_Opreorder_3, axiom, (preorder @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Orderings_Oorder_4, axiom, (order @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Orderings_Oord_5, axiom, (ord @ nat)).
% 2.37/1.50 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.37/1.50 thf(tcon_Set_Oset___Orderings_Oorder__top_6, axiom, (! [A4:$tType]: ((order_top @ (set @ A4))))).
% 2.37/1.50 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A4:$tType]: ((preorder @ (set @ A4))))).
% 2.37/1.50 thf(tcon_Set_Oset___Orderings_Oorder_8, axiom, (! [A4:$tType]: ((order @ (set @ A4))))).
% 2.37/1.50 thf(tcon_Set_Oset___Orderings_Otop_9, axiom, (! [A4:$tType]: ((top @ (set @ A4))))).
% 2.37/1.50 thf(tcon_Set_Oset___Orderings_Oord_10, axiom, (! [A4:$tType]: ((ord @ (set @ A4))))).
% 2.37/1.50 thf(tcon_Set_Oset___Groups_Ozero_11, axiom, (! [A4:$tType]: (((zero @ A4) => (zero @ (set @ A4)))))).
% 2.37/1.50 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_12, axiom, (topolo503727757_space @ $o)).
% 2.37/1.50 thf(tcon_HOL_Obool___Orderings_Oorder__top_13, axiom, (order_top @ $o)).
% 2.37/1.50 thf(tcon_HOL_Obool___Orderings_Opreorder_14, axiom, (preorder @ $o)).
% 2.37/1.50 thf(tcon_HOL_Obool___Orderings_Olinorder_15, axiom, (linorder @ $o)).
% 2.37/1.50 thf(tcon_HOL_Obool___Orderings_Oorder_16, axiom, (order @ $o)).
% 2.37/1.50 thf(tcon_HOL_Obool___Orderings_Otop_17, axiom, (top @ $o)).
% 2.37/1.50 thf(tcon_HOL_Obool___Orderings_Oord_18, axiom, (ord @ $o)).
% 2.37/1.50 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19, axiom, (condit1037483654norder @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum, axiom, (condit1656338222tinuum @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, axiom, (real_V55928688vector @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_20, axiom, (topolo503727757_space @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_21, axiom, (linord1659791738miring @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space, axiom, (real_V1759440057_space @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space, axiom, (topolo890362671_space @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space, axiom, (real_V2090557954_space @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Odense__linorder, axiom, (dense_linorder @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Odense__order, axiom, (dense_order @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Opreorder_22, axiom, (preorder @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Olinorder_23, axiom, (linorder @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Ono__top_24, axiom, (no_top @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Ono__bot, axiom, (no_bot @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Oorder_25, axiom, (order @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Orderings_Oord_26, axiom, (ord @ real)).
% 2.37/1.50 thf(tcon_Real_Oreal___Groups_Ozero_27, axiom, (zero @ real)).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_28, axiom, (! [A4:$tType,A9:$tType]: ((((real_V55928688vector @ A4) & (real_V55928688vector @ A9)) => (real_V55928688vector @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_29, axiom, (! [A4:$tType,A9:$tType]: ((((topolo503727757_space @ A4) & (topolo503727757_space @ A9)) => (topolo503727757_space @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ocomplete__space_30, axiom, (! [A4:$tType,A9:$tType]: ((((real_V1759440057_space @ A4) & (real_V1759440057_space @ A9)) => (real_V1759440057_space @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Topological__Spaces_Operfect__space_31, axiom, (! [A4:$tType,A9:$tType]: ((((euclid925273238_space @ A4) & (euclid925273238_space @ A9)) => (topolo890362671_space @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_32, axiom, (! [A4:$tType,A9:$tType]: ((((real_V2090557954_space @ A4) & (real_V2090557954_space @ A9)) => (real_V2090557954_space @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_33, axiom, (! [A4:$tType,A9:$tType]: ((((order_top @ A4) & (order_top @ A9)) => (order_top @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_34, axiom, (! [A4:$tType,A9:$tType]: ((((preorder @ A4) & (preorder @ A9)) => (preorder @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Orderings_Oorder_35, axiom, (! [A4:$tType,A9:$tType]: ((((order @ A4) & (order @ A9)) => (order @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Orderings_Otop_36, axiom, (! [A4:$tType,A9:$tType]: ((((top @ A4) & (top @ A9)) => (top @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Orderings_Oord_37, axiom, (! [A4:$tType,A9:$tType]: ((((ord @ A4) & (ord @ A9)) => (ord @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tcon_Product__Type_Oprod___Groups_Ozero_38, axiom, (! [A4:$tType,A9:$tType]: ((((zero @ A4) & (zero @ A9)) => (zero @ ((product_prod @ A4) @ A9)))))).
% 2.37/1.50 thf(tfree_0, hypothesis, (real_V2090557954_space @ a)).
% 2.37/1.50 thf(conj_0, hypothesis, (! [A10:a,B8:real]: ((((((member @ ((product_prod @ a) @ real)) @ ((((product_Pair @ a) @ real) @ A10) @ B8)) @ (((lower_261390618igraph @ a) @ (top_top @ (set @ a))) @ f)) & (((ord_less @ real) @ (((real_V2000881966t_dist @ ((product_prod @ a) @ real)) @ ((((product_Pair @ a) @ real) @ A10) @ B8)) @ ((((product_Pair @ a) @ real) @ x) @ y))) @ e)) => thesis)))).
% 2.37/1.50 thf(conj_1, conjecture, thesis).
% 2.37/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.65X5awu5pu/DTF2DTF27033.p
% 2.37/1.50 ------------------------
% 2.64/1.54 ---- Cleaned THF ---
% 2.64/1.54 thf(ty_t_Extended__Real_Oereal,type,
% 2.64/1.54 extended_ereal: $tType ).
% 2.64/1.54
% 2.64/1.54 thf(ty_t_Product__Type_Oprod,type,
% 2.64/1.54 product_prod: $tType > $tType > $tType ).
% 2.64/1.54
% 2.64/1.54 thf(ty_t_Real_Oreal,type,
% 2.64/1.54 real: $tType ).
% 2.64/1.54
% 2.64/1.54 thf(ty_t_Set_Oset,type,
% 2.64/1.54 set: $tType > $tType ).
% 2.64/1.54
% 2.64/1.54 thf(ty_t_Nat_Onat,type,
% 2.64/1.54 nat: $tType ).
% 2.64/1.54
% 2.64/1.54 thf(ty_tf_a,type,
% 2.64/1.54 a: $tType ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
% 2.64/1.54 euclid925273238_space:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_HOL_Otype,type,
% 2.64/1.54 type:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Groups_Ozero,type,
% 2.64/1.54 zero:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Oord,type,
% 2.64/1.54 ord:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Otop,type,
% 2.64/1.54 top:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Oorder,type,
% 2.64/1.54 order:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Ono__bot,type,
% 2.64/1.54 no_bot:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Ono__top,type,
% 2.64/1.54 no_top:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Olinorder,type,
% 2.64/1.54 linorder:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Opreorder,type,
% 2.64/1.54 preorder:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Oorder__top,type,
% 2.64/1.54 order_top:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Owellorder,type,
% 2.64/1.54 wellorder:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Odense__order,type,
% 2.64/1.54 dense_order:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Fields_Olinordered__field,type,
% 2.64/1.54 linordered_field:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.64/1.54 dense_linorder:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
% 2.64/1.54 real_V2090557954_space:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 2.64/1.54 topolo890362671_space:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
% 2.64/1.54 real_V1759440057_space:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.64/1.54 linord1659791738miring:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Topological__Spaces_Otopological__space,type,
% 2.64/1.54 topolo503727757_space:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.64/1.54 canoni770627133id_add:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
% 2.64/1.54 real_V55928688vector:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.64/1.54 condit1656338222tinuum:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.64/1.54 condit1037483654norder:
% 2.64/1.54 !>[A: $tType] : $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Elementary__Topology_Oclosure,type,
% 2.64/1.54 elementary_closure:
% 2.64/1.54 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.64/1.54 zero_zero:
% 2.64/1.54 !>[A: $tType] : A ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_OEpigraph,type,
% 2.64/1.54 lower_261390618igraph:
% 2.64/1.54 !>[A: $tType] : ( ( set @ A ) > ( A > extended_ereal ) > ( set @ ( product_prod @ A @ real ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.64/1.54 ord_less:
% 2.64/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.64/1.54 ord_less_eq:
% 2.64/1.54 !>[A: $tType] : ( A > A > $o ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.64/1.54 top_top:
% 2.64/1.54 !>[A: $tType] : A ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Product__Type_OPair,type,
% 2.64/1.54 product_Pair:
% 2.64/1.54 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.64/1.54 product_rec_prod:
% 2.64/1.54 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
% 2.64/1.54 real_V2000881966t_dist:
% 2.64/1.54 !>[A: $tType] : ( A > A > real ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Set_OCollect,type,
% 2.64/1.54 collect:
% 2.64/1.54 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_Uniform__Limit_Ouniformly__Cauchy__on,type,
% 2.64/1.54 unifor760693226chy_on:
% 2.64/1.54 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( nat > A > B ) > $o ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_c_member,type,
% 2.64/1.54 member:
% 2.64/1.54 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.64/1.54
% 2.64/1.54 thf(sy_v_e____,type,
% 2.64/1.54 e: real ).
% 2.64/1.54
% 2.64/1.54 thf(sy_v_f,type,
% 2.64/1.54 f: a > extended_ereal ).
% 2.64/1.54
% 2.64/1.54 thf(sy_v_thesis____,type,
% 2.64/1.54 thesis: $o ).
% 2.64/1.54
% 2.64/1.54 thf(sy_v_x____,type,
% 2.64/1.54 x: a ).
% 2.64/1.54
% 2.64/1.54 thf(sy_v_y____,type,
% 2.64/1.54 y: real ).
% 2.64/1.54
% 2.64/1.54 thf(sy_v_z____,type,
% 2.64/1.54 z: real ).
% 2.64/1.54
% 2.64/1.54 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,
% 2.64/1.54 ? [X: product_prod @ a @ real] :
% 2.64/1.54 ( ( member @ ( product_prod @ a @ real ) @ X @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
% 2.64/1.54 & ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ X @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_1__092_060open_0620_A_060_Ae_092_060close_062,axiom,
% 2.64/1.54 ord_less @ real @ ( zero_zero @ real ) @ e ).
% 2.64/1.54
% 2.64/1.54 thf(fact_2_xy,axiom,
% 2.64/1.54 ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ x @ y ) @ ( elementary_closure @ ( product_prod @ a @ real ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) ) )
% 2.64/1.54 & ( ord_less_eq @ real @ y @ z ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_3_UNIV__I,axiom,
% 2.64/1.54 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_4_iso__tuple__UNIV__I,axiom,
% 2.64/1.54 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_5_prod_Oinject,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.64/1.54 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.64/1.54 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.64/1.54 = ( ( X1 = Y1 )
% 2.64/1.54 & ( X22 = Y2 ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_6_old_Oprod_Oinject,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.64/1.54 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.64/1.54 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.64/1.54 = ( ( A2 = A3 )
% 2.64/1.54 & ( B2 = B3 ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_7_top__apply,axiom,
% 2.64/1.54 ! [C: $tType,D: $tType] :
% 2.64/1.54 ( ( top @ C )
% 2.64/1.54 => ( ( top_top @ ( D > C ) )
% 2.64/1.54 = ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_8_dist__commute__lessI,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A )
% 2.64/1.54 => ! [Y: A,X2: A,E: real] :
% 2.64/1.54 ( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X2 ) @ E )
% 2.64/1.54 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ E ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_9_top__prod__def,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType] :
% 2.64/1.54 ( ( ( top @ B )
% 2.64/1.54 & ( top @ A ) )
% 2.64/1.54 => ( ( top_top @ ( product_prod @ A @ B ) )
% 2.64/1.54 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_10_top_Oextremum__strict,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order_top @ A )
% 2.64/1.54 => ! [A2: A] :
% 2.64/1.54 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_11_top_Onot__eq__extremum,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order_top @ A )
% 2.64/1.54 => ! [A2: A] :
% 2.64/1.54 ( ( A2
% 2.64/1.54 != ( top_top @ A ) )
% 2.64/1.54 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_12_epigraph__mono,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A )
% 2.64/1.54 => ! [X2: A,Y: real,F: A > extended_ereal,Z: real] :
% 2.64/1.54 ( ( ( member @ ( product_prod @ A @ real ) @ ( product_Pair @ A @ real @ X2 @ Y ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) )
% 2.64/1.54 & ( ord_less_eq @ real @ Y @ Z ) )
% 2.64/1.54 => ( member @ ( product_prod @ A @ real ) @ ( product_Pair @ A @ real @ X2 @ Z ) @ ( lower_261390618igraph @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_13_metric__eq__thm,axiom,
% 2.64/1.54 ! [A4: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A4 )
% 2.64/1.54 => ! [X4: A4,S: set @ A4,Y3: A4] :
% 2.64/1.54 ( ( member @ A4 @ X4 @ S )
% 2.64/1.54 => ( ( member @ A4 @ Y3 @ S )
% 2.64/1.54 => ( ( X4 = Y3 )
% 2.64/1.54 = ( ! [X3: A4] :
% 2.64/1.54 ( ( member @ A4 @ X3 @ S )
% 2.64/1.54 => ( ( real_V2000881966t_dist @ A4 @ X4 @ X3 )
% 2.64/1.54 = ( real_V2000881966t_dist @ A4 @ Y3 @ X3 ) ) ) ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_14_order__refl,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( preorder @ A )
% 2.64/1.54 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_15_Pair__le,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType] :
% 2.64/1.54 ( ( ( ord @ B )
% 2.64/1.54 & ( ord @ A ) )
% 2.64/1.54 => ! [A2: A,B2: B,C2: A,D2: B] :
% 2.64/1.54 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D2 ) )
% 2.64/1.54 = ( ( ord_less_eq @ A @ A2 @ C2 )
% 2.64/1.54 & ( ord_less_eq @ B @ B2 @ D2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_16_dist__eq__0__iff,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ( ( real_V2000881966t_dist @ A @ X2 @ Y )
% 2.64/1.54 = ( zero_zero @ real ) )
% 2.64/1.54 = ( X2 = Y ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_17_dist__self,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A )
% 2.64/1.54 => ! [X2: A] :
% 2.64/1.54 ( ( real_V2000881966t_dist @ A @ X2 @ X2 )
% 2.64/1.54 = ( zero_zero @ real ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_18_dist__le__zero__iff,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ ( zero_zero @ real ) )
% 2.64/1.54 = ( X2 = Y ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_19_zero__less__dist__iff,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( real_V2090557954_space @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) )
% 2.64/1.54 = ( X2 != Y ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_20_le__funD,axiom,
% 2.64/1.54 ! [B: $tType,A: $tType] :
% 2.64/1.54 ( ( ord @ B )
% 2.64/1.54 => ! [F: A > B,G: A > B,X2: A] :
% 2.64/1.54 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.64/1.54 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_21_le__funE,axiom,
% 2.64/1.54 ! [B: $tType,A: $tType] :
% 2.64/1.54 ( ( ord @ B )
% 2.64/1.54 => ! [F: A > B,G: A > B,X2: A] :
% 2.64/1.54 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.64/1.54 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_22_le__funI,axiom,
% 2.64/1.54 ! [B: $tType,A: $tType] :
% 2.64/1.54 ( ( ord @ B )
% 2.64/1.54 => ! [F: A > B,G: A > B] :
% 2.64/1.54 ( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
% 2.64/1.54 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_23_le__fun__def,axiom,
% 2.64/1.54 ! [B: $tType,A: $tType] :
% 2.64/1.54 ( ( ord @ B )
% 2.64/1.54 => ( ( ord_less_eq @ ( A > B ) )
% 2.64/1.54 = ( ^ [F2: A > B,G2: A > B] :
% 2.64/1.54 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_24_order__subst1,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType] :
% 2.64/1.54 ( ( ( order @ B )
% 2.64/1.54 & ( order @ A ) )
% 2.64/1.54 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.64/1.54 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.64/1.54 => ( ! [X: B,Y4: B] :
% 2.64/1.54 ( ( ord_less_eq @ B @ X @ Y4 )
% 2.64/1.54 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.54 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_25_order__subst2,axiom,
% 2.64/1.54 ! [A: $tType,C: $tType] :
% 2.64/1.54 ( ( ( order @ C )
% 2.64/1.54 & ( order @ A ) )
% 2.64/1.54 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.54 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.64/1.54 => ( ! [X: A,Y4: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.64/1.54 => ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.54 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_26_Pair__mono,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType] :
% 2.64/1.54 ( ( ( ord @ B )
% 2.64/1.54 & ( ord @ A ) )
% 2.64/1.54 => ! [X2: A,X5: A,Y: B,Y5: B] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X2 @ X5 )
% 2.64/1.54 => ( ( ord_less_eq @ B @ Y @ Y5 )
% 2.64/1.54 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_27_ord__eq__le__subst,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType] :
% 2.64/1.54 ( ( ( ord @ B )
% 2.64/1.54 & ( ord @ A ) )
% 2.64/1.54 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.64/1.54 ( ( A2
% 2.64/1.54 = ( F @ B2 ) )
% 2.64/1.54 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.64/1.54 => ( ! [X: B,Y4: B] :
% 2.64/1.54 ( ( ord_less_eq @ B @ X @ Y4 )
% 2.64/1.54 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.54 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_28_ord__le__eq__subst,axiom,
% 2.64/1.54 ! [A: $tType,B: $tType] :
% 2.64/1.54 ( ( ( ord @ B )
% 2.64/1.54 & ( ord @ A ) )
% 2.64/1.54 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.54 => ( ( ( F @ B2 )
% 2.64/1.54 = C2 )
% 2.64/1.54 => ( ! [X: A,Y4: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.64/1.54 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.54 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_29_eq__iff,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 2.64/1.54 = ( ^ [X3: A,Y7: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 2.64/1.54 & ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_30_antisym,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.54 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.64/1.54 => ( X2 = Y ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_31_linear,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( linorder @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.54 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_32_eq__refl,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( preorder @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ( X2 = Y )
% 2.64/1.54 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_33_le__cases,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( linorder @ A )
% 2.64/1.54 => ! [X2: A,Y: A] :
% 2.64/1.54 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.54 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_34_order_Otrans,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.54 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.64/1.54 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_35_le__cases3,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( linorder @ A )
% 2.64/1.54 => ! [X2: A,Y: A,Z: A] :
% 2.64/1.54 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.54 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.64/1.54 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.64/1.54 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 2.64/1.54 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 2.64/1.54 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.64/1.54 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.64/1.54 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.64/1.54 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.64/1.54 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 2.64/1.54 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 2.64/1.54 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_36_antisym__conv,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ! [Y: A,X2: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.64/1.54 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.54 = ( X2 = Y ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_37_order__class_Oorder_Oeq__iff,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 2.64/1.54 = ( ^ [A5: A,B4: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.64/1.54 & ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_38_ord__eq__le__trans,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( ord @ A )
% 2.64/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.54 ( ( A2 = B2 )
% 2.64/1.54 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.64/1.54 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_39_ord__le__eq__trans,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( ord @ A )
% 2.64/1.54 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.54 => ( ( B2 = C2 )
% 2.64/1.54 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_40_order__class_Oorder_Oantisym,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ! [A2: A,B2: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.54 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.64/1.54 => ( A2 = B2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_41_order__trans,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( preorder @ A )
% 2.64/1.54 => ! [X2: A,Y: A,Z: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.54 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.64/1.54 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_42_dual__order_Orefl,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( order @ A )
% 2.64/1.54 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_43_linorder__wlog,axiom,
% 2.64/1.54 ! [A: $tType] :
% 2.64/1.54 ( ( linorder @ A )
% 2.64/1.54 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.64/1.54 ( ! [A6: A,B5: A] :
% 2.64/1.54 ( ( ord_less_eq @ A @ A6 @ B5 )
% 2.64/1.54 => ( P @ A6 @ B5 ) )
% 2.64/1.54 => ( ! [A6: A,B5: A] :
% 2.64/1.54 ( ( P @ B5 @ A6 )
% 2.64/1.54 => ( P @ A6 @ B5 ) )
% 2.64/1.54 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.64/1.54
% 2.64/1.54 thf(fact_44_dual__order_Otrans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A,C2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.64/1.55 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.64/1.55 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 2.64/1.55 ! [A: $tType,A2: A,P: A > $o] :
% 2.64/1.55 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.64/1.55 = ( P @ A2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A] :
% 2.64/1.55 ( ( collect @ A
% 2.64/1.55 @ ^ [X3: A] : ( member @ A @ X3 @ A7 ) )
% 2.64/1.55 = A7 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_47_Collect__cong,axiom,
% 2.64/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( P @ X )
% 2.64/1.55 = ( Q @ X ) )
% 2.64/1.55 => ( ( collect @ A @ P )
% 2.64/1.55 = ( collect @ A @ Q ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_48_ext,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( F @ X )
% 2.64/1.55 = ( G @ X ) )
% 2.64/1.55 => ( F = G ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_49_dual__order_Oeq__iff,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 2.64/1.55 = ( ^ [A5: A,B4: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.64/1.55 & ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_50_dual__order_Oantisym,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.64/1.55 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.55 => ( A2 = B2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_51_zero__le__dist,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ! [X2: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_52_top__set__def,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( top_top @ ( set @ A ) )
% 2.64/1.55 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_53_order_Onot__eq__order__implies__strict,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A] :
% 2.64/1.55 ( ( A2 != B2 )
% 2.64/1.55 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_54_dual__order_Ostrict__implies__order,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A] :
% 2.64/1.55 ( ( ord_less @ A @ B2 @ A2 )
% 2.64/1.55 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_55_dual__order_Ostrict__iff__order,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ord_less @ A )
% 2.64/1.55 = ( ^ [B4: A,A5: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.64/1.55 & ( A5 != B4 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_56_dual__order_Oorder__iff__strict,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ord_less_eq @ A )
% 2.64/1.55 = ( ^ [B4: A,A5: A] :
% 2.64/1.55 ( ( ord_less @ A @ B4 @ A5 )
% 2.64/1.55 | ( A5 = B4 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_57_order_Ostrict__implies__order,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_58_dense__le__bounded,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( dense_linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ! [W: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ W )
% 2.64/1.55 => ( ( ord_less @ A @ W @ Y )
% 2.64/1.55 => ( ord_less_eq @ A @ W @ Z ) ) )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_59_dense__ge__bounded,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( dense_linorder @ A )
% 2.64/1.55 => ! [Z: A,X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ Z @ X2 )
% 2.64/1.55 => ( ! [W: A] :
% 2.64/1.55 ( ( ord_less @ A @ Z @ W )
% 2.64/1.55 => ( ( ord_less @ A @ W @ X2 )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_60_dual__order_Ostrict__trans2,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A,C2: A] :
% 2.64/1.55 ( ( ord_less @ A @ B2 @ A2 )
% 2.64/1.55 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.64/1.55 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_61_dual__order_Ostrict__trans1,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A,C2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.64/1.55 => ( ( ord_less @ A @ C2 @ B2 )
% 2.64/1.55 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_62_order_Ostrict__iff__order,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ord_less @ A )
% 2.64/1.55 = ( ^ [A5: A,B4: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.64/1.55 & ( A5 != B4 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_63_order_Oorder__iff__strict,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ord_less_eq @ A )
% 2.64/1.55 = ( ^ [A5: A,B4: A] :
% 2.64/1.55 ( ( ord_less @ A @ A5 @ B4 )
% 2.64/1.55 | ( A5 = B4 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_64_order_Ostrict__trans2,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_65_order_Ostrict__trans1,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ord_less @ A @ B2 @ C2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_66_not__le__imp__less,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [Y: A,X2: A] :
% 2.64/1.55 ( ~ ( ord_less_eq @ A @ Y @ X2 )
% 2.64/1.55 => ( ord_less @ A @ X2 @ Y ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_67_less__le__not__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ( ( ord_less @ A )
% 2.64/1.55 = ( ^ [X3: A,Y7: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 2.64/1.55 & ~ ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_68_le__imp__less__or__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 | ( X2 = Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_69_le__less__linear,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.55 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_70_dense__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( dense_linorder @ A )
% 2.64/1.55 => ! [Y: A,Z: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( ord_less @ A @ X @ Y )
% 2.64/1.55 => ( ord_less_eq @ A @ X @ Z ) )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_71_dense__ge,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( dense_linorder @ A )
% 2.64/1.55 => ! [Z: A,Y: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( ord_less @ A @ Z @ X )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ X ) )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_72_less__le__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.64/1.55 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_73_le__less__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ord_less @ A @ Y @ Z )
% 2.64/1.55 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_74_less__imp__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_75_antisym__conv2,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.64/1.55 = ( X2 = Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_76_antisym__conv1,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.64/1.55 = ( X2 = Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_77_le__neq__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( A2 != B2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_78_not__less,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.64/1.55 = ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_79_not__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
% 2.64/1.55 = ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_80_order__less__le__subst2,axiom,
% 2.64/1.55 ! [A: $tType,C: $tType] :
% 2.64/1.55 ( ( ( order @ C )
% 2.64/1.55 & ( order @ A ) )
% 2.64/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.64/1.55 => ( ! [X: A,Y4: A] :
% 2.64/1.55 ( ( ord_less @ A @ X @ Y4 )
% 2.64/1.55 => ( ord_less @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_81_order__less__le__subst1,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( ( order @ B )
% 2.64/1.55 & ( order @ A ) )
% 2.64/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.64/1.55 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.64/1.55 => ( ! [X: B,Y4: B] :
% 2.64/1.55 ( ( ord_less_eq @ B @ X @ Y4 )
% 2.64/1.55 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_82_order__le__less__subst2,axiom,
% 2.64/1.55 ! [A: $tType,C: $tType] :
% 2.64/1.55 ( ( ( order @ C )
% 2.64/1.55 & ( order @ A ) )
% 2.64/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.64/1.55 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 2.64/1.55 => ( ! [X: A,Y4: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.64/1.55 => ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_83_order__le__less__subst1,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( ( order @ B )
% 2.64/1.55 & ( order @ A ) )
% 2.64/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.64/1.55 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.64/1.55 => ( ( ord_less @ B @ B2 @ C2 )
% 2.64/1.55 => ( ! [X: B,Y4: B] :
% 2.64/1.55 ( ( ord_less @ B @ X @ Y4 )
% 2.64/1.55 => ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_84_less__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ord_less @ A )
% 2.64/1.55 = ( ^ [X3: A,Y7: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 2.64/1.55 & ( X3 != Y7 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_85_le__less,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ( ( ord_less_eq @ A )
% 2.64/1.55 = ( ^ [X3: A,Y7: A] :
% 2.64/1.55 ( ( ord_less @ A @ X3 @ Y7 )
% 2.64/1.55 | ( X3 = Y7 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_86_leI,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_87_leD,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [Y: A,X2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.64/1.55 => ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_88_top_Oextremum__uniqueI,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order_top @ A )
% 2.64/1.55 => ! [A2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.64/1.55 => ( A2
% 2.64/1.55 = ( top_top @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_89_top_Oextremum__unique,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order_top @ A )
% 2.64/1.55 => ! [A2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.64/1.55 = ( A2
% 2.64/1.55 = ( top_top @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_90_top__greatest,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order_top @ A )
% 2.64/1.55 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_91_dist__not__less__zero,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ~ ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ Y ) @ ( zero_zero @ real ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_92_dist__pos__lt,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( X2 != Y )
% 2.64/1.55 => ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X2 @ Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_93_dual__order_Ostrict__implies__not__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A] :
% 2.64/1.55 ( ( ord_less @ A @ B2 @ A2 )
% 2.64/1.55 => ( A2 != B2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_94_order_Ostrict__implies__not__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( A2 != B2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_95_not__less__iff__gr__or__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.64/1.55 = ( ( ord_less @ A @ Y @ X2 )
% 2.64/1.55 | ( X2 = Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_96_dual__order_Ostrict__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A,C2: A] :
% 2.64/1.55 ( ( ord_less @ A @ B2 @ A2 )
% 2.64/1.55 => ( ( ord_less @ A @ C2 @ B2 )
% 2.64/1.55 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_97_linorder__less__wlog,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.64/1.55 ( ! [A6: A,B5: A] :
% 2.64/1.55 ( ( ord_less @ A @ A6 @ B5 )
% 2.64/1.55 => ( P @ A6 @ B5 ) )
% 2.64/1.55 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 2.64/1.55 => ( ! [A6: A,B5: A] :
% 2.64/1.55 ( ( P @ B5 @ A6 )
% 2.64/1.55 => ( P @ A6 @ B5 ) )
% 2.64/1.55 => ( P @ A2 @ B2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_98_exists__least__iff,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( wellorder @ A )
% 2.64/1.55 => ( ( ^ [P2: A > $o] :
% 2.64/1.55 ? [X6: A] : ( P2 @ X6 ) )
% 2.64/1.55 = ( ^ [P3: A > $o] :
% 2.64/1.55 ? [N: A] :
% 2.64/1.55 ( ( P3 @ N )
% 2.64/1.55 & ! [M: A] :
% 2.64/1.55 ( ( ord_less @ A @ M @ N )
% 2.64/1.55 => ~ ( P3 @ M ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_99_less__imp__not__less,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_100_order_Ostrict__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ord_less @ A @ B2 @ C2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_101_dual__order_Oirrefl,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A] :
% 2.64/1.55 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_102_linorder__cases,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ( X2 != Y )
% 2.64/1.55 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_103_less__imp__triv,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A,P: $o] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ord_less @ A @ Y @ X2 )
% 2.64/1.55 => P ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_104_less__imp__not__eq2,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( Y != X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_105_antisym__conv3,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [Y: A,X2: A] :
% 2.64/1.55 ( ~ ( ord_less @ A @ Y @ X2 )
% 2.64/1.55 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.64/1.55 = ( X2 = Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_106_less__induct,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( wellorder @ A )
% 2.64/1.55 => ! [P: A > $o,A2: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ! [Y8: A] :
% 2.64/1.55 ( ( ord_less @ A @ Y8 @ X )
% 2.64/1.55 => ( P @ Y8 ) )
% 2.64/1.55 => ( P @ X ) )
% 2.64/1.55 => ( P @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_107_less__not__sym,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_108_less__imp__not__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( X2 != Y ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_109_dual__order_Oasym,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [B2: A,A2: A] :
% 2.64/1.55 ( ( ord_less @ A @ B2 @ A2 )
% 2.64/1.55 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_110_ord__less__eq__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord @ A )
% 2.64/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( B2 = C2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_111_ord__eq__less__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord @ A )
% 2.64/1.55 => ! [A2: A,B2: A,C2: A] :
% 2.64/1.55 ( ( A2 = B2 )
% 2.64/1.55 => ( ( ord_less @ A @ B2 @ C2 )
% 2.64/1.55 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_112_less__irrefl,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A] :
% 2.64/1.55 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_113_less__linear,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 | ( X2 = Y )
% 2.64/1.55 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_114_less__trans,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A,Z: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ( ord_less @ A @ Y @ Z )
% 2.64/1.55 => ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_115_less__asym_H,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [A2: A,B2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_116_less__asym,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( preorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_117_less__imp__neq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( X2 != Y ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_118_dense,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( dense_order @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ? [Z3: A] :
% 2.64/1.55 ( ( ord_less @ A @ X2 @ Z3 )
% 2.64/1.55 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_119_order_Oasym,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( order @ A )
% 2.64/1.55 => ! [A2: A,B2: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_120_neq__iff,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( X2 != Y )
% 2.64/1.55 = ( ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_121_neqE,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [X2: A,Y: A] :
% 2.64/1.55 ( ( X2 != Y )
% 2.64/1.55 => ( ~ ( ord_less @ A @ X2 @ Y )
% 2.64/1.55 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_122_gt__ex,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( no_top @ A )
% 2.64/1.55 => ! [X2: A] :
% 2.64/1.55 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_123_lt__ex,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( no_bot @ A )
% 2.64/1.55 => ! [X2: A] :
% 2.64/1.55 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_124_order__less__subst2,axiom,
% 2.64/1.55 ! [A: $tType,C: $tType] :
% 2.64/1.55 ( ( ( order @ C )
% 2.64/1.55 & ( order @ A ) )
% 2.64/1.55 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 2.64/1.55 => ( ! [X: A,Y4: A] :
% 2.64/1.55 ( ( ord_less @ A @ X @ Y4 )
% 2.64/1.55 => ( ord_less @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_125_order__less__subst1,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( ( order @ B )
% 2.64/1.55 & ( order @ A ) )
% 2.64/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.64/1.55 => ( ( ord_less @ B @ B2 @ C2 )
% 2.64/1.55 => ( ! [X: B,Y4: B] :
% 2.64/1.55 ( ( ord_less @ B @ X @ Y4 )
% 2.64/1.55 => ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_126_ord__less__eq__subst,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( ( ord @ B )
% 2.64/1.55 & ( ord @ A ) )
% 2.64/1.55 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( ( F @ B2 )
% 2.64/1.55 = C2 )
% 2.64/1.55 => ( ! [X: A,Y4: A] :
% 2.64/1.55 ( ( ord_less @ A @ X @ Y4 )
% 2.64/1.55 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_127_ord__eq__less__subst,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( ( ord @ B )
% 2.64/1.55 & ( ord @ A ) )
% 2.64/1.55 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.64/1.55 ( ( A2
% 2.64/1.55 = ( F @ B2 ) )
% 2.64/1.55 => ( ( ord_less @ B @ B2 @ C2 )
% 2.64/1.55 => ( ! [X: B,Y4: B] :
% 2.64/1.55 ( ( ord_less @ B @ X @ Y4 )
% 2.64/1.55 => ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.64/1.55 => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_128_old_Oprod_Oinducts,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.64/1.55 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 2.64/1.55 => ( P @ Prod ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_129_old_Oprod_Oexhaust,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.64/1.55 ~ ! [A6: A,B5: B] :
% 2.64/1.55 ( Y
% 2.64/1.55 != ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_130_prod__induct7,axiom,
% 2.64/1.55 ! [G3: $tType,F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 2.64/1.55 ( ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_131_prod__induct6,axiom,
% 2.64/1.55 ! [F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 2.64/1.55 ( ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D3 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_132_prod__induct5,axiom,
% 2.64/1.55 ! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 2.64/1.55 ( ! [A6: A,B5: B,C3: C,D3: D,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_133_prod__induct4,axiom,
% 2.64/1.55 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.64/1.55 ( ! [A6: A,B5: B,C3: C,D3: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_134_prod__induct3,axiom,
% 2.64/1.55 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.64/1.55 ( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_135_prod__cases7,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 2.64/1.55 ~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3,G4: G3] :
% 2.64/1.55 ( Y
% 2.64/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_136_prod__cases6,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 2.64/1.55 ~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2,F4: F3] :
% 2.64/1.55 ( Y
% 2.64/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D3 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_137_prod__cases5,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 2.64/1.55 ~ ! [A6: A,B5: B,C3: C,D3: D,E3: E2] :
% 2.64/1.55 ( Y
% 2.64/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_138_prod__cases4,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.64/1.55 ~ ! [A6: A,B5: B,C3: C,D3: D] :
% 2.64/1.55 ( Y
% 2.64/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_139_prod__cases3,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.64/1.55 ~ ! [A6: A,B5: B,C3: C] :
% 2.64/1.55 ( Y
% 2.64/1.55 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_140_Pair__inject,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.64/1.55 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.64/1.55 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.64/1.55 => ~ ( ( A2 = A3 )
% 2.64/1.55 => ( B2 != B3 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_141_prod__cases,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
% 2.64/1.55 ( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
% 2.64/1.55 => ( P @ P4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_142_surj__pair,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
% 2.64/1.55 ? [X: A,Y4: B] :
% 2.64/1.55 ( P4
% 2.64/1.55 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_143_UNIV__witness,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_144_UNIV__eq__I,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A] :
% 2.64/1.55 ( ! [X: A] : ( member @ A @ X @ A7 )
% 2.64/1.55 => ( ( top_top @ ( set @ A ) )
% 2.64/1.55 = A7 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_145_dist__commute,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ( ( real_V2000881966t_dist @ A )
% 2.64/1.55 = ( ^ [X3: A,Y7: A] : ( real_V2000881966t_dist @ A @ Y7 @ X3 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_146_closure__UNIV,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( topolo503727757_space @ A )
% 2.64/1.55 => ( ( elementary_closure @ A @ ( top_top @ ( set @ A ) ) )
% 2.64/1.55 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_147_closure__approachable__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ! [X2: A,S2: set @ A] :
% 2.64/1.55 ( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
% 2.64/1.55 = ( ! [E4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 2.64/1.55 => ? [X3: A] :
% 2.64/1.55 ( ( member @ A @ X3 @ S2 )
% 2.64/1.55 & ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X3 @ X2 ) @ E4 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_148_not__gr__zero,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [N2: A] :
% 2.64/1.55 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
% 2.64/1.55 = ( N2
% 2.64/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_149_le__zero__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [N2: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
% 2.64/1.55 = ( N2
% 2.64/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_150_closure__approachableD,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ! [X2: A,S2: set @ A,E: real] :
% 2.64/1.55 ( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
% 2.64/1.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
% 2.64/1.55 => ? [X: A] :
% 2.64/1.55 ( ( member @ A @ X @ S2 )
% 2.64/1.55 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X2 @ X ) @ E ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_151_closure__approachable,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ! [X2: A,S2: set @ A] :
% 2.64/1.55 ( ( member @ A @ X2 @ ( elementary_closure @ A @ S2 ) )
% 2.64/1.55 = ( ! [E4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 2.64/1.55 => ? [X3: A] :
% 2.64/1.55 ( ( member @ A @ X3 @ S2 )
% 2.64/1.55 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X3 @ X2 ) @ E4 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_152_old_Oprod_Orec,axiom,
% 2.64/1.55 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.64/1.55 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.64/1.55 = ( F1 @ A2 @ B2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_153_perfect__choose__dist,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ( real_V2090557954_space @ A )
% 2.64/1.55 & ( topolo890362671_space @ A ) )
% 2.64/1.55 => ! [R: real,X2: A] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 2.64/1.55 => ? [A6: A] :
% 2.64/1.55 ( ( A6 != X2 )
% 2.64/1.55 & ( ord_less @ real @ ( real_V2000881966t_dist @ A @ A6 @ X2 ) @ R ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_154_vector__choose__dist,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ( real_V55928688vector @ A )
% 2.64/1.55 & ( topolo890362671_space @ A ) )
% 2.64/1.55 => ! [C2: real,X2: A] :
% 2.64/1.55 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
% 2.64/1.55 => ~ ! [Y4: A] :
% 2.64/1.55 ( ( real_V2000881966t_dist @ A @ X2 @ Y4 )
% 2.64/1.55 != C2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_155_approachable__lt__le2,axiom,
% 2.64/1.55 ! [A: $tType,Q: A > $o,F: A > real,P: A > $o] :
% 2.64/1.55 ( ( ? [D4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 2.64/1.55 & ! [X3: A] :
% 2.64/1.55 ( ( Q @ X3 )
% 2.64/1.55 => ( ( ord_less @ real @ ( F @ X3 ) @ D4 )
% 2.64/1.55 => ( P @ X3 ) ) ) ) )
% 2.64/1.55 = ( ? [D4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
% 2.64/1.55 & ! [X3: A] :
% 2.64/1.55 ( ( ord_less_eq @ real @ ( F @ X3 ) @ D4 )
% 2.64/1.55 => ( ( Q @ X3 )
% 2.64/1.55 => ( P @ X3 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_156_closure__closure,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( topolo503727757_space @ A )
% 2.64/1.55 => ! [S2: set @ A] :
% 2.64/1.55 ( ( elementary_closure @ A @ ( elementary_closure @ A @ S2 ) )
% 2.64/1.55 = ( elementary_closure @ A @ S2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_157_top1I,axiom,
% 2.64/1.55 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_158_less__fun__def,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( ord @ B )
% 2.64/1.55 => ( ( ord_less @ ( A > B ) )
% 2.64/1.55 = ( ^ [F2: A > B,G2: A > B] :
% 2.64/1.55 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.64/1.55 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_159_less__prod__def,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( ( ord @ A )
% 2.64/1.55 & ( ord @ B ) )
% 2.64/1.55 => ( ( ord_less @ ( product_prod @ A @ B ) )
% 2.64/1.55 = ( ^ [X3: product_prod @ A @ B,Y7: product_prod @ A @ B] :
% 2.64/1.55 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X3 @ Y7 )
% 2.64/1.55 & ~ ( ord_less_eq @ ( product_prod @ A @ B ) @ Y7 @ X3 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_160_closure__subset,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( topolo503727757_space @ A )
% 2.64/1.55 => ! [S2: set @ A] : ( ord_less_eq @ ( set @ A ) @ S2 @ ( elementary_closure @ A @ S2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_161_closure__mono,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( topolo503727757_space @ A )
% 2.64/1.55 => ! [S2: set @ A,T2: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T2 )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ ( elementary_closure @ A @ S2 ) @ ( elementary_closure @ A @ T2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_162_subset__UNIV,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_163_zero__reorient,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( zero @ A )
% 2.64/1.55 => ! [X2: A] :
% 2.64/1.55 ( ( ( zero_zero @ A )
% 2.64/1.55 = X2 )
% 2.64/1.55 = ( X2
% 2.64/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_164_zero__le,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_165_gr__zeroI,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [N2: A] :
% 2.64/1.55 ( ( N2
% 2.64/1.55 != ( zero_zero @ A ) )
% 2.64/1.55 => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_166_not__less__zero,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [N2: A] :
% 2.64/1.55 ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_167_gr__implies__not__zero,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [M2: A,N2: A] :
% 2.64/1.55 ( ( ord_less @ A @ M2 @ N2 )
% 2.64/1.55 => ( N2
% 2.64/1.55 != ( zero_zero @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_168_zero__less__iff__neq__zero,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( canoni770627133id_add @ A )
% 2.64/1.55 => ! [N2: A] :
% 2.64/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
% 2.64/1.55 = ( N2
% 2.64/1.55 != ( zero_zero @ A ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_169_uniformly__convergent__eq__cauchy,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( real_V1759440057_space @ A )
% 2.64/1.55 => ! [P: B > $o,S3: nat > B > A] :
% 2.64/1.55 ( ( ? [L: B > A] :
% 2.64/1.55 ! [E4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 2.64/1.55 => ? [N3: nat] :
% 2.64/1.55 ! [N: nat,X3: B] :
% 2.64/1.55 ( ( ( ord_less_eq @ nat @ N3 @ N )
% 2.64/1.55 & ( P @ X3 ) )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ ( S3 @ N @ X3 ) @ ( L @ X3 ) ) @ E4 ) ) ) )
% 2.64/1.55 = ( ! [E4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 2.64/1.55 => ? [N3: nat] :
% 2.64/1.55 ! [M: nat,N: nat,X3: B] :
% 2.64/1.55 ( ( ( ord_less_eq @ nat @ N3 @ M )
% 2.64/1.55 & ( ord_less_eq @ nat @ N3 @ N )
% 2.64/1.55 & ( P @ X3 ) )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ A @ ( S3 @ M @ X3 ) @ ( S3 @ N @ X3 ) ) @ E4 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_170_uniformly__cauchy__imp__uniformly__convergent,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( real_V1759440057_space @ B )
% 2.64/1.55 => ! [P: A > $o,S3: nat > A > B,L2: A > B] :
% 2.64/1.55 ( ! [E3: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 2.64/1.55 => ? [N4: nat] :
% 2.64/1.55 ! [M3: nat,N5: nat,X: A] :
% 2.64/1.55 ( ( ( ord_less_eq @ nat @ N4 @ M3 )
% 2.64/1.55 & ( ord_less_eq @ nat @ N4 @ N5 )
% 2.64/1.55 & ( P @ X ) )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ M3 @ X ) @ ( S3 @ N5 @ X ) ) @ E3 ) ) )
% 2.64/1.55 => ( ! [X: A] :
% 2.64/1.55 ( ( P @ X )
% 2.64/1.55 => ! [E3: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 2.64/1.55 => ? [N4: nat] :
% 2.64/1.55 ! [N5: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ N4 @ N5 )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ N5 @ X ) @ ( L2 @ X ) ) @ E3 ) ) ) )
% 2.64/1.55 => ! [E5: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E5 )
% 2.64/1.55 => ? [N6: nat] :
% 2.64/1.55 ! [N7: nat,X7: A] :
% 2.64/1.55 ( ( ( ord_less_eq @ nat @ N6 @ N7 )
% 2.64/1.55 & ( P @ X7 ) )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( S3 @ N7 @ X7 ) @ ( L2 @ X7 ) ) @ E5 ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_171_top__empty__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( top_top @ ( A > $o ) )
% 2.64/1.55 = ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_172_subsetI,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( member @ A @ X @ A7 )
% 2.64/1.55 => ( member @ A @ X @ B6 ) )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_173_psubsetI,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( A7 != B6 )
% 2.64/1.55 => ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_174_subset__antisym,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A7 )
% 2.64/1.55 => ( A7 = B6 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_175_in__mono,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,X2: A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( member @ A @ X2 @ A7 )
% 2.64/1.55 => ( member @ A @ X2 @ B6 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_176_subsetD,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,C2: A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( member @ A @ C2 @ A7 )
% 2.64/1.55 => ( member @ A @ C2 @ B6 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_177_psubsetE,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_178_equalityE,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( A7 = B6 )
% 2.64/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_179_subset__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.64/1.55 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.64/1.55 ! [X3: A] :
% 2.64/1.55 ( ( member @ A @ X3 @ A8 )
% 2.64/1.55 => ( member @ A @ X3 @ B7 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_180_equalityD1,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( A7 = B6 )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_181_equalityD2,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( A7 = B6 )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_182_psubset__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) )
% 2.64/1.55 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 2.64/1.55 & ( A8 != B7 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_183_subset__iff,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.64/1.55 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.64/1.55 ! [T3: A] :
% 2.64/1.55 ( ( member @ A @ T3 @ A8 )
% 2.64/1.55 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_184_subset__refl,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ A7 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_185_Collect__mono,axiom,
% 2.64/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( P @ X )
% 2.64/1.55 => ( Q @ X ) )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_186_subset__trans,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_187_set__eq__subset,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ^ [Y6: set @ A,Z2: set @ A] : ( Y6 = Z2 ) )
% 2.64/1.55 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 2.64/1.55 & ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_188_Collect__mono__iff,axiom,
% 2.64/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.64/1.55 = ( ! [X3: A] :
% 2.64/1.55 ( ( P @ X3 )
% 2.64/1.55 => ( Q @ X3 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_189_psubset__imp__subset,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ord_less_eq @ ( set @ A ) @ A7 @ B6 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_190_psubset__subset__trans,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.64/1.55 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_191_subset__not__subset__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) )
% 2.64/1.55 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
% 2.64/1.55 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_192_subset__psubset__trans,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 2.64/1.55 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_193_subset__iff__psubset__eq,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.64/1.55 = ( ^ [A8: set @ A,B7: set @ A] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) @ A8 @ B7 )
% 2.64/1.55 | ( A8 = B7 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_194_subrelI,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.64/1.55 ( ! [X: A,Y4: B] :
% 2.64/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
% 2.64/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S3 ) )
% 2.64/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_195_seq__mono__lemma,axiom,
% 2.64/1.55 ! [M2: nat,D2: nat > real,E: nat > real] :
% 2.64/1.55 ( ! [N5: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M2 @ N5 )
% 2.64/1.55 => ( ord_less @ real @ ( D2 @ N5 ) @ ( E @ N5 ) ) )
% 2.64/1.55 => ( ! [N5: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M2 @ N5 )
% 2.64/1.55 => ( ord_less_eq @ real @ ( E @ N5 ) @ ( E @ M2 ) ) )
% 2.64/1.55 => ! [N7: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M2 @ N7 )
% 2.64/1.55 => ( ord_less @ real @ ( D2 @ N7 ) @ ( E @ M2 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_196_le0,axiom,
% 2.64/1.55 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_197_bot__nat__0_Oextremum,axiom,
% 2.64/1.55 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_198_neq0__conv,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ( ( N2
% 2.64/1.55 != ( zero_zero @ nat ) )
% 2.64/1.55 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_199_less__nat__zero__code,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_200_bot__nat__0_Onot__eq__extremum,axiom,
% 2.64/1.55 ! [A2: nat] :
% 2.64/1.55 ( ( A2
% 2.64/1.55 != ( zero_zero @ nat ) )
% 2.64/1.55 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_201_gr0I,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ( ( N2
% 2.64/1.55 != ( zero_zero @ nat ) )
% 2.64/1.55 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_202_not__gr0,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
% 2.64/1.55 = ( N2
% 2.64/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_203_not__less0,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_204_less__zeroE,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_205_gr__implies__not0,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M2 @ N2 )
% 2.64/1.55 => ( N2
% 2.64/1.55 != ( zero_zero @ nat ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_206_infinite__descent0,axiom,
% 2.64/1.55 ! [P: nat > $o,N2: nat] :
% 2.64/1.55 ( ( P @ ( zero_zero @ nat ) )
% 2.64/1.55 => ( ! [N5: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
% 2.64/1.55 => ( ~ ( P @ N5 )
% 2.64/1.55 => ? [M4: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M4 @ N5 )
% 2.64/1.55 & ~ ( P @ M4 ) ) ) )
% 2.64/1.55 => ( P @ N2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_207_bot__nat__0_Oextremum__strict,axiom,
% 2.64/1.55 ! [A2: nat] :
% 2.64/1.55 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_208_infinite__descent0__measure,axiom,
% 2.64/1.55 ! [A: $tType,V: A > nat,P: A > $o,X2: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( ( V @ X )
% 2.64/1.55 = ( zero_zero @ nat ) )
% 2.64/1.55 => ( P @ X ) )
% 2.64/1.55 => ( ! [X: A] :
% 2.64/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
% 2.64/1.55 => ( ~ ( P @ X )
% 2.64/1.55 => ? [Y8: A] :
% 2.64/1.55 ( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X ) )
% 2.64/1.55 & ~ ( P @ Y8 ) ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_209_psubset__trans,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,C4: set @ A] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 2.64/1.55 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_210_psubsetD,axiom,
% 2.64/1.55 ! [A: $tType,A7: set @ A,B6: set @ A,C2: A] :
% 2.64/1.55 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 2.64/1.55 => ( ( member @ A @ C2 @ A7 )
% 2.64/1.55 => ( member @ A @ C2 @ B6 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_211_measure__induct,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( wellorder @ B )
% 2.64/1.55 => ! [F: A > B,P: A > $o,A2: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ! [Y8: A] :
% 2.64/1.55 ( ( ord_less @ B @ ( F @ Y8 ) @ ( F @ X ) )
% 2.64/1.55 => ( P @ Y8 ) )
% 2.64/1.55 => ( P @ X ) )
% 2.64/1.55 => ( P @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_212_measure__induct__rule,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( wellorder @ B )
% 2.64/1.55 => ! [F: A > B,P: A > $o,A2: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ! [Y8: A] :
% 2.64/1.55 ( ( ord_less @ B @ ( F @ Y8 ) @ ( F @ X ) )
% 2.64/1.55 => ( P @ Y8 ) )
% 2.64/1.55 => ( P @ X ) )
% 2.64/1.55 => ( P @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_213_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.64/1.55 ! [A2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.64/1.55 => ( A2
% 2.64/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_214_bot__nat__0_Oextremum__unique,axiom,
% 2.64/1.55 ! [A2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.64/1.55 = ( A2
% 2.64/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_215_less__mono__imp__le__mono,axiom,
% 2.64/1.55 ! [F: nat > nat,I: nat,J: nat] :
% 2.64/1.55 ( ! [I2: nat,J2: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ I2 @ J2 )
% 2.64/1.55 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.64/1.55 => ( ( ord_less_eq @ nat @ I @ J )
% 2.64/1.55 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_216_le__neq__implies__less,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.64/1.55 => ( ( M2 != N2 )
% 2.64/1.55 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_217_Nat_Oex__has__greatest__nat,axiom,
% 2.64/1.55 ! [P: nat > $o,K: nat,B2: nat] :
% 2.64/1.55 ( ( P @ K )
% 2.64/1.55 => ( ! [Y4: nat] :
% 2.64/1.55 ( ( P @ Y4 )
% 2.64/1.55 => ( ord_less_eq @ nat @ Y4 @ B2 ) )
% 2.64/1.55 => ? [X: nat] :
% 2.64/1.55 ( ( P @ X )
% 2.64/1.55 & ! [Y8: nat] :
% 2.64/1.55 ( ( P @ Y8 )
% 2.64/1.55 => ( ord_less_eq @ nat @ Y8 @ X ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_218_less__or__eq__imp__le,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( ( ord_less @ nat @ M2 @ N2 )
% 2.64/1.55 | ( M2 = N2 ) )
% 2.64/1.55 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_219_le__eq__less__or__eq,axiom,
% 2.64/1.55 ( ( ord_less_eq @ nat )
% 2.64/1.55 = ( ^ [M: nat,N: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M @ N )
% 2.64/1.55 | ( M = N ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_220_less__imp__le__nat,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M2 @ N2 )
% 2.64/1.55 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_221_ex__least__nat__le,axiom,
% 2.64/1.55 ! [P: nat > $o,N2: nat] :
% 2.64/1.55 ( ( P @ N2 )
% 2.64/1.55 => ( ~ ( P @ ( zero_zero @ nat ) )
% 2.64/1.55 => ? [K2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ K2 @ N2 )
% 2.64/1.55 & ! [I3: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ I3 @ K2 )
% 2.64/1.55 => ~ ( P @ I3 ) )
% 2.64/1.55 & ( P @ K2 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_222_nat__le__linear,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.64/1.55 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_223_nat__less__le,axiom,
% 2.64/1.55 ( ( ord_less @ nat )
% 2.64/1.55 = ( ^ [M: nat,N: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.64/1.55 & ( M != N ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_224_le__antisym,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.64/1.55 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 2.64/1.55 => ( M2 = N2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_225_eq__imp__le,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( M2 = N2 )
% 2.64/1.55 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_226_le__trans,axiom,
% 2.64/1.55 ! [I: nat,J: nat,K: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ I @ J )
% 2.64/1.55 => ( ( ord_less_eq @ nat @ J @ K )
% 2.64/1.55 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_227_le__refl,axiom,
% 2.64/1.55 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_228_le__0__eq,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
% 2.64/1.55 = ( N2
% 2.64/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_229_less__eq__nat_Osimps_I1_J,axiom,
% 2.64/1.55 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_230_uniformly__Cauchy__onI,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ B )
% 2.64/1.55 => ! [X8: set @ A,F: nat > A > B] :
% 2.64/1.55 ( ! [E3: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 2.64/1.55 => ? [M5: nat] :
% 2.64/1.55 ! [X: A] :
% 2.64/1.55 ( ( member @ A @ X @ X8 )
% 2.64/1.55 => ! [M3: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M5 @ M3 )
% 2.64/1.55 => ! [N5: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M5 @ N5 )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F @ M3 @ X ) @ ( F @ N5 @ X ) ) @ E3 ) ) ) ) )
% 2.64/1.55 => ( unifor760693226chy_on @ A @ B @ X8 @ F ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_231_uniformly__Cauchy__on__def,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ B )
% 2.64/1.55 => ( ( unifor760693226chy_on @ A @ B )
% 2.64/1.55 = ( ^ [X9: set @ A,F2: nat > A > B] :
% 2.64/1.55 ! [E4: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
% 2.64/1.55 => ? [M6: nat] :
% 2.64/1.55 ! [X3: A] :
% 2.64/1.55 ( ( member @ A @ X3 @ X9 )
% 2.64/1.55 => ! [M: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M6 @ M )
% 2.64/1.55 => ! [N: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M6 @ N )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F2 @ M @ X3 ) @ ( F2 @ N @ X3 ) ) @ E4 ) ) ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_232_infinite__descent__measure,axiom,
% 2.64/1.55 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ~ ( P @ X )
% 2.64/1.55 => ? [Y8: A] :
% 2.64/1.55 ( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X ) )
% 2.64/1.55 & ~ ( P @ Y8 ) ) )
% 2.64/1.55 => ( P @ X2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_233_linorder__neqE__nat,axiom,
% 2.64/1.55 ! [X2: nat,Y: nat] :
% 2.64/1.55 ( ( X2 != Y )
% 2.64/1.55 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 2.64/1.55 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_234_infinite__descent,axiom,
% 2.64/1.55 ! [P: nat > $o,N2: nat] :
% 2.64/1.55 ( ! [N5: nat] :
% 2.64/1.55 ( ~ ( P @ N5 )
% 2.64/1.55 => ? [M4: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M4 @ N5 )
% 2.64/1.55 & ~ ( P @ M4 ) ) )
% 2.64/1.55 => ( P @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_235_nat__less__induct,axiom,
% 2.64/1.55 ! [P: nat > $o,N2: nat] :
% 2.64/1.55 ( ! [N5: nat] :
% 2.64/1.55 ( ! [M4: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M4 @ N5 )
% 2.64/1.55 => ( P @ M4 ) )
% 2.64/1.55 => ( P @ N5 ) )
% 2.64/1.55 => ( P @ N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_236_less__irrefl__nat,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_237_less__not__refl3,axiom,
% 2.64/1.55 ! [S3: nat,T4: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ S3 @ T4 )
% 2.64/1.55 => ( S3 != T4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_238_less__not__refl2,axiom,
% 2.64/1.55 ! [N2: nat,M2: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ N2 @ M2 )
% 2.64/1.55 => ( M2 != N2 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_239_less__not__refl,axiom,
% 2.64/1.55 ! [N2: nat] :
% 2.64/1.55 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_240_nat__neq__iff,axiom,
% 2.64/1.55 ! [M2: nat,N2: nat] :
% 2.64/1.55 ( ( M2 != N2 )
% 2.64/1.55 = ( ( ord_less @ nat @ M2 @ N2 )
% 2.64/1.55 | ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_241_uniformly__Cauchy__onI_H,axiom,
% 2.64/1.55 ! [B: $tType,A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ B )
% 2.64/1.55 => ! [X8: set @ A,F: nat > A > B] :
% 2.64/1.55 ( ! [E3: real] :
% 2.64/1.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
% 2.64/1.55 => ? [M5: nat] :
% 2.64/1.55 ! [X: A] :
% 2.64/1.55 ( ( member @ A @ X @ X8 )
% 2.64/1.55 => ! [M3: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ M5 @ M3 )
% 2.64/1.55 => ! [N5: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ M3 @ N5 )
% 2.64/1.55 => ( ord_less @ real @ ( real_V2000881966t_dist @ B @ ( F @ M3 @ X ) @ ( F @ N5 @ X ) ) @ E3 ) ) ) ) )
% 2.64/1.55 => ( unifor760693226chy_on @ A @ B @ X8 @ F ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_242_nat__descend__induct,axiom,
% 2.64/1.55 ! [N2: nat,P: nat > $o,M2: nat] :
% 2.64/1.55 ( ! [K2: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ N2 @ K2 )
% 2.64/1.55 => ( P @ K2 ) )
% 2.64/1.55 => ( ! [K2: nat] :
% 2.64/1.55 ( ( ord_less_eq @ nat @ K2 @ N2 )
% 2.64/1.55 => ( ! [I3: nat] :
% 2.64/1.55 ( ( ord_less @ nat @ K2 @ I3 )
% 2.64/1.55 => ( P @ I3 ) )
% 2.64/1.55 => ( P @ K2 ) ) )
% 2.64/1.55 => ( P @ M2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_243_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.64/1.55 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 2.64/1.55 ( ( P @ K )
% 2.64/1.55 => ( ! [Y4: A] :
% 2.64/1.55 ( ( P @ Y4 )
% 2.64/1.55 => ( ord_less @ nat @ ( F @ Y4 ) @ B2 ) )
% 2.64/1.55 => ? [X: A] :
% 2.64/1.55 ( ( P @ X )
% 2.64/1.55 & ! [Y8: A] :
% 2.64/1.55 ( ( P @ Y8 )
% 2.64/1.55 => ( ord_less_eq @ nat @ ( F @ Y8 ) @ ( F @ X ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_244_ex__has__least__nat,axiom,
% 2.64/1.55 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 2.64/1.55 ( ( P @ K )
% 2.64/1.55 => ? [X: A] :
% 2.64/1.55 ( ( P @ X )
% 2.64/1.55 & ! [Y8: A] :
% 2.64/1.55 ( ( P @ Y8 )
% 2.64/1.55 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ Y8 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_245_less__eq__real__def,axiom,
% 2.64/1.55 ( ( ord_less_eq @ real )
% 2.64/1.55 = ( ^ [X3: real,Y7: real] :
% 2.64/1.55 ( ( ord_less @ real @ X3 @ Y7 )
% 2.64/1.55 | ( X3 = Y7 ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_246_zero__prod__def,axiom,
% 2.64/1.55 ! [A: $tType,B: $tType] :
% 2.64/1.55 ( ( ( zero @ B )
% 2.64/1.55 & ( zero @ A ) )
% 2.64/1.55 => ( ( zero_zero @ ( product_prod @ A @ B ) )
% 2.64/1.55 = ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_247_complete__real,axiom,
% 2.64/1.55 ! [S2: set @ real] :
% 2.64/1.55 ( ? [X7: real] : ( member @ real @ X7 @ S2 )
% 2.64/1.55 => ( ? [Z4: real] :
% 2.64/1.55 ! [X: real] :
% 2.64/1.55 ( ( member @ real @ X @ S2 )
% 2.64/1.55 => ( ord_less_eq @ real @ X @ Z4 ) )
% 2.64/1.55 => ? [Y4: real] :
% 2.64/1.55 ( ! [X7: real] :
% 2.64/1.55 ( ( member @ real @ X7 @ S2 )
% 2.64/1.55 => ( ord_less_eq @ real @ X7 @ Y4 ) )
% 2.64/1.55 & ! [Z4: real] :
% 2.64/1.55 ( ! [X: real] :
% 2.64/1.55 ( ( member @ real @ X @ S2 )
% 2.64/1.55 => ( ord_less_eq @ real @ X @ Z4 ) )
% 2.64/1.55 => ( ord_less_eq @ real @ Y4 @ Z4 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_248_field__lbound__gt__zero,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linordered_field @ A )
% 2.64/1.55 => ! [D1: A,D22: A] :
% 2.64/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 2.64/1.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
% 2.64/1.55 => ? [E3: A] :
% 2.64/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
% 2.64/1.55 & ( ord_less @ A @ E3 @ D1 )
% 2.64/1.55 & ( ord_less @ A @ E3 @ D22 ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_249_less__numeral__extra_I3_J,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linord1659791738miring @ A )
% 2.64/1.55 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_250_complete__interval,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( condit1037483654norder @ A )
% 2.64/1.55 => ! [A2: A,B2: A,P: A > $o] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B2 )
% 2.64/1.55 => ( ( P @ A2 )
% 2.64/1.55 => ( ~ ( P @ B2 )
% 2.64/1.55 => ? [C3: A] :
% 2.64/1.55 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.64/1.55 & ( ord_less_eq @ A @ C3 @ B2 )
% 2.64/1.55 & ! [X7: A] :
% 2.64/1.55 ( ( ( ord_less_eq @ A @ A2 @ X7 )
% 2.64/1.55 & ( ord_less @ A @ X7 @ C3 ) )
% 2.64/1.55 => ( P @ X7 ) )
% 2.64/1.55 & ! [D5: A] :
% 2.64/1.55 ( ! [X: A] :
% 2.64/1.55 ( ( ( ord_less_eq @ A @ A2 @ X )
% 2.64/1.55 & ( ord_less @ A @ X @ D5 ) )
% 2.64/1.55 => ( P @ X ) )
% 2.64/1.55 => ( ord_less_eq @ A @ D5 @ C3 ) ) ) ) ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_251_ex__gt__or__lt,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( condit1656338222tinuum @ A )
% 2.64/1.55 => ! [A2: A] :
% 2.64/1.55 ? [B5: A] :
% 2.64/1.55 ( ( ord_less @ A @ A2 @ B5 )
% 2.64/1.55 | ( ord_less @ A @ B5 @ A2 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_252_le__numeral__extra_I3_J,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linord1659791738miring @ A )
% 2.64/1.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(fact_253_minf_I8_J,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( linorder @ A )
% 2.64/1.55 => ! [T4: A] :
% 2.64/1.55 ? [Z3: A] :
% 2.64/1.55 ! [X7: A] :
% 2.64/1.55 ( ( ord_less @ A @ X7 @ Z3 )
% 2.64/1.55 => ~ ( ord_less_eq @ A @ T4 @ X7 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(subcl_Real__Vector__Spaces_Ometric__space___HOL_Otype,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ( type @ A ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(subcl_Real__Vector__Spaces_Ometric__space___Topological__Spaces_Otopological__space,axiom,
% 2.64/1.55 ! [A: $tType] :
% 2.64/1.55 ( ( real_V2090557954_space @ A )
% 2.64/1.55 => ( topolo503727757_space @ A ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( euclid925273238_space @ A4 )
% 2.64/1.55 & ( euclid925273238_space @ A9 ) )
% 2.64/1.55 => ( euclid925273238_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space_1,axiom,
% 2.64/1.55 euclid925273238_space @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( topolo503727757_space @ A9 )
% 2.64/1.55 => ( topolo503727757_space @ ( A4 > A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( order_top @ A9 )
% 2.64/1.55 => ( order_top @ ( A4 > A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( preorder @ A9 )
% 2.64/1.55 => ( preorder @ ( A4 > A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( order @ A9 )
% 2.64/1.55 => ( order @ ( A4 > A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_fun___Orderings_Otop,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( top @ A9 )
% 2.64/1.55 => ( top @ ( A4 > A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_fun___Orderings_Oord,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ord @ A9 )
% 2.64/1.55 => ( ord @ ( A4 > A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.64/1.55 condit1037483654norder @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.64/1.55 canoni770627133id_add @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_2,axiom,
% 2.64/1.55 topolo503727757_space @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.64/1.55 linord1659791738miring @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.64/1.55 wellorder @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
% 2.64/1.55 preorder @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.64/1.55 linorder @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.64/1.55 no_top @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
% 2.64/1.55 order @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
% 2.64/1.55 ord @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.64/1.55 zero @ nat ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Set_Oset___Orderings_Oorder__top_6,axiom,
% 2.64/1.55 ! [A4: $tType] : ( order_top @ ( set @ A4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 2.64/1.55 ! [A4: $tType] : ( preorder @ ( set @ A4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 2.64/1.55 ! [A4: $tType] : ( order @ ( set @ A4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Set_Oset___Orderings_Otop_9,axiom,
% 2.64/1.55 ! [A4: $tType] : ( top @ ( set @ A4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 2.64/1.55 ! [A4: $tType] : ( ord @ ( set @ A4 ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Set_Oset___Groups_Ozero_11,axiom,
% 2.64/1.55 ! [A4: $tType] :
% 2.64/1.55 ( ( zero @ A4 )
% 2.64/1.55 => ( zero @ ( set @ A4 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_12,axiom,
% 2.64/1.55 topolo503727757_space @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
% 2.64/1.55 order_top @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 2.64/1.55 preorder @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
% 2.64/1.55 linorder @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 2.64/1.55 order @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
% 2.64/1.55 top @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
% 2.64/1.55 ord @ $o ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
% 2.64/1.55 condit1037483654norder @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
% 2.64/1.55 condit1656338222tinuum @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
% 2.64/1.55 real_V55928688vector @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_20,axiom,
% 2.64/1.55 topolo503727757_space @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_21,axiom,
% 2.64/1.55 linord1659791738miring @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
% 2.64/1.55 real_V1759440057_space @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
% 2.64/1.55 topolo890362671_space @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
% 2.64/1.55 real_V2090557954_space @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 2.64/1.55 dense_linorder @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.64/1.55 linordered_field @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 2.64/1.55 dense_order @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Opreorder_22,axiom,
% 2.64/1.55 preorder @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Olinorder_23,axiom,
% 2.64/1.55 linorder @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Ono__top_24,axiom,
% 2.64/1.55 no_top @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
% 2.64/1.55 no_bot @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Oorder_25,axiom,
% 2.64/1.55 order @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Orderings_Oord_26,axiom,
% 2.64/1.55 ord @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
% 2.64/1.55 zero @ real ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_28,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( real_V55928688vector @ A4 )
% 2.64/1.55 & ( real_V55928688vector @ A9 ) )
% 2.64/1.55 => ( real_V55928688vector @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_29,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( topolo503727757_space @ A4 )
% 2.64/1.55 & ( topolo503727757_space @ A9 ) )
% 2.64/1.55 => ( topolo503727757_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ocomplete__space_30,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( real_V1759440057_space @ A4 )
% 2.64/1.55 & ( real_V1759440057_space @ A9 ) )
% 2.64/1.55 => ( real_V1759440057_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Topological__Spaces_Operfect__space_31,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( euclid925273238_space @ A4 )
% 2.64/1.55 & ( euclid925273238_space @ A9 ) )
% 2.64/1.55 => ( topolo890362671_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_32,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( real_V2090557954_space @ A4 )
% 2.64/1.55 & ( real_V2090557954_space @ A9 ) )
% 2.64/1.55 => ( real_V2090557954_space @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_33,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( order_top @ A4 )
% 2.64/1.55 & ( order_top @ A9 ) )
% 2.64/1.55 => ( order_top @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_34,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( preorder @ A4 )
% 2.64/1.55 & ( preorder @ A9 ) )
% 2.64/1.55 => ( preorder @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Orderings_Oorder_35,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( order @ A4 )
% 2.64/1.55 & ( order @ A9 ) )
% 2.64/1.55 => ( order @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Orderings_Otop_36,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( top @ A4 )
% 2.64/1.55 & ( top @ A9 ) )
% 2.64/1.55 => ( top @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Orderings_Oord_37,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( ord @ A4 )
% 2.64/1.55 & ( ord @ A9 ) )
% 2.64/1.55 => ( ord @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tcon_Product__Type_Oprod___Groups_Ozero_38,axiom,
% 2.64/1.55 ! [A4: $tType,A9: $tType] :
% 2.64/1.55 ( ( ( zero @ A4 )
% 2.64/1.55 & ( zero @ A9 ) )
% 2.64/1.55 => ( zero @ ( product_prod @ A4 @ A9 ) ) ) ).
% 2.64/1.55
% 2.64/1.55 thf(tfree_0,hypothesis,
% 2.64/1.55 real_V2090557954_space @ a ).
% 2.64/1.55
% 2.64/1.55 thf(conj_0,hypothesis,
% 2.64/1.55 ! [A10: a,B8: real] :
% 2.64/1.55 ( ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A10 @ B8 ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
% 2.64/1.55 & ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A10 @ B8 ) @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) )
% 2.64/1.55 => thesis ) ).
% 2.64/1.55
% 2.64/1.55 thf(conj_1,conjecture,
% 2.64/1.55 thesis ).
% 2.64/1.55 ------------------------
% 2.64/1.57 % (27229)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/2Mi)
% 2.64/1.57 % (27225)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/183Mi)
% 2.64/1.57 % (27229)Instruction limit reached!
% 2.64/1.57 % (27229)------------------------------
% 2.64/1.57 % (27229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (27229)Termination reason: Unknown
% 2.64/1.57 % (27229)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (27229)Memory used [KB]: 1407
% 2.64/1.57 % (27229)Time elapsed: 0.002 s
% 2.64/1.57 % (27229)Instructions burned: 2 (million)
% 2.64/1.57 % (27229)------------------------------
% 2.64/1.57 % (27229)------------------------------
% 2.64/1.57 % (27230)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/275Mi)
% 2.64/1.57 % (27231)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/18Mi)
% 2.64/1.57 % (27228)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/2Mi)
% 2.64/1.57 % (27226)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/4Mi)
% 2.64/1.58 % (27228)Instruction limit reached!
% 2.64/1.58 % (27228)------------------------------
% 2.64/1.58 % (27228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (27228)Termination reason: Unknown
% 2.64/1.58 % (27228)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (27228)Memory used [KB]: 1407
% 2.64/1.58 % (27228)Time elapsed: 0.004 s
% 2.64/1.58 % (27228)Instructions burned: 3 (million)
% 2.64/1.58 % (27228)------------------------------
% 2.64/1.58 % (27228)------------------------------
% 2.64/1.58 % (27226)Instruction limit reached!
% 2.64/1.58 % (27226)------------------------------
% 2.64/1.58 % (27226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (27226)Termination reason: Unknown
% 2.64/1.58 % (27226)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (27226)Memory used [KB]: 1407
% 2.64/1.58 % (27226)Time elapsed: 0.006 s
% 2.64/1.58 % (27226)Instructions burned: 4 (million)
% 2.64/1.58 % (27226)------------------------------
% 2.64/1.58 % (27226)------------------------------
% 2.64/1.58 % (27227)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/27Mi)
% 2.64/1.58 % (27232)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/3Mi)
% 2.64/1.58 % (27232)Instruction limit reached!
% 2.64/1.58 % (27232)------------------------------
% 2.64/1.58 % (27232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (27232)Termination reason: Unknown
% 2.64/1.58 % (27232)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (27232)Memory used [KB]: 1535
% 2.64/1.58 % (27232)Time elapsed: 0.003 s
% 2.64/1.58 % (27232)Instructions burned: 5 (million)
% 2.64/1.58 % (27232)------------------------------
% 2.64/1.58 % (27232)------------------------------
% 2.64/1.58 % (27231)Instruction limit reached!
% 2.64/1.58 % (27231)------------------------------
% 2.64/1.58 % (27231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (27231)Termination reason: Unknown
% 2.64/1.58 % (27231)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (27231)Memory used [KB]: 1791
% 2.64/1.58 % (27231)Time elapsed: 0.011 s
% 2.64/1.58 % (27231)Instructions burned: 18 (million)
% 2.64/1.58 % (27231)------------------------------
% 2.64/1.58 % (27231)------------------------------
% 2.64/1.59 % (27233)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/37Mi)
% 2.64/1.59 % (27235)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/3Mi)
% 2.64/1.59 % (27235)Instruction limit reached!
% 2.64/1.59 % (27235)------------------------------
% 2.64/1.59 % (27235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (27235)Termination reason: Unknown
% 2.64/1.59 % (27235)Termination phase: shuffling
% 2.64/1.59
% 2.64/1.59 % (27235)Memory used [KB]: 1407
% 2.64/1.59 % (27235)Time elapsed: 0.002 s
% 2.64/1.59 % (27235)Instructions burned: 3 (million)
% 2.64/1.59 % (27235)------------------------------
% 2.64/1.59 % (27235)------------------------------
% 2.64/1.59 % (27227)Instruction limit reached!
% 2.64/1.59 % (27227)------------------------------
% 2.64/1.59 % (27227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (27227)Termination reason: Unknown
% 2.64/1.59 % (27227)Termination phase: Property scanning
% 2.64/1.59
% 2.64/1.59 % (27227)Memory used [KB]: 2046
% 2.64/1.59 % (27227)Time elapsed: 0.017 s
% 2.64/1.59 % (27227)Instructions burned: 28 (million)
% 2.64/1.59 % (27227)------------------------------
% 2.64/1.59 % (27227)------------------------------
% 2.70/1.60 % (27236)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/1041Mi)
% 2.70/1.60 % (27233)Instruction limit reached!
% 2.70/1.60 % (27233)------------------------------
% 2.70/1.60 % (27233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (27233)Termination reason: Unknown
% 2.70/1.60 % (27233)Termination phase: Property scanning
% 2.70/1.60
% 2.70/1.60 % (27233)Memory used [KB]: 2174
% 2.70/1.60 % (27233)Time elapsed: 0.014 s
% 2.70/1.60 % (27233)Instructions burned: 39 (million)
% 2.70/1.60 % (27233)------------------------------
% 2.70/1.60 % (27233)------------------------------
% 2.70/1.60 % (27237)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/7Mi)
% 2.70/1.60 % (27234)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/15Mi)
% 2.70/1.60 % (27237)Instruction limit reached!
% 2.70/1.60 % (27237)------------------------------
% 2.70/1.60 % (27237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (27237)Termination reason: Unknown
% 2.70/1.61 % (27237)Termination phase: shuffling
% 2.70/1.61
% 2.70/1.61 % (27237)Memory used [KB]: 1535
% 2.70/1.61 % (27237)Time elapsed: 0.004 s
% 2.70/1.61 % (27237)Instructions burned: 9 (million)
% 2.70/1.61 % (27237)------------------------------
% 2.70/1.61 % (27237)------------------------------
% 2.70/1.61 % (27238)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/16Mi)
% 2.70/1.61 % (27234)Instruction limit reached!
% 2.70/1.61 % (27234)------------------------------
% 2.70/1.61 % (27234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (27234)Termination reason: Unknown
% 2.70/1.61 % (27234)Termination phase: shuffling
% 2.70/1.61
% 2.70/1.61 % (27234)Memory used [KB]: 1791
% 2.70/1.61 % (27234)Time elapsed: 0.008 s
% 2.70/1.61 % (27234)Instructions burned: 15 (million)
% 2.70/1.61 % (27234)------------------------------
% 2.70/1.61 % (27234)------------------------------
% 2.70/1.61 % (27238)Instruction limit reached!
% 2.70/1.61 % (27238)------------------------------
% 2.70/1.61 % (27238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (27238)Termination reason: Unknown
% 2.70/1.61 % (27238)Termination phase: shuffling
% 2.70/1.61
% 2.70/1.61 % (27238)Memory used [KB]: 1791
% 2.70/1.61 % (27238)Time elapsed: 0.007 s
% 2.70/1.61 % (27238)Instructions burned: 16 (million)
% 2.70/1.61 % (27238)------------------------------
% 2.70/1.61 % (27238)------------------------------
% 2.70/1.61 % (27239)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/3Mi)
% 2.70/1.61 % (27239)Instruction limit reached!
% 2.70/1.61 % (27239)------------------------------
% 2.70/1.61 % (27239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (27239)Termination reason: Unknown
% 2.70/1.61 % (27239)Termination phase: shuffling
% 2.70/1.61
% 2.70/1.61 % (27239)Memory used [KB]: 1407
% 2.70/1.61 % (27239)Time elapsed: 0.002 s
% 2.70/1.61 % (27239)Instructions burned: 3 (million)
% 2.70/1.61 % (27239)------------------------------
% 2.70/1.61 % (27239)------------------------------
% 2.70/1.61 % (27240)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/3Mi)
% 2.70/1.62 % (27240)Instruction limit reached!
% 2.70/1.62 % (27240)------------------------------
% 2.70/1.62 % (27240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (27240)Termination reason: Unknown
% 2.70/1.62 % (27240)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (27240)Memory used [KB]: 1535
% 2.70/1.62 % (27240)Time elapsed: 0.003 s
% 2.70/1.62 % (27240)Instructions burned: 5 (million)
% 2.70/1.62 % (27240)------------------------------
% 2.70/1.62 % (27240)------------------------------
% 2.70/1.62 % (27230)First to succeed.
% 2.70/1.62 % (27241)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27033 for (2999ds/7Mi)
% 2.70/1.62 % (27230)Refutation found. Thanks to Tanya!
% 2.70/1.62 % SZS status Theorem for DTF2THF_27033
% 2.70/1.62 % SZS output start Proof for DTF2THF_27033
% 2.70/1.62 thf(type_def_5, type, set: ($tType) > $tType).
% 2.70/1.62 thf(type_def_7, type, extended_ereal: $tType).
% 2.70/1.62 thf(type_def_8, type, real: $tType).
% 2.70/1.62 thf(type_def_9, type, product_prod: ($tType * $tType) > $tType).
% 2.70/1.62 thf(type_def_10, type, nat: $tType).
% 2.70/1.62 thf(type_def_11, type, a: $tType).
% 2.70/1.62 thf(func_def_0, type, extended_ereal: $tType).
% 2.70/1.62 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 2.70/1.62 thf(func_def_2, type, real: $tType).
% 2.70/1.62 thf(func_def_3, type, set: ($tType) > $tType).
% 2.70/1.62 thf(func_def_4, type, nat: $tType).
% 2.70/1.62 thf(func_def_5, type, a: $tType).
% 2.70/1.62 thf(func_def_6, type, euclid925273238_space: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_8, type, zero: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_9, type, ord: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_10, type, top: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_11, type, order: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_12, type, no_bot: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_13, type, no_top: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_14, type, linorder: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_15, type, preorder: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_16, type, order_top: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_17, type, wellorder: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_18, type, dense_order: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_19, type, linordered_field: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_20, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_21, type, real_V2090557954_space: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_22, type, topolo890362671_space: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_23, type, real_V1759440057_space: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_24, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_25, type, topolo503727757_space: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_27, type, real_V55928688vector: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_28, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_29, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.70/1.62 thf(func_def_30, type, elementary_closure: !>[X0: $tType]:(set @ X0 > set @ X0)).
% 2.70/1.62 thf(func_def_31, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.70/1.62 thf(func_def_32, type, lower_261390618igraph: !>[X0: $tType]:(set @ X0 > (X0 > extended_ereal) > set @ product_prod @ X0 @ real)).
% 2.70/1.62 thf(func_def_33, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.70/1.62 thf(func_def_34, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.70/1.62 thf(func_def_35, type, top_top: !>[X0: $tType]:(X0)).
% 2.70/1.62 thf(func_def_36, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.70/1.62 thf(func_def_37, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.70/1.62 thf(func_def_38, type, real_V2000881966t_dist: !>[X0: $tType]:(X0 > X0 > real)).
% 2.70/1.62 thf(func_def_39, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.70/1.62 thf(func_def_40, type, unifor760693226chy_on: !>[X0: $tType, X1: $tType]:(set @ X0 > (nat > X0 > X1) > $o)).
% 2.70/1.62 thf(func_def_41, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.70/1.62 thf(func_def_42, type, e: real).
% 2.70/1.62 thf(func_def_43, type, f: a > extended_ereal).
% 2.70/1.62 thf(func_def_45, type, x: a).
% 2.70/1.62 thf(func_def_46, type, y: real).
% 2.70/1.62 thf(func_def_47, type, z: real).
% 2.70/1.62 thf(func_def_65, type, sP0: !>[X0: $tType]:((X0 > real) > (X0 > $o) > (X0 > $o) > $o)).
% 2.70/1.62 thf(func_def_66, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_67, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_68, type, sK3: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_69, type, sK4: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_70, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_71, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_72, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.70/1.62 thf(func_def_73, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.70/1.62 thf(func_def_74, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.70/1.62 thf(func_def_75, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.70/1.62 thf(func_def_76, type, sK11: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_77, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_78, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.70/1.62 thf(func_def_79, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.70/1.62 thf(func_def_80, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.70/1.62 thf(func_def_81, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.70/1.62 thf(func_def_82, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.70/1.62 thf(func_def_83, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_84, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_85, type, sK20: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_86, type, sK21: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_87, type, sK22: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_88, type, sK23: (nat > $o) > nat > nat).
% 2.70/1.62 thf(func_def_89, type, sK24: (nat > $o) > nat > nat).
% 2.70/1.62 thf(func_def_90, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.70/1.62 thf(func_def_91, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.70/1.62 thf(func_def_92, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.70/1.62 thf(func_def_93, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.70/1.62 thf(func_def_94, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.70/1.62 thf(func_def_95, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.70/1.62 thf(func_def_96, type, sK31: (nat > $o) > nat).
% 2.70/1.62 thf(func_def_97, type, sK32: product_prod @ a @ real).
% 2.70/1.62 thf(func_def_98, type, sK33: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.70/1.62 thf(func_def_99, type, sK34: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.70/1.62 thf(func_def_100, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_101, type, sK36: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_102, type, sK37: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_103, type, sK38: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_104, type, sK39: (nat > $o) > nat).
% 2.70/1.62 thf(func_def_105, type, sK40: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.70/1.62 thf(func_def_106, type, sK41: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.70/1.62 thf(func_def_107, type, sK42: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.70/1.62 thf(func_def_108, type, sK43: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.70/1.62 thf(func_def_109, type, sK44: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_110, type, sK45: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_111, type, sK46: (nat > $o) > nat).
% 2.70/1.62 thf(func_def_112, type, sK47: nat > (nat > $o) > nat).
% 2.70/1.62 thf(func_def_113, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.70/1.62 thf(func_def_114, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.70/1.62 thf(func_def_115, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.70/1.62 thf(func_def_116, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_117, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_118, type, sK53: !>[X0: $tType]:(real > (X0 > real) > (X0 > $o) > (X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_119, type, sK54: !>[X0: $tType]:((X0 > real) > (X0 > $o) > (X0 > $o) > real)).
% 2.70/1.62 thf(func_def_120, type, sK55: !>[X0: $tType]:((X0 > $o) > (X0 > real) > (X0 > $o) > real)).
% 2.70/1.62 thf(func_def_121, type, sK56: !>[X0: $tType]:(real > (X0 > $o) > (X0 > real) > (X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_122, type, sK57: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_123, type, sK58: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_124, type, sK59: (nat > real) > (nat > real) > nat > nat).
% 2.70/1.62 thf(func_def_125, type, sK60: (nat > real) > nat > nat).
% 2.70/1.62 thf(func_def_126, type, sK61: !>[X0: $tType]:(X0 > set @ X0 > X0 > X0)).
% 2.70/1.62 thf(func_def_127, type, sK62: (nat > nat) > nat).
% 2.70/1.62 thf(func_def_128, type, sK63: (nat > nat) > nat).
% 2.70/1.62 thf(func_def_129, type, sK64: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_130, type, sK65: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_131, type, sK66: (nat > $o) > nat).
% 2.70/1.62 thf(func_def_132, type, sK67: !>[X0: $tType]:(X0)).
% 2.70/1.62 thf(func_def_133, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.70/1.62 thf(func_def_134, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_135, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_136, type, sK71: real > set @ real > real).
% 2.70/1.62 thf(func_def_137, type, sK72: set @ real > real).
% 2.70/1.62 thf(func_def_138, type, sK73: real > set @ real > real).
% 2.70/1.62 thf(func_def_139, type, sK74: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.70/1.62 thf(func_def_140, type, sK75: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.70/1.62 thf(func_def_141, type, sK76: nat > (nat > $o) > nat).
% 2.70/1.62 thf(func_def_142, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.70/1.62 thf(func_def_143, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.70/1.62 thf(func_def_144, type, sK79: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.70/1.62 thf(func_def_145, type, sK80: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.70/1.62 thf(func_def_146, type, sK81: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_147, type, sK82: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_148, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.70/1.62 thf(func_def_149, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.70/1.62 thf(func_def_150, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.70/1.62 thf(func_def_151, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.70/1.62 thf(func_def_152, type, sK87: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.70/1.62 thf(func_def_153, type, sK88: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_154, type, sK89: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.70/1.62 thf(func_def_155, type, sK90: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_156, type, sK91: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.70/1.62 thf(func_def_157, type, sK92: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.70/1.62 thf(func_def_158, type, sK93: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.70/1.62 thf(func_def_159, type, sK94: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.70/1.62 thf(func_def_160, type, sK95: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.70/1.62 thf(func_def_161, type, sK96: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.70/1.62 thf(func_def_162, type, sK97: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.70/1.62 thf(func_def_163, type, sK98: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.70/1.62 thf(func_def_164, type, sK99: !>[X0: $tType]:(set @ X0 > X0)).
% 2.70/1.62 thf(func_def_165, type, sK100: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.70/1.62 thf(func_def_166, type, sK101: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.70/1.62 thf(func_def_167, type, sK102: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.70/1.62 thf(func_def_168, type, sK103: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.70/1.62 thf(func_def_169, type, sK104: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.70/1.62 thf(func_def_170, type, sK105: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.70/1.62 thf(func_def_171, type, sK106: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.70/1.62 thf(func_def_172, type, sK107: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.70/1.62 thf(func_def_174, type, ph109: !>[X0: $tType]:(X0)).
% 2.70/1.62 thf(f1602,plain,(
% 2.70/1.62 $false),
% 2.70/1.62 inference(subsumption_resolution,[status(thm)],[f1586,f1297])).
% 2.70/1.62 thf(f1297,plain,(
% 2.70/1.62 ($true = (member @ product_prod @ a @ real @ sK32 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)))),
% 2.70/1.62 inference(cnf_transformation,[status(thm)],[f1160])).
% 2.70/1.62 thf(f1160,plain,(
% 2.70/1.62 ((ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ sK32 @ (product_Pair @ a @ real @ x @ y)) @ e) = $true) & ($true = (member @ product_prod @ a @ real @ sK32 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)))),
% 2.70/1.62 inference(skolemisation,[status(esa),new_symbols(skolem,[sK32])],[f590,f1159])).
% 2.70/1.62 thf(f1159,plain,(
% 2.70/1.62 ? [X0 : product_prod @ a @ real] : (($true = (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ X0 @ (product_Pair @ a @ real @ x @ y)) @ e)) & ((member @ product_prod @ a @ real @ X0 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)) = $true)) => (((ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ sK32 @ (product_Pair @ a @ real @ x @ y)) @ e) = $true) & ($true = (member @ product_prod @ a @ real @ sK32 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))))),
% 2.70/1.62 introduced(definition,[],[choice_axiom])).
% 2.70/1.62 thf(f590,plain,(
% 2.70/1.62 ? [X0 : product_prod @ a @ real] : (($true = (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ X0 @ (product_Pair @ a @ real @ x @ y)) @ e)) & ((member @ product_prod @ a @ real @ X0 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)) = $true))),
% 2.70/1.62 inference(fool_elimination,[status(thm)],[f589])).
% 2.70/1.62 thf(f589,plain,(
% 2.70/1.62 ? [X0 : product_prod @ a @ real] : ((member @ product_prod @ a @ real @ X0 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)) & (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ X0 @ (product_Pair @ a @ real @ x @ y)) @ e))),
% 2.70/1.62 inference(rectify,[status(thm)],[f1])).
% 2.70/1.62 thf(f1,axiom,(
% 2.70/1.62 ? [X3 : product_prod @ a @ real] : ((member @ product_prod @ a @ real @ X3 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)) & (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ X3 @ (product_Pair @ a @ real @ x @ y)) @ e))),
% 2.70/1.62 file('/export/starexec/sandbox/tmp/tmp.65X5awu5pu/DTF2THF_27033.p',fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062)).
% 2.70/1.62 thf(f1586,plain,(
% 2.70/1.62 ($true != (member @ product_prod @ a @ real @ sK32 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)))),
% 2.70/1.62 inference(trivial_inequality_removal,[status(thm)],[f1572])).
% 2.70/1.62 thf(f1572,plain,(
% 2.70/1.62 ($true != (member @ product_prod @ a @ real @ sK32 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) | ($true != $true)),
% 2.70/1.62 inference(superposition,[status(thm)],[f1512,f1298])).
% 2.70/1.62 thf(f1298,plain,(
% 2.70/1.62 ((ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ sK32 @ (product_Pair @ a @ real @ x @ y)) @ e) = $true)),
% 2.70/1.62 inference(cnf_transformation,[status(thm)],[f1160])).
% 2.70/1.62 thf(f1512,plain,(
% 2.70/1.62 ( ! [X0 : product_prod @ a @ real] : (($true != (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ X0 @ (product_Pair @ a @ real @ x @ y)) @ e)) | ((member @ product_prod @ a @ real @ X0 @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)) != $true)) )),
% 2.70/1.62 inference(superposition,[status(thm)],[f1510,f1325])).
% 2.70/1.62 thf(f1325,plain,(
% 2.70/1.62 ( ! [X1 : $tType,X0 : $tType,X2 : product_prod @ X0 @ X1] : (((product_Pair @ X0 @ X1 @ (sK40 @ X0 @ X1 @ X2) @ (sK41 @ X0 @ X1 @ X2)) = X2)) )),
% 2.70/1.62 inference(cnf_transformation,[status(thm)],[f1173])).
% 2.70/1.62 thf(f1173,plain,(
% 2.70/1.62 ! [X0 : $tType,X1 : $tType,X2 : product_prod @ X0 @ X1] : ((product_Pair @ X0 @ X1 @ (sK40 @ X0 @ X1 @ X2) @ (sK41 @ X0 @ X1 @ X2)) = X2)),
% 2.70/1.62 inference(skolemisation,[status(esa),new_symbols(skolem,[sK40,sK41])],[f1017,f1172])).
% 2.70/1.62 thf(f1172,plain,(
% 2.70/1.62 ! [X1 : $tType,X0 : $tType,X2 : product_prod @ X0 @ X1] : (? [X3 : X0,X4 : X1] : ((product_Pair @ X0 @ X1 @ X3 @ X4) = X2) => ((product_Pair @ X0 @ X1 @ (sK40 @ X0 @ X1 @ X2) @ (sK41 @ X0 @ X1 @ X2)) = X2))),
% 2.70/1.62 introduced(definition,[],[choice_axiom])).
% 2.70/1.62 thf(f1017,plain,(
% 2.70/1.62 ! [X0 : $tType,X1 : $tType,X2 : product_prod @ X0 @ X1] : ? [X3 : X0,X4 : X1] : ((product_Pair @ X0 @ X1 @ X3 @ X4) = X2)),
% 2.70/1.62 inference(ennf_transformation,[status(thm)],[f939])).
% 2.70/1.62 thf(f939,plain,(
% 2.70/1.62 ! [X0 : $tType,X1 : $tType,X2 : product_prod @ X0 @ X1] : ~! [X3 : X0,X4 : X1] : ((product_Pair @ X0 @ X1 @ X3 @ X4) != X2)),
% 2.70/1.62 inference(rectify,[status(thm)],[f130])).
% 2.70/1.62 thf(f130,axiom,(
% 2.70/1.62 ! [X0 : $tType,X1 : $tType,X16 : product_prod @ X0 @ X1] : ~! [X38 : X0,X39 : X1] : ((product_Pair @ X0 @ X1 @ X38 @ X39) != X16)),
% 2.70/1.62 file('/export/starexec/sandbox/tmp/tmp.65X5awu5pu/DTF2THF_27033.p',fact_129_old_Oprod_Oexhaust)).
% 2.70/1.62 thf(f1510,plain,(
% 2.70/1.62 ( ! [X0 : a,X1 : real] : (($true != (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (product_Pair @ a @ real @ x @ y)) @ e)) | ($true != (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f)))) )),
% 2.70/1.62 inference(subsumption_resolution,[status(thm)],[f1418,f1422])).
% 2.70/1.62 thf(f1422,plain,(
% 2.70/1.62 (thesis != $true)),
% 2.70/1.62 inference(cnf_transformation,[status(thm)],[f937])).
% 2.70/1.62 thf(f937,plain,(
% 2.70/1.62 (thesis != $true)),
% 2.70/1.62 inference(flattening,[status(thm)],[f662])).
% 2.70/1.62 thf(f662,plain,(
% 2.70/1.62 ~(thesis = $true)),
% 2.70/1.62 inference(fool_elimination,[status(thm)],[f661])).
% 2.70/1.62 thf(f661,plain,(
% 2.70/1.62 ~thesis),
% 2.70/1.62 inference(rectify,[status(thm)],[f321])).
% 2.70/1.62 thf(f321,negated_conjecture,(
% 2.70/1.62 ~thesis),
% 2.70/1.62 inference(negated_conjecture,[status(cth)],[f320])).
% 2.70/1.62 thf(f320,conjecture,(
% 2.70/1.62 thesis),
% 2.70/1.62 file('/export/starexec/sandbox/tmp/tmp.65X5awu5pu/DTF2THF_27033.p',conj_1)).
% 2.70/1.62 thf(f1418,plain,(
% 2.70/1.62 ( ! [X0 : a,X1 : real] : (($true != (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) | ($true != (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (product_Pair @ a @ real @ x @ y)) @ e)) | (thesis = $true)) )),
% 2.70/1.62 inference(cnf_transformation,[status(thm)],[f1065])).
% 2.70/1.62 thf(f1065,plain,(
% 2.70/1.62 ! [X0 : a,X1 : real] : (($true != (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) | ($true != (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (product_Pair @ a @ real @ x @ y)) @ e)) | (thesis = $true))),
% 2.70/1.62 inference(flattening,[status(thm)],[f1064])).
% 2.70/1.62 thf(f1064,plain,(
% 2.70/1.62 ! [X0 : a,X1 : real] : ((thesis = $true) | (($true != (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) | ($true != (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (product_Pair @ a @ real @ x @ y)) @ e))))),
% 2.70/1.62 inference(ennf_transformation,[status(thm)],[f712])).
% 2.70/1.62 thf(f712,plain,(
% 2.70/1.62 ! [X0 : a,X1 : real] : ((($true = (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) & ($true = (ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (product_Pair @ a @ real @ x @ y)) @ e))) => (thesis = $true))),
% 2.70/1.62 inference(fool_elimination,[status(thm)],[f711])).
% 2.70/1.62 thf(f711,plain,(
% 2.70/1.62 ! [X0 : a,X1 : real] : (((ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (product_Pair @ a @ real @ x @ y)) @ e) & (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X0 @ X1) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) => thesis)),
% 2.70/1.62 inference(rectify,[status(thm)],[f319])).
% 2.70/1.62 thf(f319,axiom,(
% 2.70/1.62 ! [X104 : a,X105 : real] : (((ord_less @ real @ (real_V2000881966t_dist @ product_prod @ a @ real @ (product_Pair @ a @ real @ X104 @ X105) @ (product_Pair @ a @ real @ x @ y)) @ e) & (member @ product_prod @ a @ real @ (product_Pair @ a @ real @ X104 @ X105) @ (lower_261390618igraph @ a @ top_top @ set @ a @ f))) => thesis)),
% 2.70/1.62 file('/export/starexec/sandbox/tmp/tmp.65X5awu5pu/DTF2THF_27033.p',conj_0)).
% 2.70/1.62 % SZS output end Proof for DTF2THF_27033
% 2.70/1.62 % (27230)------------------------------
% 2.70/1.62 % (27230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (27230)Termination reason: Refutation
% 2.70/1.62
% 2.70/1.62 % (27230)Memory used [KB]: 7291
% 2.70/1.62 % (27230)Time elapsed: 0.050 s
% 2.70/1.62 % (27230)Instructions burned: 125 (million)
% 2.70/1.62 % (27230)------------------------------
% 2.70/1.62 % (27230)------------------------------
% 2.70/1.62 % (27224)Success in time 0.076 s
%------------------------------------------------------------------------------