%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP093^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n023.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 08:55:30 AM UTC 2026
% Result : Theorem 2.94s 1.71s
% Output : Refutation 2.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP093^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n023.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 08:26:49 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : ITP093^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.46 % Domain : Interactive Theorem Proving
% 0.29/0.46 % Problem : Sledgehammer Kuratowski problem prob_43__5514550_1
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Des21]
% 0.29/0.46 % Names : Kuratowski/prob_43__5514550_1 [Des21]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.00 v7.5.0
% 0.29/0.46 % Syntax : Number of formulae : 338 ( 123 unt; 48 typ; 0 def)
% 0.29/0.46 % Number of atoms : 760 ( 210 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.29/0.46 % Number of connectives : 4141 ( 81 ~; 12 |; 64 &;3618 @)
% 0.29/0.46 % ( 0 <=>; 366 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 28 ( 8 avg)
% 0.29/0.46 % Number of types : 4 ( 3 usr)
% 0.29/0.46 % Number of type conns : 266 ( 266 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 48 ( 45 usr; 4 con; 0-5 aty)
% 0.29/0.46 % Number of variables : 1258 ( 120 ^;1065 !; 18 ?;1258 :)
% 0.29/0.46 % ( 55 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.46 % SPC : TH1_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:29:07.273
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % Could-be-implicit typings (6)
% 0.29/0.46 thf(ty_t_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
% 0.29/0.46 pair_p1731315293ph_ext: $tType > $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Product__Type_Ounit,type,
% 0.29/0.46 product_unit: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.29/0.46 product_prod: $tType > $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Set_Oset,type,
% 0.29/0.46 set: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Nat_Onat,type,
% 0.29/0.46 nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_tf_a,type,
% 0.29/0.46 a: $tType ).
% 0.29/0.46
% 0.29/0.46 % Explicit typings (42)
% 0.29/0.46 thf(sy_cl_HOL_Otype,type,
% 0.29/0.46 type:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.29/0.46 bot:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.46 ord:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.46 order:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.29/0.46 finite_finite:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.46 linorder:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.46 preorder:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.29/0.46 order_bot:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.29/0.46 bNF_Ca1785829860lChain:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_OFpow,type,
% 0.29/0.46 finite_Fpow:
% 0.29/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ocard,type,
% 0.29/0.46 finite_card:
% 0.29/0.46 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Finite__Set_Ofinite,type,
% 0.29/0.46 finite_finite2:
% 0.29/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.29/0.46 bot_bot:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.46 ord_less_eq:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__bidirected__digraph,type,
% 0.29/0.46 pair_p1033978749igraph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__digraph,type,
% 0.29/0.46 pair_pair_digraph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph,type,
% 0.29/0.46 pair_p953238171igraph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms,type,
% 0.29/0.46 pair_p1373739070axioms:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__graph,type,
% 0.29/0.46 pair_pair_graph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__loopfree__digraph,type,
% 0.29/0.46 pair_p1282390076igraph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__loopfree__digraph__axioms,type,
% 0.29/0.46 pair_p814123999axioms:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore,type,
% 0.29/0.46 pair_pair_pre_more:
% 0.29/0.46 !>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > Z ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore__update,type,
% 0.29/0.46 pair_p1108258481update:
% 0.29/0.46 !>[Z: $tType,A: $tType] : ( ( Z > Z ) > ( pair_p1731315293ph_ext @ A @ Z ) > ( pair_p1731315293ph_ext @ A @ Z ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
% 0.29/0.46 pair_p1152105390ph_ext:
% 0.29/0.46 !>[A: $tType,Z: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > Z > ( pair_p1731315293ph_ext @ A @ Z ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs,type,
% 0.29/0.46 pair_pair_pre_parcs:
% 0.29/0.46 !>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts,type,
% 0.29/0.46 pair_pair_pre_pverts:
% 0.29/0.46 !>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__pseudo__graph,type,
% 0.29/0.46 pair_p389585203_graph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__sym__digraph,type,
% 0.29/0.46 pair_p460180511igraph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__sym__digraph__axioms,type,
% 0.29/0.46 pair_p477239746axioms:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Opair__wf__digraph,type,
% 0.29/0.46 pair_pair_wf_digraph:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pair__Digraph_Osubdivide,type,
% 0.29/0.46 pair_subdivide:
% 0.29/0.46 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > ( pair_p1731315293ph_ext @ A @ product_unit ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.29/0.46 product_Pair:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OSigma,type,
% 0.29/0.46 product_Sigma:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.29/0.46 produc2004651681e_prod:
% 0.29/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.29/0.46 product_rec_prod:
% 0.29/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.29/0.46 product_case_prod:
% 0.29/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oproduct,type,
% 0.29/0.46 product_product:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect,type,
% 0.29/0.46 collect:
% 0.29/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.29/0.46 is_empty:
% 0.29/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member,type,
% 0.29/0.46 member:
% 0.29/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_G,type,
% 0.29/0.46 g: pair_p1731315293ph_ext @ a @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_n,type,
% 0.29/0.46 n: nat ).
% 0.29/0.46
% 0.29/0.46 % Relevant facts (256)
% 0.29/0.46 thf(fact_0_finite__code,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( finite_finite @ A )
% 0.29/0.46 => ( ( finite_finite2 @ A )
% 0.29/0.46 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_code
% 0.29/0.46 thf(fact_1__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062,axiom,
% 0.29/0.46 ( finite_finite2 @ ( product_prod @ a @ a )
% 0.29/0.46 @ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
% 0.29/0.46 @ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>finite (pverts G \<times> pverts G)\<close>
% 0.29/0.46 thf(fact_2__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062,axiom,
% 0.29/0.46 ( ord_less_eq @ ( set @ ( product_prod @ a @ a ) ) @ ( pair_pair_pre_parcs @ a @ product_unit @ g )
% 0.29/0.46 @ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
% 0.29/0.46 @ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>parcs G \<subseteq> pverts G \<times> pverts G\<close>
% 0.29/0.46 thf(fact_3_finite__set__choice,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 => ? [X_1: B] : ( P @ X @ X_1 ) )
% 0.29/0.46 => ? [F: A > B] :
% 0.29/0.46 ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A3 )
% 0.29/0.46 => ( P @ X2 @ ( F @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_set_choice
% 0.29/0.46 thf(fact_4_finite,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( finite_finite @ A )
% 0.29/0.46 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite
% 0.29/0.46 thf(fact_5_pair__fin__digraph__axioms__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_p1373739070axioms @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G ) )
% 0.29/0.46 & ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms_def
% 0.29/0.46 thf(fact_6_pair__fin__digraph__axioms_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_p1373739070axioms @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_axioms.intro
% 0.29/0.46 thf(fact_7_pair__fin__digraph_Opair__finite__arcs,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_arcs
% 0.29/0.46 thf(fact_8_finite__subset,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( finite_finite2 @ A @ B2 )
% 0.29/0.46 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_subset
% 0.29/0.46 thf(fact_9_infinite__super,axiom,
% 0.29/0.46 ! [A: $tType,S: set @ A,T2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
% 0.29/0.46 => ( ~ ( finite_finite2 @ A @ S )
% 0.29/0.46 => ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_super
% 0.29/0.46 thf(fact_10_rev__finite__subset,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_finite_subset
% 0.29/0.46 thf(fact_11_finite__has__maximal2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: set @ A,A4: A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ( member @ A @ A4 @ A3 )
% 0.29/0.46 => ? [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 & ( ord_less_eq @ A @ A4 @ X )
% 0.29/0.46 & ! [Xa: A] :
% 0.29/0.46 ( ( member @ A @ Xa @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Xa )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_maximal2
% 0.29/0.46 thf(fact_12_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.29/0.46 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.29/0.46 => ( finite_finite2 @ A
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 & ( Q @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_conjI
% 0.29/0.46 thf(fact_13_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( finite_finite2 @ A
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 | ( Q @ X3 ) ) ) )
% 0.29/0.46 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.29/0.46 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_disjI
% 0.29/0.46 thf(fact_14_finite__Collect__subsets,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( finite_finite2 @ ( set @ A )
% 0.29/0.46 @ ( collect @ ( set @ A )
% 0.29/0.46 @ ^ [B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_subsets
% 0.29/0.46 thf(fact_15_finite__SigmaI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ! [A5: A] :
% 0.29/0.46 ( ( member @ A @ A5 @ A3 )
% 0.29/0.46 => ( finite_finite2 @ B @ ( B2 @ A5 ) ) )
% 0.29/0.46 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI
% 0.29/0.46 thf(fact_16_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p1373739070axioms @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.axioms(2)
% 0.29/0.46 thf(fact_17_not__finite__existsD,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.29/0.46 => ? [X_12: A] : ( P @ X_12 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_finite_existsD
% 0.29/0.46 thf(fact_18_pigeonhole__infinite__rel,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,R: A > B > $o] :
% 0.29/0.46 ( ~ ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ( finite_finite2 @ B @ B2 )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 => ? [Xa: B] :
% 0.29/0.46 ( ( member @ B @ Xa @ B2 )
% 0.29/0.46 & ( R @ X @ Xa ) ) )
% 0.29/0.46 => ? [X: B] :
% 0.29/0.46 ( ( member @ B @ X @ B2 )
% 0.29/0.46 & ~ ( finite_finite2 @ A
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [A6: A] :
% 0.29/0.46 ( ( member @ A @ A6 @ A3 )
% 0.29/0.46 & ( R @ A6 @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pigeonhole_infinite_rel
% 0.29/0.46 thf(fact_19_pair__fin__digraph_Opair__fin__digraph,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p953238171igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_fin_digraph
% 0.29/0.46 thf(fact_20_finite__cartesian__product,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ( finite_finite2 @ B @ B2 )
% 0.29/0.46 => ( finite_finite2 @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product
% 0.29/0.46 thf(fact_21_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.pair_finite_verts
% 0.29/0.46 thf(fact_22_finite__has__minimal2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: set @ A,A4: A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ( member @ A @ A4 @ A3 )
% 0.29/0.46 => ? [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 & ( ord_less_eq @ A @ X @ A4 )
% 0.29/0.46 & ! [Xa: A] :
% 0.29/0.46 ( ( member @ A @ Xa @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Xa @ X )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_minimal2
% 0.29/0.46 thf(fact_23_infinite__cartesian__product,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ~ ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ~ ( finite_finite2 @ B @ B2 )
% 0.29/0.46 => ~ ( finite_finite2 @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_cartesian_product
% 0.29/0.46 thf(fact_24_Times__subset__cancel2,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,X4: A,C2: set @ A,A3: set @ B,B2: set @ B] :
% 0.29/0.46 ( ( member @ A @ X4 @ C2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.29/0.46 @ ( product_Sigma @ B @ A @ A3
% 0.29/0.46 @ ^ [Uu: B] : C2 )
% 0.29/0.46 @ ( product_Sigma @ B @ A @ B2
% 0.29/0.46 @ ^ [Uu: B] : C2 ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ B ) @ A3 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_subset_cancel2
% 0.29/0.46 thf(fact_25_Sigma__mono,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,C2: set @ A,B2: A > ( set @ B ),D: A > ( set @ B )] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C2 )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X ) @ ( D @ X ) ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ C2 @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_mono
% 0.29/0.46 thf(fact_26_subsetI,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 => ( member @ A @ X @ B2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_27_subset__antisym,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.29/0.46 => ( A3 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_28_order__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_29_pair__pre__digraph_Oequality,axiom,
% 0.29/0.46 ! [Z: $tType,A: $tType,R2: pair_p1731315293ph_ext @ A @ Z,R3: pair_p1731315293ph_ext @ A @ Z] :
% 0.29/0.46 ( ( ( pair_pair_pre_pverts @ A @ Z @ R2 )
% 0.29/0.46 = ( pair_pair_pre_pverts @ A @ Z @ R3 ) )
% 0.29/0.46 => ( ( ( pair_pair_pre_parcs @ A @ Z @ R2 )
% 0.29/0.46 = ( pair_pair_pre_parcs @ A @ Z @ R3 ) )
% 0.29/0.46 => ( ( ( pair_pair_pre_more @ A @ Z @ R2 )
% 0.29/0.46 = ( pair_pair_pre_more @ A @ Z @ R3 ) )
% 0.29/0.46 => ( R2 = R3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.equality
% 0.29/0.46 thf(fact_30_Sigma__cong,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ A,C2: A > ( set @ B ),D: A > ( set @ B )] :
% 0.29/0.46 ( ( A3 = B2 )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ B2 )
% 0.29/0.46 => ( ( C2 @ X )
% 0.29/0.46 = ( D @ X ) ) )
% 0.29/0.46 => ( ( product_Sigma @ A @ B @ A3 @ C2 )
% 0.29/0.46 = ( product_Sigma @ A @ B @ B2 @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_cong
% 0.29/0.46 thf(fact_31_Times__eq__cancel2,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,X4: A,C2: set @ A,A3: set @ B,B2: set @ B] :
% 0.29/0.46 ( ( member @ A @ X4 @ C2 )
% 0.29/0.46 => ( ( ( product_Sigma @ B @ A @ A3
% 0.29/0.46 @ ^ [Uu: B] : C2 )
% 0.29/0.46 = ( product_Sigma @ B @ A @ B2
% 0.29/0.46 @ ^ [Uu: B] : C2 ) )
% 0.29/0.46 = ( A3 = B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_eq_cancel2
% 0.29/0.46 thf(fact_32_Collect__subset,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,P: A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( set @ A )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A3 )
% 0.29/0.46 & ( P @ X3 ) ) )
% 0.29/0.46 @ A3 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_subset
% 0.29/0.46 thf(fact_33_less__eq__set__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.29/0.46 ( ord_less_eq @ ( A > $o )
% 0.29/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A2 )
% 0.29/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_set_def
% 0.29/0.46 thf(fact_34_predicate1I,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X: A] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( Q @ X ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate1I
% 0.29/0.46 thf(fact_35_predicate1D,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o,X4: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.29/0.46 => ( ( P @ X4 )
% 0.29/0.46 => ( Q @ X4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate1D
% 0.29/0.46 thf(fact_36_rev__predicate1D,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X4: A,Q: A > $o] :
% 0.29/0.46 ( ( P @ X4 )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.29/0.46 => ( Q @ X4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_predicate1D
% 0.29/0.46 thf(fact_37_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B4: A,A4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_38_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.29/0.46 = ( ^ [A6: A,B5: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.29/0.46 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_39_dual__order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B4: A,A4: A,C3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C3 @ B4 )
% 0.29/0.46 => ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_40_linorder__wlog,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.29/0.46 ( ! [A5: A,B6: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A5 @ B6 )
% 0.29/0.46 => ( P @ A5 @ B6 ) )
% 0.29/0.46 => ( ! [A5: A,B6: A] :
% 0.29/0.46 ( ( P @ B6 @ A5 )
% 0.29/0.46 => ( P @ A5 @ B6 ) )
% 0.29/0.46 => ( P @ A4 @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_41_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_42_order__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X4: A,Y2: A,Z3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y2 @ Z3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X4 @ Z3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_43_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A4: A,B4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.29/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_44_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 => ( ( B4 = C3 )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.29/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.29/0.46 = ( P @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( collect @ A
% 0.29/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.29/0.46 = A3 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_47_Collect__cong,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X: A] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 = ( Q @ X ) )
% 0.29/0.46 => ( ( collect @ A @ P )
% 0.29/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_48_ext,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F2: A > B,G3: A > B] :
% 0.29/0.46 ( ! [X: A] :
% 0.29/0.46 ( ( F2 @ X )
% 0.29/0.46 = ( G3 @ X ) )
% 0.29/0.46 => ( F2 = G3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ext
% 0.29/0.46 thf(fact_49_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.29/0.46 ( ( A4 = B4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B4 @ C3 )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_50_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.29/0.46 = ( ^ [A6: A,B5: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.29/0.46 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_51_antisym__conv,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [Y2: A,X4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 = ( X4 = Y2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_52_le__cases3,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X4: A,Y2: A,Z3: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y2 @ Z3 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X4 @ Z3 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ X4 @ Z3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z3 @ Y2 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Z3 @ Y2 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y2 @ Z3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z3 @ X4 ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ Z3 @ X4 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_53_order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B4 @ C3 )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_54_le__cases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X4: A,Y2: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_55_eq__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X4: A,Y2: A] :
% 0.29/0.46 ( ( X4 = Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_56_linear,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 | ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_57_antisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.29/0.46 => ( X4 = Y2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_58_eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.29/0.46 = ( ^ [X3: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.29/0.46 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_59_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B )
% 0.29/0.46 & ( ord @ A ) )
% 0.29/0.46 => ! [A4: A,B4: A,F2: A > B,C3: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 => ( ( ( F2 @ B4 )
% 0.29/0.46 = C3 )
% 0.29/0.46 => ( ! [X: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_60_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B )
% 0.29/0.46 & ( ord @ A ) )
% 0.29/0.46 => ! [A4: A,F2: B > A,B4: B,C3: B] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( F2 @ B4 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B4 @ C3 )
% 0.29/0.46 => ( ! [X: B,Y4: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_61_order__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType] :
% 0.29/0.46 ( ( ( order @ C )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A4: A,B4: A,F2: A > C,C3: C] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.46 => ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C3 )
% 0.29/0.46 => ( ! [X: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ C @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_62_order__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B )
% 0.29/0.46 & ( order @ A ) )
% 0.29/0.46 => ! [A4: A,F2: B > A,B4: B,C3: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ ( F2 @ B4 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B4 @ C3 )
% 0.29/0.46 => ( ! [X: B,Y4: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_63_le__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.46 = ( ^ [F3: A > B,G4: A > B] :
% 0.29/0.46 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_fun_def
% 0.29/0.46 thf(fact_64_le__funI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ! [F2: A > B,G3: A > B] :
% 0.29/0.46 ( ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G3 @ X ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B ) @ F2 @ G3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funI
% 0.29/0.46 thf(fact_65_le__funE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ! [F2: A > B,G3: A > B,X4: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G3 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G3 @ X4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funE
% 0.29/0.46 thf(fact_66_le__funD,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ! [F2: A > B,G3: A > B,X4: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G3 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G3 @ X4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funD
% 0.29/0.46 thf(fact_67_Collect__mono__iff,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.29/0.46 = ( ! [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_68_set__eq__subset,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 ) )
% 0.29/0.46 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_69_subset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_70_Collect__mono,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X: A] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( Q @ X ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_71_subset__refl,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_72_subset__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.29/0.46 ! [T3: A] :
% 0.29/0.46 ( ( member @ A @ T3 @ A2 )
% 0.29/0.46 => ( member @ A @ T3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_73_equalityD2,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( A3 = B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_74_equalityD1,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( A3 = B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_75_subset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.29/0.46 ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A2 )
% 0.29/0.46 => ( member @ A @ X3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_76_equalityE,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( A3 = B2 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_77_subsetD,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C3: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( member @ A @ C3 @ A3 )
% 0.29/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetD
% 0.29/0.46 thf(fact_78_in__mono,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( member @ A @ X4 @ A3 )
% 0.29/0.46 => ( member @ A @ X4 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_mono
% 0.29/0.46 thf(fact_79_pred__subset__eq,axiom,
% 0.29/0.46 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > $o )
% 0.29/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.29/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.29/0.46
% 0.29/0.46 % pred_subset_eq
% 0.29/0.46 thf(fact_80_pair__pre__digraph_Osurjective,axiom,
% 0.29/0.46 ! [Z: $tType,A: $tType,R2: pair_p1731315293ph_ext @ A @ Z] :
% 0.29/0.46 ( R2
% 0.29/0.46 = ( pair_p1152105390ph_ext @ A @ Z @ ( pair_pair_pre_pverts @ A @ Z @ R2 ) @ ( pair_pair_pre_parcs @ A @ Z @ R2 ) @ ( pair_pair_pre_more @ A @ Z @ R2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.surjective
% 0.29/0.46 thf(fact_81_member__product,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ X4 @ ( product_product @ A @ B @ A3 @ B2 ) )
% 0.29/0.46 = ( member @ ( product_prod @ A @ B ) @ X4
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % member_product
% 0.29/0.46 thf(fact_82_Product__Type_Oproduct__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( product_product @ A @ B )
% 0.29/0.46 = ( ^ [A2: set @ A,B3: set @ B] :
% 0.29/0.46 ( product_Sigma @ A @ B @ A2
% 0.29/0.46 @ ^ [Uu: A] : B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Product_Type.product_def
% 0.29/0.46 thf(fact_83_subset__Collect__iff,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: set @ A,P: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A3 )
% 0.29/0.46 & ( P @ X3 ) ) ) )
% 0.29/0.46 = ( ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ B2 )
% 0.29/0.46 => ( P @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_Collect_iff
% 0.29/0.46 thf(fact_84_subset__CollectI,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ B2 )
% 0.29/0.46 => ( ( Q @ X )
% 0.29/0.46 => ( P @ X ) ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ B2 )
% 0.29/0.46 & ( Q @ X3 ) ) )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A3 )
% 0.29/0.46 & ( P @ X3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_CollectI
% 0.29/0.46 thf(fact_85_Collect__restrict,axiom,
% 0.29/0.46 ! [A: $tType,X5: set @ A,P: A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( set @ A )
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ X5 )
% 0.29/0.46 & ( P @ X3 ) ) )
% 0.29/0.46 @ X5 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_restrict
% 0.29/0.46 thf(fact_86_prop__restrict,axiom,
% 0.29/0.46 ! [A: $tType,X4: A,Z4: set @ A,X5: set @ A,P: A > $o] :
% 0.29/0.46 ( ( member @ A @ X4 @ Z4 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ X5 )
% 0.29/0.46 & ( P @ X3 ) ) ) )
% 0.29/0.46 => ( P @ X4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prop_restrict
% 0.29/0.46 thf(fact_87_conj__subset__def,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 & ( Q @ X3 ) ) ) )
% 0.29/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % conj_subset_def
% 0.29/0.46 thf(fact_88_pair__pre__digraph_Oext__inject,axiom,
% 0.29/0.46 ! [A: $tType,Z: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z,Pverts2: set @ A,Parcs2: set @ ( product_prod @ A @ A ),More2: Z] :
% 0.29/0.46 ( ( ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More )
% 0.29/0.46 = ( pair_p1152105390ph_ext @ A @ Z @ Pverts2 @ Parcs2 @ More2 ) )
% 0.29/0.46 = ( ( Pverts = Pverts2 )
% 0.29/0.46 & ( Parcs = Parcs2 )
% 0.29/0.46 & ( More = More2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.ext_inject
% 0.29/0.46 thf(fact_89_pair__pre__digraph_Oinduct__scheme,axiom,
% 0.29/0.46 ! [Z: $tType,A: $tType,P: ( pair_p1731315293ph_ext @ A @ Z ) > $o,R2: pair_p1731315293ph_ext @ A @ Z] :
% 0.29/0.46 ( ! [Pverts3: set @ A,Parcs3: set @ ( product_prod @ A @ A ),More3: Z] : ( P @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts3 @ Parcs3 @ More3 ) )
% 0.29/0.46 => ( P @ R2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.induct_scheme
% 0.29/0.46 thf(fact_90_pair__pre__digraph_Ocases__scheme,axiom,
% 0.29/0.46 ! [A: $tType,Z: $tType,R2: pair_p1731315293ph_ext @ A @ Z] :
% 0.29/0.46 ~ ! [Pverts3: set @ A,Parcs3: set @ ( product_prod @ A @ A ),More3: Z] :
% 0.29/0.46 ( R2
% 0.29/0.46 != ( pair_p1152105390ph_ext @ A @ Z @ Pverts3 @ Parcs3 @ More3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.cases_scheme
% 0.29/0.46 thf(fact_91_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 0.29/0.46 ! [Z: $tType,A: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.29/0.46 ( ( pair_pair_pre_parcs @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.29/0.46 = Parcs ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.select_convs(2)
% 0.29/0.46 thf(fact_92_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 0.29/0.46 ! [Z: $tType,A: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.29/0.46 ( ( pair_pair_pre_pverts @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.29/0.46 = Pverts ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.select_convs(1)
% 0.29/0.46 thf(fact_93_pair__pre__digraph_Oselect__convs_I3_J,axiom,
% 0.29/0.46 ! [A: $tType,Z: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.29/0.46 ( ( pair_pair_pre_more @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.29/0.46 = More ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.select_convs(3)
% 0.29/0.46 thf(fact_94_Fpow__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( finite_Fpow @ A )
% 0.29/0.46 = ( ^ [A2: set @ A] :
% 0.29/0.46 ( collect @ ( set @ A )
% 0.29/0.46 @ ^ [X6: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
% 0.29/0.46 & ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Fpow_def
% 0.29/0.46 thf(fact_95_pair__fin__digraph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G )
% 0.29/0.46 & ( pair_p1373739070axioms @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph_def
% 0.29/0.46 thf(fact_96_pair__fin__digraph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p1373739070axioms @ A @ G2 )
% 0.29/0.46 => ( pair_p953238171igraph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.intro
% 0.29/0.46 thf(fact_97_pair__pre__digraph_Ounfold__congs_I3_J,axiom,
% 0.29/0.46 ! [A: $tType,A7: $tType,R2: pair_p1731315293ph_ext @ A7 @ A,R3: pair_p1731315293ph_ext @ A7 @ A,V: A,F2: A > A,F4: A > A] :
% 0.29/0.46 ( ( R2 = R3 )
% 0.29/0.46 => ( ( ( pair_pair_pre_more @ A7 @ A @ R3 )
% 0.29/0.46 = V )
% 0.29/0.46 => ( ! [V2: A] :
% 0.29/0.46 ( ( V2 = V )
% 0.29/0.46 => ( ( F2 @ V2 )
% 0.29/0.46 = ( F4 @ V2 ) ) )
% 0.29/0.46 => ( ( pair_p1108258481update @ A @ A7 @ F2 @ R2 )
% 0.29/0.46 = ( pair_p1108258481update @ A @ A7 @ F4 @ R3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.unfold_congs(3)
% 0.29/0.46 thf(fact_98_pair__pre__digraph_Ofold__congs_I3_J,axiom,
% 0.29/0.46 ! [A: $tType,A7: $tType,R2: pair_p1731315293ph_ext @ A7 @ A,R3: pair_p1731315293ph_ext @ A7 @ A,V: A,F2: A > A,F4: A > A] :
% 0.29/0.46 ( ( R2 = R3 )
% 0.29/0.46 => ( ( ( pair_pair_pre_more @ A7 @ A @ R3 )
% 0.29/0.46 = V )
% 0.29/0.46 => ( ! [V2: A] :
% 0.29/0.46 ( ( V = V2 )
% 0.29/0.46 => ( ( F2 @ V2 )
% 0.29/0.46 = ( F4 @ V2 ) ) )
% 0.29/0.46 => ( ( pair_p1108258481update @ A @ A7 @ F2 @ R2 )
% 0.29/0.46 = ( pair_p1108258481update @ A @ A7 @ F4 @ R3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.fold_congs(3)
% 0.29/0.46 thf(fact_99_finite__cartesian__product__iff,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ( finite_finite2 @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 ) )
% 0.29/0.46 = ( ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( B2
% 0.29/0.46 = ( bot_bot @ ( set @ B ) ) )
% 0.29/0.46 | ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 & ( finite_finite2 @ B @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_product_iff
% 0.29/0.46 thf(fact_100_bot__apply,axiom,
% 0.29/0.46 ! [C: $tType,D2: $tType] :
% 0.29/0.46 ( ( bot @ C )
% 0.29/0.46 => ( ( bot_bot @ ( D2 > C ) )
% 0.29/0.46 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_apply
% 0.29/0.46 thf(fact_101_empty__iff,axiom,
% 0.29/0.46 ! [A: $tType,C3: A] :
% 0.29/0.46 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_iff
% 0.29/0.46 thf(fact_102_all__not__in__conv,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( ! [X3: A] :
% 0.29/0.46 ~ ( member @ A @ X3 @ A3 ) )
% 0.29/0.46 = ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % all_not_in_conv
% 0.29/0.46 thf(fact_103_Collect__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( collect @ A @ P )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( ! [X3: A] :
% 0.29/0.46 ~ ( P @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq
% 0.29/0.46 thf(fact_104_empty__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A @ P ) )
% 0.29/0.46 = ( ! [X3: A] :
% 0.29/0.46 ~ ( P @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_Collect_eq
% 0.29/0.46 thf(fact_105_empty__subsetI,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.29/0.46
% 0.29/0.46 % empty_subsetI
% 0.29/0.46 thf(fact_106_subset__empty,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_empty
% 0.29/0.46 thf(fact_107_Sigma__empty1,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_empty1
% 0.29/0.46 thf(fact_108_Times__empty,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ( ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 )
% 0.29/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.29/0.46 = ( ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( B2
% 0.29/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Times_empty
% 0.29/0.46 thf(fact_109_Sigma__empty2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 0.29/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_empty2
% 0.29/0.46 thf(fact_110_empty__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A
% 0.29/0.46 @ ^ [X3: A] : $false ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_def
% 0.29/0.46 thf(fact_111_Sigma__empty__iff,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,I: set @ A,X5: A > ( set @ B )] :
% 0.29/0.46 ( ( ( product_Sigma @ A @ B @ I @ X5 )
% 0.29/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.29/0.46 = ( ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ I )
% 0.29/0.46 => ( ( X5 @ X3 )
% 0.29/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Sigma_empty_iff
% 0.29/0.46 thf(fact_112_empty__in__Fpow,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % empty_in_Fpow
% 0.29/0.46 thf(fact_113_Fpow__not__empty,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( finite_Fpow @ A @ A3 )
% 0.29/0.46 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Fpow_not_empty
% 0.29/0.46 thf(fact_114_emptyE,axiom,
% 0.29/0.46 ! [A: $tType,A4: A] :
% 0.29/0.46 ~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % emptyE
% 0.29/0.46 thf(fact_115_equals0D,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,A4: A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ~ ( member @ A @ A4 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0D
% 0.29/0.46 thf(fact_116_equals0I,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ! [Y4: A] :
% 0.29/0.46 ~ ( member @ A @ Y4 @ A3 )
% 0.29/0.46 => ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equals0I
% 0.29/0.46 thf(fact_117_ex__in__conv,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.29/0.46 = ( A3
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_in_conv
% 0.29/0.46 thf(fact_118_bot__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( bot @ B )
% 0.29/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.29/0.46 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_fun_def
% 0.29/0.46 thf(fact_119_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 0.29/0.46 => ( A4
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_120_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 0.29/0.46 = ( A4
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_121_bot_Oextremum,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A )
% 0.29/0.46 => ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_122_finite_OemptyI,axiom,
% 0.29/0.46 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite.emptyI
% 0.29/0.46 thf(fact_123_infinite__imp__nonempty,axiom,
% 0.29/0.46 ! [A: $tType,S: set @ A] :
% 0.29/0.46 ( ~ ( finite_finite2 @ A @ S )
% 0.29/0.46 => ( S
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_imp_nonempty
% 0.29/0.46 thf(fact_124_subset__emptyI,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ( ! [X: A] :
% 0.29/0.46 ~ ( member @ A @ X @ A3 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_emptyI
% 0.29/0.46 thf(fact_125_times__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B,C2: set @ A,D: set @ B] :
% 0.29/0.46 ( ( ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 )
% 0.29/0.46 = ( product_Sigma @ A @ B @ C2
% 0.29/0.46 @ ^ [Uu: A] : D ) )
% 0.29/0.46 = ( ( ( A3 = C2 )
% 0.29/0.46 & ( B2 = D ) )
% 0.29/0.46 | ( ( ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( B2
% 0.29/0.46 = ( bot_bot @ ( set @ B ) ) ) )
% 0.29/0.46 & ( ( C2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( D
% 0.29/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % times_eq_iff
% 0.29/0.46 thf(fact_126_Fpow__mono,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Fpow_mono
% 0.29/0.46 thf(fact_127_pair__fin__digraph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_fin_digraph.axioms(1)
% 0.29/0.46 thf(fact_128_finite__has__maximal,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ( A3
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ? [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 & ! [Xa: A] :
% 0.29/0.46 ( ( member @ A @ Xa @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Xa )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_maximal
% 0.29/0.46 thf(fact_129_finite__has__minimal,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ( A3
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ? [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 & ! [Xa: A] :
% 0.29/0.46 ( ( member @ A @ Xa @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Xa @ X )
% 0.29/0.46 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_has_minimal
% 0.29/0.46 thf(fact_130_pair__pre__digraph_Oupdate__convs_I3_J,axiom,
% 0.29/0.46 ! [A: $tType,Z: $tType,More2: Z > Z,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.29/0.46 ( ( pair_p1108258481update @ Z @ A @ More2 @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.29/0.46 = ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ ( More2 @ More ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pre_digraph.update_convs(3)
% 0.29/0.46 thf(fact_131_times__subset__iff,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A3: set @ A,C2: set @ B,B2: set @ A,D: set @ B] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : C2 )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ B2
% 0.29/0.46 @ ^ [Uu: A] : D ) )
% 0.29/0.46 = ( ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 | ( C2
% 0.29/0.46 = ( bot_bot @ ( set @ B ) ) )
% 0.29/0.46 | ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ B ) @ C2 @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % times_subset_iff
% 0.29/0.46 thf(fact_132_finite__SigmaI2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( finite_finite2 @ A
% 0.29/0.46 @ ( collect @ A
% 0.29/0.46 @ ^ [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A3 )
% 0.29/0.46 & ( ( B2 @ X3 )
% 0.29/0.46 != ( bot_bot @ ( set @ B ) ) ) ) ) )
% 0.29/0.46 => ( ! [A5: A] :
% 0.29/0.46 ( ( member @ A @ A5 @ A3 )
% 0.29/0.46 => ( finite_finite2 @ B @ ( B2 @ A5 ) ) )
% 0.29/0.46 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_SigmaI2
% 0.29/0.46 thf(fact_133_finite__cartesian__productD1,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ( finite_finite2 @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 ) )
% 0.29/0.46 => ( ( B2
% 0.29/0.46 != ( bot_bot @ ( set @ B ) ) )
% 0.29/0.46 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_productD1
% 0.29/0.46 thf(fact_134_finite__cartesian__productD2,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
% 0.29/0.46 ( ( finite_finite2 @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_Sigma @ A @ B @ A3
% 0.29/0.46 @ ^ [Uu: A] : B2 ) )
% 0.29/0.46 => ( ( A3
% 0.29/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 => ( finite_finite2 @ B @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_cartesian_productD2
% 0.29/0.46 thf(fact_135_finite__transitivity__chain,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,R: A > A > $o] :
% 0.29/0.46 ( ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ~ ( R @ X @ X )
% 0.29/0.46 => ( ! [X: A,Y4: A,Z5: A] :
% 0.29/0.46 ( ( R @ X @ Y4 )
% 0.29/0.46 => ( ( R @ Y4 @ Z5 )
% 0.29/0.46 => ( R @ X @ Z5 ) ) )
% 0.29/0.46 => ( ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 => ? [Y5: A] :
% 0.29/0.46 ( ( member @ A @ Y5 @ A3 )
% 0.29/0.46 & ( R @ X @ Y5 ) ) )
% 0.29/0.46 => ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_transitivity_chain
% 0.29/0.46 thf(fact_136_pair__wf__digraph_Opair__wf__digraph__subdivide,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_pair_wf_digraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_wf_digraph.pair_wf_digraph_subdivide
% 0.29/0.46 thf(fact_137_Set_Ois__empty__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( is_empty @ A )
% 0.29/0.46 = ( ^ [A2: set @ A] :
% 0.29/0.46 ( A2
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.is_empty_def
% 0.29/0.46 thf(fact_138_pair__wf__digraph_Oin__arcsD2,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( member @ A @ V3 @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_wf_digraph.in_arcsD2
% 0.29/0.46 thf(fact_139_pair__wf__digraph_Oin__arcsD1,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( member @ A @ U @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_wf_digraph.in_arcsD1
% 0.29/0.46 thf(fact_140_old_Oprod_Oinject,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A4: A,B4: B,A8: A,B7: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ A4 @ B4 )
% 0.29/0.46 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 0.29/0.46 = ( ( A4 = A8 )
% 0.29/0.46 & ( B4 = B7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.inject
% 0.29/0.46 thf(fact_141_prod_Oinject,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.29/0.46 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.29/0.46 = ( ( X1 = Y1 )
% 0.29/0.46 & ( X22 = Y22 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod.inject
% 0.29/0.46 thf(fact_142_SigmaI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A4: A,A3: set @ A,B4: B,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( member @ A @ A4 @ A3 )
% 0.29/0.46 => ( ( member @ B @ B4 @ ( B2 @ A4 ) )
% 0.29/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % SigmaI
% 0.29/0.46 thf(fact_143_mem__Sigma__iff,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.29/0.46 = ( ( member @ A @ A4 @ A3 )
% 0.29/0.46 & ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Sigma_iff
% 0.29/0.46 thf(fact_144_bot__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.29/0.46 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq
% 0.29/0.46 thf(fact_145_bot__empty__eq2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( A > B > $o ) )
% 0.29/0.46 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq2
% 0.29/0.46 thf(fact_146_bot__set__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_set_def
% 0.29/0.46 thf(fact_147_old_Oprod_Oinducts,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.29/0.46 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.29/0.46 => ( P @ Prod ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.inducts
% 0.29/0.46 thf(fact_148_old_Oprod_Oexhaust,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
% 0.29/0.46 ~ ! [A5: A,B6: B] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( product_Pair @ A @ B @ A5 @ B6 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.exhaust
% 0.29/0.46 thf(fact_149_prod__induct7,axiom,
% 0.29/0.46 ! [G5: $tType,F5: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) )] :
% 0.29/0.46 ( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5,G6: G5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G5 ) @ E3 @ ( product_Pair @ F5 @ G5 @ F @ G6 ) ) ) ) ) ) )
% 0.29/0.46 => ( P @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct7
% 0.29/0.46 thf(fact_150_prod__induct6,axiom,
% 0.29/0.46 ! [F5: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) )] :
% 0.29/0.46 ( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F5 ) @ D3 @ ( product_Pair @ E2 @ F5 @ E3 @ F ) ) ) ) ) )
% 0.29/0.46 => ( P @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct6
% 0.29/0.46 thf(fact_151_prod__induct5,axiom,
% 0.29/0.46 ! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 0.29/0.46 ( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) )
% 0.29/0.46 => ( P @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct5
% 0.29/0.46 thf(fact_152_prod__induct4,axiom,
% 0.29/0.46 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.29/0.46 ( ! [A5: A,B6: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B6 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.29/0.46 => ( P @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct4
% 0.29/0.46 thf(fact_153_prod__induct3,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.29/0.46 ( ! [A5: A,B6: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) )
% 0.29/0.46 => ( P @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct3
% 0.29/0.46 thf(fact_154_prod__cases7,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F5: $tType,G5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) )] :
% 0.29/0.46 ~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5,G6: G5] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G5 ) @ E3 @ ( product_Pair @ F5 @ G5 @ F @ G6 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases7
% 0.29/0.46 thf(fact_155_prod__cases6,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) )] :
% 0.29/0.46 ~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F5 ) @ D3 @ ( product_Pair @ E2 @ F5 @ E3 @ F ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases6
% 0.29/0.46 thf(fact_156_prod__cases5,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 0.29/0.46 ~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases5
% 0.29/0.46 thf(fact_157_prod__cases4,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.29/0.46 ~ ! [A5: A,B6: B,C4: C,D3: D2] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B6 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases4
% 0.29/0.46 thf(fact_158_prod__cases3,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.29/0.46 ~ ! [A5: A,B6: B,C4: C] :
% 0.29/0.46 ( Y2
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases3
% 0.29/0.46 thf(fact_159_Pair__inject,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A4: A,B4: B,A8: A,B7: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ A4 @ B4 )
% 0.29/0.46 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 0.29/0.46 => ~ ( ( A4 = A8 )
% 0.29/0.46 => ( B4 != B7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Pair_inject
% 0.29/0.46 thf(fact_160_prod__cases,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.29/0.46 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.29/0.46 => ( P @ P2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases
% 0.29/0.46 thf(fact_161_surj__pair,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.29/0.46 ? [X: A,Y4: B] :
% 0.29/0.46 ( P2
% 0.29/0.46 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % surj_pair
% 0.29/0.46 thf(fact_162_subdivide_Oinduct,axiom,
% 0.29/0.46 ! [A: $tType,P: ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > $o,A0: pair_p1731315293ph_ext @ A @ product_unit,A1: product_prod @ A @ A,A22: A] :
% 0.29/0.46 ( ! [G7: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V2: A,X_12: A] : ( P @ G7 @ ( product_Pair @ A @ A @ U2 @ V2 ) @ X_12 )
% 0.29/0.46 => ( P @ A0 @ A1 @ A22 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subdivide.induct
% 0.29/0.46 thf(fact_163_subdivide_Ocases,axiom,
% 0.29/0.46 ! [A: $tType,X4: product_prod @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A )] :
% 0.29/0.46 ~ ! [G7: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V2: A,W2: A] :
% 0.29/0.46 ( X4
% 0.29/0.46 != ( product_Pair @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A ) @ G7 @ ( product_Pair @ ( product_prod @ A @ A ) @ A @ ( product_Pair @ A @ A @ U2 @ V2 ) @ W2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subdivide.cases
% 0.29/0.46 thf(fact_164_pred__equals__eq2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.29/0.46 ( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.29/0.46 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) ) )
% 0.29/0.46 = ( R = S ) ) ).
% 0.29/0.46
% 0.29/0.46 % pred_equals_eq2
% 0.29/0.46 thf(fact_165_ssubst__Pair__rhs,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
% 0.29/0.46 => ( ( S3 = S2 )
% 0.29/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ssubst_Pair_rhs
% 0.29/0.46 thf(fact_166_SigmaE,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.29/0.46 => ~ ! [X: A] :
% 0.29/0.46 ( ( member @ A @ X @ A3 )
% 0.29/0.46 => ! [Y4: B] :
% 0.29/0.46 ( ( member @ B @ Y4 @ ( B2 @ X ) )
% 0.29/0.46 => ( C3
% 0.29/0.46 != ( product_Pair @ A @ B @ X @ Y4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % SigmaE
% 0.29/0.46 thf(fact_167_SigmaD1,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.29/0.46 => ( member @ A @ A4 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % SigmaD1
% 0.29/0.46 thf(fact_168_SigmaD2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.29/0.46 => ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % SigmaD2
% 0.29/0.46 thf(fact_169_SigmaE2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.29/0.46 => ~ ( ( member @ A @ A4 @ A3 )
% 0.29/0.46 => ~ ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % SigmaE2
% 0.29/0.46 thf(fact_170_subrelI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.29/0.46 ( ! [X: A,Y4: B] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 )
% 0.29/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subrelI
% 0.29/0.46 thf(fact_171_pred__subset__eq2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o )
% 0.29/0.46 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 0.29/0.46 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.29/0.46
% 0.29/0.46 % pred_subset_eq2
% 0.29/0.46 thf(fact_172_relChain__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B )
% 0.29/0.46 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.29/0.46 = ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.29/0.46 ! [I2: A,J: A] :
% 0.29/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R4 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % relChain_def
% 0.29/0.46 thf(fact_173_old_Oprod_Orec,axiom,
% 0.29/0.46 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B4: B] :
% 0.29/0.46 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.29/0.46 = ( F1 @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.rec
% 0.29/0.46 thf(fact_174_less__by__empty,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_by_empty
% 0.29/0.46 thf(fact_175_calculation,axiom,
% 0.29/0.46 ( ( finite_finite2 @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g ) )
% 0.29/0.46 & ( ( finite_card @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g ) )
% 0.29/0.46 = n )
% 0.29/0.46 & ( ( pair_pair_pre_parcs @ a @ product_unit @ g )
% 0.29/0.46 = ( collect @ ( product_prod @ a @ a )
% 0.29/0.46 @ ( product_case_prod @ a @ a @ $o
% 0.29/0.46 @ ^ [U3: a,V4: a] :
% 0.29/0.46 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ U3 @ V4 )
% 0.29/0.46 @ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
% 0.29/0.46 @ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) )
% 0.29/0.46 & ( U3 != V4 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % calculation
% 0.29/0.46 thf(fact_176_predicate2I,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.29/0.46 ( ! [X: A,Y4: B] :
% 0.29/0.46 ( ( P @ X @ Y4 )
% 0.29/0.46 => ( Q @ X @ Y4 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2I
% 0.29/0.46 thf(fact_177_split__part,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ $o
% 0.29/0.46 @ ^ [A6: A,B5: B] :
% 0.29/0.46 ( P
% 0.29/0.46 & ( Q @ A6 @ B5 ) ) )
% 0.29/0.46 = ( ^ [Ab: product_prod @ A @ B] :
% 0.29/0.46 ( P
% 0.29/0.46 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_part
% 0.29/0.46 thf(fact_178_case__prod__conv,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A4: B,B4: C] :
% 0.29/0.46 ( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A4 @ B4 ) )
% 0.29/0.46 = ( F2 @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_conv
% 0.29/0.46 thf(fact_179_case__prodI2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
% 0.29/0.46 ( ! [A5: A,B6: B] :
% 0.29/0.46 ( ( P2
% 0.29/0.46 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.29/0.46 => ( C3 @ A5 @ B6 ) )
% 0.29/0.46 => ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodI2
% 0.29/0.46 thf(fact_180_case__prodI,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B4: B] :
% 0.29/0.46 ( ( F2 @ A4 @ B4 )
% 0.29/0.46 => ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodI
% 0.29/0.46 thf(fact_181_Collect__case__prod,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 0.29/0.46 ( ( collect @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_case_prod @ A @ B @ $o
% 0.29/0.46 @ ^ [A6: A,B5: B] :
% 0.29/0.46 ( ( P @ A6 )
% 0.29/0.46 & ( Q @ B5 ) ) ) )
% 0.29/0.46 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.29/0.46 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_case_prod
% 0.29/0.46 thf(fact_182_bot2E,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,X4: A,Y2: B] :
% 0.29/0.46 ~ ( bot_bot @ ( A > B > $o ) @ X4 @ Y2 ) ).
% 0.29/0.46
% 0.29/0.46 % bot2E
% 0.29/0.46 thf(fact_183_Collect__case__prod__mono,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A3: A > B > $o,B2: A > B > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_case_prod_mono
% 0.29/0.46 thf(fact_184_case__prodD,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B4: B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.29/0.46 => ( F2 @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodD
% 0.29/0.46 thf(fact_185_case__prodE,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
% 0.29/0.46 => ~ ! [X: A,Y4: B] :
% 0.29/0.46 ( ( P2
% 0.29/0.46 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.29/0.46 => ~ ( C3 @ X @ Y4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodE
% 0.29/0.46 thf(fact_186_case__prodE2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z3: product_prod @ B @ C] :
% 0.29/0.46 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z3 ) )
% 0.29/0.46 => ~ ! [X: B,Y4: C] :
% 0.29/0.46 ( ( Z3
% 0.29/0.46 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 0.29/0.46 => ~ ( Q @ ( P @ X @ Y4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodE2
% 0.29/0.46 thf(fact_187_case__prod__eta,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ C
% 0.29/0.46 @ ^ [X3: A,Y3: B] : ( F2 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.29/0.46 = F2 ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_eta
% 0.29/0.46 thf(fact_188_cond__case__prod__eta,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 0.29/0.46 ( ! [X: A,Y4: B] :
% 0.29/0.46 ( ( F2 @ X @ Y4 )
% 0.29/0.46 = ( G3 @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
% 0.29/0.46 => ( ( product_case_prod @ A @ B @ C @ F2 )
% 0.29/0.46 = G3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % cond_case_prod_eta
% 0.29/0.46 thf(fact_189_old_Oprod_Ocase,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.29/0.46 = ( F2 @ X1 @ X22 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.case
% 0.29/0.46 thf(fact_190_split__cong,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F2: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.29/0.46 ( ! [X: A,Y4: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ X @ Y4 )
% 0.29/0.46 = Q2 )
% 0.29/0.46 => ( ( F2 @ X @ Y4 )
% 0.29/0.46 = ( G3 @ X @ Y4 ) ) )
% 0.29/0.46 => ( ( P2 = Q2 )
% 0.29/0.46 => ( ( product_case_prod @ A @ B @ C @ F2 @ P2 )
% 0.29/0.46 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_cong
% 0.29/0.46 thf(fact_191_rev__predicate2D,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,X4: A,Y2: B,Q: A > B > $o] :
% 0.29/0.46 ( ( P @ X4 @ Y2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 => ( Q @ X4 @ Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_predicate2D
% 0.29/0.46 thf(fact_192_predicate2D,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X4: A,Y2: B] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 => ( ( P @ X4 @ Y2 )
% 0.29/0.46 => ( Q @ X4 @ Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2D
% 0.29/0.46 thf(fact_193_Collect__case__prod__Sigma,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 0.29/0.46 ( ( collect @ ( product_prod @ A @ B )
% 0.29/0.46 @ ( product_case_prod @ A @ B @ $o
% 0.29/0.46 @ ^ [X3: A,Y3: B] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 & ( Q @ X3 @ Y3 ) ) ) )
% 0.29/0.46 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.29/0.46 @ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_case_prod_Sigma
% 0.29/0.46 thf(fact_194_prod_Odisc__eq__case,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.29/0.46 ( product_case_prod @ A @ B @ $o
% 0.29/0.46 @ ^ [Uu: A,Uv: B] : $true
% 0.29/0.46 @ Prod ) ).
% 0.29/0.46
% 0.29/0.46 % prod.disc_eq_case
% 0.29/0.46 thf(fact_195_prod_Ocase__distrib,axiom,
% 0.29/0.46 ! [C: $tType,D2: $tType,B: $tType,A: $tType,H: C > D2,F2: A > B > C,Prod: product_prod @ A @ B] :
% 0.29/0.46 ( ( H @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
% 0.29/0.46 = ( product_case_prod @ A @ B @ D2
% 0.29/0.46 @ ^ [X12: A,X23: B] : ( H @ ( F2 @ X12 @ X23 ) )
% 0.29/0.46 @ Prod ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod.case_distrib
% 0.29/0.46 thf(fact_196_card__subset__eq,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ( finite_card @ A @ A3 )
% 0.29/0.46 = ( finite_card @ A @ B2 ) )
% 0.29/0.46 => ( A3 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % card_subset_eq
% 0.29/0.46 thf(fact_197_infinite__arbitrarily__large,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,N: nat] :
% 0.29/0.46 ( ~ ( finite_finite2 @ A @ A3 )
% 0.29/0.46 => ? [B8: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ B8 )
% 0.29/0.46 & ( ( finite_card @ A @ B8 )
% 0.29/0.46 = N )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ B8 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_arbitrarily_large
% 0.29/0.46 thf(fact_198_finite__if__finite__subsets__card__bdd,axiom,
% 0.29/0.46 ! [A: $tType,F6: set @ A,C2: nat] :
% 0.29/0.46 ( ! [G7: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ G7 @ F6 )
% 0.29/0.46 => ( ( finite_finite2 @ A @ G7 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( finite_card @ A @ G7 ) @ C2 ) ) )
% 0.29/0.46 => ( ( finite_finite2 @ A @ F6 )
% 0.29/0.46 & ( ord_less_eq @ nat @ ( finite_card @ A @ F6 ) @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_if_finite_subsets_card_bdd
% 0.29/0.46 thf(fact_199_card__seteq,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A3 ) )
% 0.29/0.46 => ( A3 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % card_seteq
% 0.29/0.46 thf(fact_200_card__mono,axiom,
% 0.29/0.46 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.29/0.46 ( ( finite_finite2 @ A @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % card_mono
% 0.29/0.46 thf(fact_201_obtain__subset__with__card__n,axiom,
% 0.29/0.46 ! [A: $tType,N: nat,S: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
% 0.29/0.46 => ~ ! [T4: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
% 0.29/0.46 => ( ( ( finite_card @ A @ T4 )
% 0.29/0.46 = N )
% 0.29/0.46 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % obtain_subset_with_card_n
% 0.29/0.46 thf(fact_202_Collect__empty__eq__bot,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( collect @ A @ P )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( P
% 0.29/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq_bot
% 0.29/0.46 thf(fact_203_card__Ex__subset,axiom,
% 0.29/0.46 ! [A: $tType,K: nat,M: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ M ) )
% 0.29/0.46 => ? [N2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ N2 @ M )
% 0.29/0.46 & ( ( finite_card @ A @ N2 )
% 0.29/0.46 = K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % card_Ex_subset
% 0.29/0.46 thf(fact_204_mem__case__prodI2,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z3: C,C3: A > B > ( set @ C )] :
% 0.29/0.46 ( ! [A5: A,B6: B] :
% 0.29/0.46 ( ( P2
% 0.29/0.46 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.29/0.46 => ( member @ C @ Z3 @ ( C3 @ A5 @ B6 ) ) )
% 0.29/0.46 => ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_case_prodI2
% 0.29/0.46 thf(fact_205_mem__case__prodI,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),A4: B,B4: C] :
% 0.29/0.46 ( ( member @ A @ Z3 @ ( C3 @ A4 @ B4 ) )
% 0.29/0.46 => ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_case_prodI
% 0.29/0.46 thf(fact_206_case__prodI2_H,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X4: C] :
% 0.29/0.46 ( ! [A5: A,B6: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ A5 @ B6 )
% 0.29/0.46 = P2 )
% 0.29/0.46 => ( C3 @ A5 @ B6 @ X4 ) )
% 0.29/0.46 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodI2'
% 0.29/0.46 thf(fact_207_finite__Collect__le__nat,axiom,
% 0.29/0.46 ! [K: nat] :
% 0.29/0.46 ( finite_finite2 @ nat
% 0.29/0.46 @ ( collect @ nat
% 0.29/0.46 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_Collect_le_nat
% 0.29/0.46 thf(fact_208_case__prodD_H,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A4: A,B4: B,C3: C] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A4 @ B4 ) @ C3 )
% 0.29/0.46 => ( R @ A4 @ B4 @ C3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodD'
% 0.29/0.46 thf(fact_209_case__prodE_H,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z3: C] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z3 )
% 0.29/0.46 => ~ ! [X: A,Y4: B] :
% 0.29/0.46 ( ( P2
% 0.29/0.46 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.29/0.46 => ~ ( C3 @ X @ Y4 @ Z3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prodE'
% 0.29/0.46 thf(fact_210_mem__case__prodE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.29/0.46 ( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 0.29/0.46 => ~ ! [X: B,Y4: C] :
% 0.29/0.46 ( ( P2
% 0.29/0.46 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 0.29/0.46 => ~ ( member @ A @ Z3 @ ( C3 @ X @ Y4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_case_prodE
% 0.29/0.46 thf(fact_211_finite__nat__set__iff__bounded__le,axiom,
% 0.29/0.46 ( ( finite_finite2 @ nat )
% 0.29/0.46 = ( ^ [N4: set @ nat] :
% 0.29/0.46 ? [M2: nat] :
% 0.29/0.46 ! [X3: nat] :
% 0.29/0.46 ( ( member @ nat @ X3 @ N4 )
% 0.29/0.46 => ( ord_less_eq @ nat @ X3 @ M2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_nat_set_iff_bounded_le
% 0.29/0.46 thf(fact_212_infinite__nat__iff__unbounded__le,axiom,
% 0.29/0.46 ! [S: set @ nat] :
% 0.29/0.46 ( ( ~ ( finite_finite2 @ nat @ S ) )
% 0.29/0.46 = ( ! [M2: nat] :
% 0.29/0.46 ? [N3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 0.29/0.46 & ( member @ nat @ N3 @ S ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_nat_iff_unbounded_le
% 0.29/0.46 thf(fact_213_bounded__Max__nat,axiom,
% 0.29/0.46 ! [P: nat > $o,X4: nat,M: nat] :
% 0.29/0.46 ( ( P @ X4 )
% 0.29/0.46 => ( ! [X: nat] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( ord_less_eq @ nat @ X @ M ) )
% 0.29/0.46 => ~ ! [M3: nat] :
% 0.29/0.46 ( ( P @ M3 )
% 0.29/0.46 => ~ ! [X2: nat] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bounded_Max_nat
% 0.29/0.46 thf(fact_214_finite__less__ub,axiom,
% 0.29/0.46 ! [F2: nat > nat,U: nat] :
% 0.29/0.46 ( ! [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( F2 @ N5 ) )
% 0.29/0.46 => ( finite_finite2 @ nat
% 0.29/0.46 @ ( collect @ nat
% 0.29/0.46 @ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F2 @ N3 ) @ U ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % finite_less_ub
% 0.29/0.46 thf(fact_215_card__le__if__inj__on__rel,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,B2: set @ A,A3: set @ B,R2: B > A > $o] :
% 0.29/0.46 ( ( finite_finite2 @ A @ B2 )
% 0.29/0.46 => ( ! [A5: B] :
% 0.29/0.46 ( ( member @ B @ A5 @ A3 )
% 0.29/0.46 => ? [B9: A] :
% 0.29/0.46 ( ( member @ A @ B9 @ B2 )
% 0.29/0.46 & ( R2 @ A5 @ B9 ) ) )
% 0.29/0.46 => ( ! [A12: B,A23: B,B6: A] :
% 0.29/0.46 ( ( member @ B @ A12 @ A3 )
% 0.29/0.46 => ( ( member @ B @ A23 @ A3 )
% 0.29/0.46 => ( ( member @ A @ B6 @ B2 )
% 0.29/0.46 => ( ( R2 @ A12 @ B6 )
% 0.29/0.46 => ( ( R2 @ A23 @ B6 )
% 0.29/0.46 => ( A12 = A23 ) ) ) ) ) )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % card_le_if_inj_on_rel
% 0.29/0.46 thf(fact_216_pair__pseudo__graph_Opair__pseudo__graph__subdivide,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.29/0.46 ( ( pair_p389585203_graph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_p389585203_graph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pseudo_graph.pair_pseudo_graph_subdivide
% 0.29/0.46 thf(fact_217_pair__graph_Opair__graph__subdivide,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.29/0.46 ( ( pair_pair_graph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_pair_graph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_graph.pair_graph_subdivide
% 0.29/0.46 thf(fact_218_pair__graph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_graph @ A @ G2 )
% 0.29/0.46 => ( pair_p389585203_graph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_graph.axioms(2)
% 0.29/0.46 thf(fact_219_pair__graph_Opair__graph,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_graph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_graph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_graph.pair_graph
% 0.29/0.46 thf(fact_220_pair__pseudo__graph_Opair__pseudo__graph,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p389585203_graph @ A @ G2 )
% 0.29/0.46 => ( pair_p389585203_graph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pseudo_graph.pair_pseudo_graph
% 0.29/0.46 thf(fact_221_pair__pseudo__graph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p389585203_graph @ A @ G2 )
% 0.29/0.46 => ( pair_p953238171igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pseudo_graph.axioms(1)
% 0.29/0.46 thf(fact_222_case__prod__app,axiom,
% 0.29/0.46 ! [A: $tType,D2: $tType,C: $tType,B: $tType] :
% 0.29/0.46 ( ( product_case_prod @ B @ C @ ( D2 > A ) )
% 0.29/0.46 = ( ^ [F3: B > C > D2 > A,X3: product_prod @ B @ C,Y3: D2] :
% 0.29/0.46 ( product_case_prod @ B @ C @ A
% 0.29/0.46 @ ^ [L: B,R4: C] : ( F3 @ L @ R4 @ Y3 )
% 0.29/0.46 @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_app
% 0.29/0.46 thf(fact_223_pair__bidirected__digraph_Opair__bidirected__digraph__subdivide,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.29/0.46 ( ( pair_p1033978749igraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_p1033978749igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_bidirected_digraph.pair_bidirected_digraph_subdivide
% 0.29/0.46 thf(fact_224_predicate2D__conj,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X4: A,Y2: B] :
% 0.29/0.46 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 & R )
% 0.29/0.46 => ( R
% 0.29/0.46 & ( ( P @ X4 @ Y2 )
% 0.29/0.46 => ( Q @ X4 @ Y2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2D_conj
% 0.29/0.46 thf(fact_225_case__prod__Pair__iden,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.29/0.46 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 0.29/0.46 = P2 ) ).
% 0.29/0.46
% 0.29/0.46 % case_prod_Pair_iden
% 0.29/0.46 thf(fact_226_eq__subset,axiom,
% 0.29/0.46 ! [A: $tType,P: A > A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( A > A > $o )
% 0.29/0.46 @ ^ [Y: A,Z2: A] : ( Y = Z2 )
% 0.29/0.46 @ ^ [A6: A,B5: A] :
% 0.29/0.46 ( ( P @ A6 @ B5 )
% 0.29/0.46 | ( A6 = B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_subset
% 0.29/0.46 thf(fact_227_pair__loopfree__digraph_Opair__loopfree__digraph__subdivide,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.29/0.46 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_p1282390076igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_loopfree_digraph.pair_loopfree_digraph_subdivide
% 0.29/0.46 thf(fact_228_pair__graph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_digraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p389585203_graph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_graph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_graph.intro
% 0.29/0.46 thf(fact_229_pair__bidirected__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p1033978749igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p1282390076igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_bidirected_digraph.axioms(2)
% 0.29/0.46 thf(fact_230_pair__loopfree__digraph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_loopfree_digraph.axioms(1)
% 0.29/0.46 thf(fact_231_pair__digraph_Opair__wf__digraph,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_digraph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_digraph.pair_wf_digraph
% 0.29/0.46 thf(fact_232_pair__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_digraph @ A @ G2 )
% 0.29/0.46 => ( pair_p1282390076igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_digraph.axioms(2)
% 0.29/0.46 thf(fact_233_pair__digraph_Opair__digraph,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_digraph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_digraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_digraph.pair_digraph
% 0.29/0.46 thf(fact_234_pair__digraph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_pair_digraph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G )
% 0.29/0.46 & ( pair_p1282390076igraph @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_digraph_def
% 0.29/0.46 thf(fact_235_pair__digraph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p1282390076igraph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_digraph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_digraph.intro
% 0.29/0.46 thf(fact_236_pair__digraph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_digraph @ A @ G2 )
% 0.29/0.46 => ( pair_p953238171igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_digraph.axioms(1)
% 0.29/0.46 thf(fact_237_pair__graph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_graph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_digraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_graph.axioms(1)
% 0.29/0.46 thf(fact_238_pair__loopfree__digraph_Ono__loops_H,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
% 0.29/0.46 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( U != V3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_loopfree_digraph.no_loops'
% 0.29/0.46 thf(fact_239_pair__graph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_pair_graph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_digraph @ A @ G )
% 0.29/0.46 & ( pair_p389585203_graph @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_graph_def
% 0.29/0.46 thf(fact_240_pair__loopfree__digraph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p814123999axioms @ A @ G2 )
% 0.29/0.46 => ( pair_p1282390076igraph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_loopfree_digraph.intro
% 0.29/0.46 thf(fact_241_pair__loopfree__digraph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_p1282390076igraph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G )
% 0.29/0.46 & ( pair_p814123999axioms @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_loopfree_digraph_def
% 0.29/0.46 thf(fact_242_pair__loopfree__digraph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p814123999axioms @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_loopfree_digraph.axioms(2)
% 0.29/0.46 thf(fact_243_pair__sym__digraph_Opair__sym__digraph__subdivide,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.29/0.46 ( ( pair_p460180511igraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( pair_p460180511igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_sym_digraph.pair_sym_digraph_subdivide
% 0.29/0.46 thf(fact_244_internal__case__prod__def,axiom,
% 0.29/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.29/0.46 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.29/0.46 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.29/0.46
% 0.29/0.46 % internal_case_prod_def
% 0.29/0.46 thf(fact_245_pair__sym__digraph_Oarcs__symmetric,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,A4: A,B4: A] :
% 0.29/0.46 ( ( pair_p460180511igraph @ A @ G2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.29/0.46 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_sym_digraph.arcs_symmetric
% 0.29/0.46 thf(fact_246_pair__bidirected__digraph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p1033978749igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p460180511igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_bidirected_digraph.axioms(1)
% 0.29/0.46 thf(fact_247_pair__sym__digraph_Oaxioms_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p460180511igraph @ A @ G2 )
% 0.29/0.46 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_sym_digraph.axioms(1)
% 0.29/0.46 thf(fact_248_pair__pseudo__graph_Oaxioms_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p389585203_graph @ A @ G2 )
% 0.29/0.46 => ( pair_p460180511igraph @ A @ G2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pseudo_graph.axioms(2)
% 0.29/0.46 thf(fact_249_pair__bidirected__digraph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p460180511igraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p1282390076igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p1033978749igraph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_bidirected_digraph.intro
% 0.29/0.46 thf(fact_250_pair__bidirected__digraph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_p1033978749igraph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p460180511igraph @ A @ G )
% 0.29/0.46 & ( pair_p1282390076igraph @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_bidirected_digraph_def
% 0.29/0.46 thf(fact_251_pair__pseudo__graph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_p389585203_graph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G )
% 0.29/0.46 & ( pair_p460180511igraph @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pseudo_graph_def
% 0.29/0.46 thf(fact_252_pair__pseudo__graph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_p953238171igraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p460180511igraph @ A @ G2 )
% 0.29/0.46 => ( pair_p389585203_graph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_pseudo_graph.intro
% 0.29/0.46 thf(fact_253_internal__case__prod__conv,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A4: B,B4: C] :
% 0.29/0.46 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A4 @ B4 ) )
% 0.29/0.46 = ( C3 @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % internal_case_prod_conv
% 0.29/0.46 thf(fact_254_pair__sym__digraph_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.29/0.46 => ( ( pair_p477239746axioms @ A @ G2 )
% 0.29/0.46 => ( pair_p460180511igraph @ A @ G2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_sym_digraph.intro
% 0.29/0.46 thf(fact_255_pair__sym__digraph__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( pair_p460180511igraph @ A )
% 0.29/0.46 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.29/0.46 ( ( pair_pair_wf_digraph @ A @ G )
% 0.29/0.46 & ( pair_p477239746axioms @ A @ G ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pair_sym_digraph_def
% 0.29/0.46
% 0.29/0.46 % Type constructors (33)
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( order_bot @ A9 )
% 0.29/0.46 => ( order_bot @ ( A7 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( preorder @ A9 )
% 0.29/0.46 => ( preorder @ ( A7 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A7 )
% 0.29/0.46 & ( finite_finite @ A9 ) )
% 0.29/0.46 => ( finite_finite @ ( A7 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( order @ A9 )
% 0.29/0.46 => ( order @ ( A7 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( ord @ A9 )
% 0.29/0.46 => ( ord @ ( A7 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( bot @ A9 )
% 0.29/0.46 => ( bot @ ( A7 > A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.29/0.46 order_bot @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.29/0.46 preorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.29/0.46 order @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.29/0.46 ord @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.29/0.46 bot @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.29/0.46 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.29/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 0.29/0.46 ! [A7: $tType] :
% 0.29/0.46 ( ( finite_finite @ A7 )
% 0.29/0.46 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.29/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.29/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.29/0.46 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
% 0.29/0.46 order_bot @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
% 0.29/0.46 preorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
% 0.29/0.46 linorder @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
% 0.29/0.46 finite_finite @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.29/0.46 order @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
% 0.29/0.46 ord @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
% 0.29/0.46 bot @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_19,axiom,
% 0.29/0.46 ! [A7: $tType,A9: $tType] :
% 0.29/0.46 ( ( ( finite_finite @ A7 )
% 0.29/0.46 & ( finite_finite @ A9 ) )
% 0.29/0.46 => ( finite_finite @ ( product_prod @ A7 @ A9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_20,axiom,
% 0.29/0.46 order_bot @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_21,axiom,
% 0.29/0.46 preorder @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_22,axiom,
% 0.29/0.46 linorder @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_23,axiom,
% 0.29/0.46 finite_finite @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_24,axiom,
% 0.29/0.46 order @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oord_25,axiom,
% 0.29/0.46 ord @ product_unit ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Product__Type_Ounit___Orderings_Obot_26,axiom,
% 0.29/0.46 bot @ product_unit ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 finite_finite2 @ ( product_prod @ a @ a ) @ ( pair_pair_pre_parcs @ a @ product_unit @ g ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 0.30/1.51 ---- Embedded in THF ---
% 0.30/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.51 %%% Generated on Tue Feb 24 08:26:50 EST 2026
% 0.30/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.51 %%% Logic specification used:
% 0.30/1.51 %%% thf(spec, logic, $$dhol).
% 0.30/1.51
% 0.30/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.XcU2giCvCK/DTF2DTF14783.p
% 0.30/1.51 thf(ty_t_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext, type, pair_p1731315293ph_ext: ($tType > ($tType > $tType))).
% 0.30/1.51 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 0.30/1.51 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 0.30/1.51 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.30/1.51 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.30/1.51 thf(ty_tf_a, type, a: $tType).
% 0.30/1.51 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 0.30/1.51 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 0.30/1.51 thf(sy_c_Finite__Set_OFpow, type, finite_Fpow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 0.30/1.51 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 0.30/1.51 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.30/1.51 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 0.30/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__bidirected__digraph, type, pair_p1033978749igraph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__digraph, type, pair_pair_digraph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__fin__digraph, type, pair_p953238171igraph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms, type, pair_p1373739070axioms: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__graph, type, pair_pair_graph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__loopfree__digraph, type, pair_p1282390076igraph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__loopfree__digraph__axioms, type, pair_p814123999axioms: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore, type, pair_pair_pre_more: (!> [A:$tType,Z:$tType]: ((((pair_p1731315293ph_ext @ A) @ Z) > Z)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore__update, type, pair_p1108258481update: (!> [Z:$tType,A:$tType]: (((Z > Z) > (((pair_p1731315293ph_ext @ A) @ Z) > ((pair_p1731315293ph_ext @ A) @ Z)))))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext, type, pair_p1152105390ph_ext: (!> [A:$tType,Z:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > (Z > ((pair_p1731315293ph_ext @ A) @ Z))))))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs, type, pair_pair_pre_parcs: (!> [A:$tType,Z:$tType]: ((((pair_p1731315293ph_ext @ A) @ Z) > (set @ ((product_prod @ A) @ A)))))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts, type, pair_pair_pre_pverts: (!> [A:$tType,Z:$tType]: ((((pair_p1731315293ph_ext @ A) @ Z) > (set @ A))))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__pseudo__graph, type, pair_p389585203_graph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__sym__digraph, type, pair_p460180511igraph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__sym__digraph__axioms, type, pair_p477239746axioms: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Opair__wf__digraph, type, pair_pair_wf_digraph: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > $o)))).
% 0.30/1.51 thf(sy_c_Pair__Digraph_Osubdivide, type, pair_subdivide: (!> [A:$tType]: ((((pair_p1731315293ph_ext @ A) @ product_unit) > (((product_prod @ A) @ A) > (A > ((pair_p1731315293ph_ext @ A) @ product_unit))))))).
% 0.30/1.51 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 0.30/1.51 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 0.30/1.51 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.30/1.51 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 0.30/1.51 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.30/1.51 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 0.30/1.51 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.30/1.51 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.30/1.51 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.30/1.51 thf(sy_v_G, type, g: ((pair_p1731315293ph_ext @ a) @ product_unit)).
% 0.30/1.51 thf(sy_v_n, type, n: nat).
% 0.30/1.51 thf(fact_0_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A2:(set @ A)]: ($true))))))).
% 0.30/1.51 thf(fact_1__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062, axiom, ((finite_finite2 @ ((product_prod @ a) @ a)) @ ((((product_Sigma @ a) @ a) @ (((pair_pair_pre_pverts @ a) @ product_unit) @ g)) @ (^ [Uu:a]: ((((pair_pair_pre_pverts @ a) @ product_unit) @ g)))))).
% 0.30/1.51 thf(fact_2__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062, axiom, (((ord_less_eq @ (set @ ((product_prod @ a) @ a))) @ (((pair_pair_pre_parcs @ a) @ product_unit) @ g)) @ ((((product_Sigma @ a) @ a) @ (((pair_pair_pre_pverts @ a) @ product_unit) @ g)) @ (^ [Uu:a]: ((((pair_pair_pre_pverts @ a) @ product_unit) @ g)))))).
% 0.30/1.51 thf(fact_3_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (? [X_1:B]: (((P @ X) @ X_1)))))) => (? [F:(A > B)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => ((P @ X2) @ (F @ X2)))))))))))).
% 0.30/1.51 thf(fact_4_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 0.30/1.51 thf(fact_5_pair__fin__digraph__axioms__def, axiom, (! [A:$tType]: (((pair_p1373739070axioms @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((finite_finite2 @ A) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G)) & ((finite_finite2 @ ((product_prod @ A) @ A)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G))))))))).
% 0.30/1.51 thf(fact_6_pair__fin__digraph__axioms_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((finite_finite2 @ A) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)) => (((finite_finite2 @ ((product_prod @ A) @ A)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((pair_p1373739070axioms @ A) @ G2)))))).
% 0.30/1.51 thf(fact_7_pair__fin__digraph_Opair__finite__arcs, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => ((finite_finite2 @ ((product_prod @ A) @ A)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)))))).
% 0.30/1.51 thf(fact_8_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((finite_finite2 @ A) @ B2) => ((finite_finite2 @ A) @ A3)))))).
% 0.30/1.51 thf(fact_9_infinite__super, axiom, (! [A:$tType,S:(set @ A),T2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T2) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T2)))))))).
% 0.30/1.51 thf(fact_10_rev__finite__subset, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((finite_finite2 @ A) @ A3)))))).
% 0.30/1.51 thf(fact_11_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A4:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A4) @ A3) => (? [X:A]: (((((member @ A) @ X) @ A3) & ((((ord_less_eq @ A) @ A4) @ X) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X) @ Xa) => (X = Xa))))))))))))))))).
% 0.30/1.51 thf(fact_12_finite__Collect__conjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3)))))))))).
% 0.30/1.51 thf(fact_13_finite__Collect__disjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) | (Q @ X3)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))).
% 0.30/1.51 thf(fact_14_finite__Collect__subsets, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B3) @ A3))))))))).
% 0.30/1.51 thf(fact_15_finite__SigmaI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((finite_finite2 @ B) @ (B2 @ A5))))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2))))))).
% 0.30/1.51 thf(fact_16_pair__fin__digraph_Oaxioms_I2_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => ((pair_p1373739070axioms @ A) @ G2))))).
% 0.30/1.51 thf(fact_17_not__finite__existsD, axiom, (! [A:$tType,P:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P)))) => (? [X_12:A]: ((P @ X_12))))))).
% 0.30/1.51 thf(fact_18_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ B),R:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B2) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (? [Xa:B]: (((((member @ B) @ Xa) @ B2) & ((R @ X) @ Xa))))))) => (? [X:B]: (((((member @ B) @ X) @ B2) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A6:A]: (((((member @ A) @ A6) @ A3) & ((R @ A6) @ X))))))))))))))))).
% 0.30/1.51 thf(fact_19_pair__fin__digraph_Opair__fin__digraph, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => ((pair_p953238171igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_20_finite__cartesian__product, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B2) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2))))))))).
% 0.30/1.51 thf(fact_21_pair__fin__digraph_Opair__finite__verts, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => ((finite_finite2 @ A) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))))).
% 0.30/1.51 thf(fact_22_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A4:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A4) @ A3) => (? [X:A]: (((((member @ A) @ X) @ A3) & ((((ord_less_eq @ A) @ X) @ A4) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X) => (X = Xa))))))))))))))))).
% 0.30/1.51 thf(fact_23_infinite__cartesian__product, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ B)]: (((~ (((finite_finite2 @ A) @ A3))) => ((~ (((finite_finite2 @ B) @ B2))) => (~ (((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2))))))))))).
% 0.30/1.51 thf(fact_24_Times__subset__cancel2, axiom, (! [A:$tType,B:$tType,X4:A,C2:(set @ A),A3:(set @ B),B2:(set @ B)]: (((((member @ A) @ X4) @ C2) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (C2)))) @ ((((product_Sigma @ B) @ A) @ B2) @ (^ [Uu:B]: (C2)))) = (((ord_less_eq @ (set @ B)) @ A3) @ B2)))))).
% 0.30/1.51 thf(fact_25_Sigma__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),C2:(set @ A),B2:(A > (set @ B)),D:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A3) @ C2) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ (set @ B)) @ (B2 @ X)) @ (D @ X))))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ B2)) @ ((((product_Sigma @ A) @ B) @ C2) @ D))))))).
% 0.30/1.51 thf(fact_26_subsetI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 0.30/1.51 thf(fact_27_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A3) => (A3 = B2)))))).
% 0.30/1.51 thf(fact_28_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 0.30/1.51 thf(fact_29_pair__pre__digraph_Oequality, axiom, (! [Z:$tType,A:$tType,R2:((pair_p1731315293ph_ext @ A) @ Z),R3:((pair_p1731315293ph_ext @ A) @ Z)]: ((((((pair_pair_pre_pverts @ A) @ Z) @ R2) = (((pair_pair_pre_pverts @ A) @ Z) @ R3)) => (((((pair_pair_pre_parcs @ A) @ Z) @ R2) = (((pair_pair_pre_parcs @ A) @ Z) @ R3)) => (((((pair_pair_pre_more @ A) @ Z) @ R2) = (((pair_pair_pre_more @ A) @ Z) @ R3)) => (R2 = R3))))))).
% 0.30/1.51 thf(fact_30_Sigma__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ A),C2:(A > (set @ B)),D:(A > (set @ B))]: (((A3 = B2) => ((! [X:A]: (((((member @ A) @ X) @ B2) => ((C2 @ X) = (D @ X))))) => (((((product_Sigma @ A) @ B) @ A3) @ C2) = ((((product_Sigma @ A) @ B) @ B2) @ D))))))).
% 0.30/1.51 thf(fact_31_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X4:A,C2:(set @ A),A3:(set @ B),B2:(set @ B)]: (((((member @ A) @ X4) @ C2) => ((((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (C2))) = ((((product_Sigma @ B) @ A) @ B2) @ (^ [Uu:B]: (C2)))) = (A3 = B2)))))).
% 0.30/1.51 thf(fact_32_Collect__subset, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A3) & (P @ X3)))))) @ A3)))).
% 0.30/1.51 thf(fact_33_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A2:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A2)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B3)))))))))).
% 0.30/1.51 thf(fact_34_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) => (Q @ X)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 0.30/1.51 thf(fact_35_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X4:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X4) => (Q @ X4)))))).
% 0.30/1.51 thf(fact_36_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X4:A,Q:(A > $o)]: (((P @ X4) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X4)))))).
% 0.30/1.51 thf(fact_37_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ A4) @ B4) => (A4 = B4))))))))).
% 0.30/1.51 thf(fact_38_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A6) & (((ord_less_eq @ A) @ A6) @ B5))))))))).
% 0.30/1.51 thf(fact_39_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A,C3:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C3) @ B4) => (((ord_less_eq @ A) @ C3) @ A4))))))))).
% 0.30/1.51 thf(fact_40_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B4:A]: (((! [A5:A,B6:A]: (((((ord_less_eq @ A) @ A5) @ B6) => ((P @ A5) @ B6)))) => ((! [A5:A,B6:A]: ((((P @ B6) @ A5) => ((P @ A5) @ B6)))) => ((P @ A4) @ B4))))))))).
% 0.30/1.51 thf(fact_41_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 0.30/1.51 thf(fact_42_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y2:A,Z3:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z3) => (((ord_less_eq @ A) @ X4) @ Z3))))))))).
% 0.30/1.51 thf(fact_43_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ A4) => (A4 = B4))))))))).
% 0.30/1.51 thf(fact_44_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C3:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((B4 = C3) => (((ord_less_eq @ A) @ A4) @ C3))))))))).
% 0.30/1.51 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 0.30/1.51 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 0.30/1.51 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.30/1.51 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G3:(A > B)]: (((! [X:A]: (((F2 @ X) = (G3 @ X)))) => (F2 = G3))))).
% 0.30/1.51 thf(fact_49_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C3:A]: (((A4 = B4) => ((((ord_less_eq @ A) @ B4) @ C3) => (((ord_less_eq @ A) @ A4) @ C3))))))))).
% 0.30/1.51 thf(fact_50_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) & (((ord_less_eq @ A) @ B5) @ A6))))))))).
% 0.30/1.51 thf(fact_51_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X4:A]: (((((ord_less_eq @ A) @ Y2) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y2) = (X4 = Y2))))))))).
% 0.30/1.51 thf(fact_52_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y2:A,Z3:A]: ((((((ord_less_eq @ A) @ X4) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z3)))) => (((((ord_less_eq @ A) @ Y2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z3)))) => (((((ord_less_eq @ A) @ X4) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y2)))) => (((((ord_less_eq @ A) @ Z3) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X4)))) => (((((ord_less_eq @ A) @ Y2) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y2))))))))))))))))).
% 0.30/1.51 thf(fact_53_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A,C3:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C3) => (((ord_less_eq @ A) @ A4) @ C3))))))))).
% 0.30/1.51 thf(fact_54_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X4)))))))).
% 0.30/1.51 thf(fact_55_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y2:A]: (((X4 = Y2) => (((ord_less_eq @ A) @ X4) @ Y2)))))))).
% 0.30/1.51 thf(fact_56_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X4)))))))).
% 0.30/1.51 thf(fact_57_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X4) => (X4 = Y2))))))))).
% 0.30/1.51 thf(fact_58_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 0.30/1.51 thf(fact_59_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B4:A,F2:(A > B),C3:B]: (((((ord_less_eq @ A) @ A4) @ B4) => (((F2 @ B4) = C3) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X)) @ (F2 @ Y4))))) => (((ord_less_eq @ B) @ (F2 @ A4)) @ C3)))))))))).
% 0.30/1.51 thf(fact_60_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F2:(B > A),B4:B,C3:B]: (((A4 = (F2 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C3) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A4) @ (F2 @ C3))))))))))).
% 0.30/1.51 thf(fact_61_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A4:A,B4:A,F2:(A > C),C3:C]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ C) @ (F2 @ B4)) @ C3) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ C) @ (F2 @ X)) @ (F2 @ Y4))))) => (((ord_less_eq @ C) @ (F2 @ A4)) @ C3)))))))))).
% 0.30/1.51 thf(fact_62_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F2:(B > A),B4:B,C3:B]: (((((ord_less_eq @ A) @ A4) @ (F2 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C3) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A4) @ (F2 @ C3))))))))))).
% 0.30/1.51 thf(fact_63_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G4:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F3 @ X3)) @ (G4 @ X3))))))))))).
% 0.30/1.51 thf(fact_64_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G3:(A > B)]: (((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G3 @ X)))) => (((ord_less_eq @ (A > B)) @ F2) @ G3)))))))).
% 0.30/1.51 thf(fact_65_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G3:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G3) => (((ord_less_eq @ B) @ (F2 @ X4)) @ (G3 @ X4))))))))).
% 0.30/1.51 thf(fact_66_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G3:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G3) => (((ord_less_eq @ B) @ (F2 @ X4)) @ (G3 @ X4))))))))).
% 0.30/1.51 thf(fact_67_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 0.30/1.51 thf(fact_68_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2))) = (^ [A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) & (((ord_less_eq @ (set @ A)) @ B3) @ A2)))))))).
% 0.30/1.51 thf(fact_69_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C2) => (((ord_less_eq @ (set @ A)) @ A3) @ C2)))))).
% 0.30/1.51 thf(fact_70_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) => (Q @ X)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 0.30/1.51 thf(fact_71_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 0.30/1.51 thf(fact_72_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A2:(set @ A),B3:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A2) => (((member @ A) @ T3) @ B3)))))))))).
% 0.30/1.51 thf(fact_73_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A3))))).
% 0.30/1.51 thf(fact_74_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 0.30/1.51 thf(fact_75_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A2:(set @ A),B3:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B3)))))))))).
% 0.30/1.51 thf(fact_76_equalityE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))))).
% 0.30/1.51 thf(fact_77_subsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C3) @ A3) => (((member @ A) @ C3) @ B2)))))).
% 0.30/1.51 thf(fact_78_in__mono, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ X4) @ A3) => (((member @ A) @ X4) @ B2)))))).
% 0.30/1.51 thf(fact_79_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 0.30/1.51 thf(fact_80_pair__pre__digraph_Osurjective, axiom, (! [Z:$tType,A:$tType,R2:((pair_p1731315293ph_ext @ A) @ Z)]: ((R2 = (((((pair_p1152105390ph_ext @ A) @ Z) @ (((pair_pair_pre_pverts @ A) @ Z) @ R2)) @ (((pair_pair_pre_parcs @ A) @ Z) @ R2)) @ (((pair_pair_pre_more @ A) @ Z) @ R2)))))).
% 0.30/1.51 thf(fact_81_member__product, axiom, (! [B:$tType,A:$tType,X4:((product_prod @ A) @ B),A3:(set @ A),B2:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X4) @ ((((product_product @ A) @ B) @ A3) @ B2)) = (((member @ ((product_prod @ A) @ B)) @ X4) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2)))))))).
% 0.30/1.51 thf(fact_82_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A2:(set @ A),B3:(set @ B)]: (((((product_Sigma @ A) @ B) @ A2) @ (^ [Uu:A]: (B3))))))))).
% 0.30/1.51 thf(fact_83_subset__Collect__iff, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A3) => ((((ord_less_eq @ (set @ A)) @ B2) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A3) & (P @ X3)))))) = (! [X3:A]: (((((member @ A) @ X3) @ B2) => (P @ X3))))))))).
% 0.30/1.51 thf(fact_84_subset__CollectI, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A3) => ((! [X:A]: (((((member @ A) @ X) @ B2) => ((Q @ X) => (P @ X))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ B2) & (Q @ X3)))))) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A3) & (P @ X3))))))))))).
% 0.30/1.51 thf(fact_85_Collect__restrict, axiom, (! [A:$tType,X5:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X5) & (P @ X3)))))) @ X5)))).
% 0.30/1.51 thf(fact_86_prop__restrict, axiom, (! [A:$tType,X4:A,Z4:(set @ A),X5:(set @ A),P:(A > $o)]: (((((member @ A) @ X4) @ Z4) => ((((ord_less_eq @ (set @ A)) @ Z4) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ X5) & (P @ X3)))))) => (P @ X4)))))).
% 0.30/1.51 thf(fact_87_conj__subset__def, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3)))))) = ((((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ P)) & (((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ Q))))))).
% 0.30/1.51 thf(fact_88_pair__pre__digraph_Oext__inject, axiom, (! [A:$tType,Z:$tType,Pverts:(set @ A),Parcs:(set @ ((product_prod @ A) @ A)),More:Z,Pverts2:(set @ A),Parcs2:(set @ ((product_prod @ A) @ A)),More2:Z]: ((((((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts) @ Parcs) @ More) = (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts2) @ Parcs2) @ More2)) = ((Pverts = Pverts2) & ((Parcs = Parcs2) & (More = More2))))))).
% 0.30/1.51 thf(fact_89_pair__pre__digraph_Oinduct__scheme, axiom, (! [Z:$tType,A:$tType,P:(((pair_p1731315293ph_ext @ A) @ Z) > $o),R2:((pair_p1731315293ph_ext @ A) @ Z)]: (((! [Pverts3:(set @ A),Parcs3:(set @ ((product_prod @ A) @ A)),More3:Z]: ((P @ (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts3) @ Parcs3) @ More3)))) => (P @ R2))))).
% 0.30/1.51 thf(fact_90_pair__pre__digraph_Ocases__scheme, axiom, (! [A:$tType,Z:$tType,R2:((pair_p1731315293ph_ext @ A) @ Z)]: ((~ ((! [Pverts3:(set @ A),Parcs3:(set @ ((product_prod @ A) @ A)),More3:Z]: ((R2 != (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts3) @ Parcs3) @ More3))))))))).
% 0.30/1.51 thf(fact_91_pair__pre__digraph_Oselect__convs_I2_J, axiom, (! [Z:$tType,A:$tType,Pverts:(set @ A),Parcs:(set @ ((product_prod @ A) @ A)),More:Z]: (((((pair_pair_pre_parcs @ A) @ Z) @ (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts) @ Parcs) @ More)) = Parcs)))).
% 0.30/1.51 thf(fact_92_pair__pre__digraph_Oselect__convs_I1_J, axiom, (! [Z:$tType,A:$tType,Pverts:(set @ A),Parcs:(set @ ((product_prod @ A) @ A)),More:Z]: (((((pair_pair_pre_pverts @ A) @ Z) @ (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts) @ Parcs) @ More)) = Pverts)))).
% 0.30/1.51 thf(fact_93_pair__pre__digraph_Oselect__convs_I3_J, axiom, (! [A:$tType,Z:$tType,Pverts:(set @ A),Parcs:(set @ ((product_prod @ A) @ A)),More:Z]: (((((pair_pair_pre_more @ A) @ Z) @ (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts) @ Parcs) @ More)) = More)))).
% 0.30/1.51 thf(fact_94_Fpow__def, axiom, (! [A:$tType]: (((finite_Fpow @ A) = (^ [A2:(set @ A)]: (((collect @ (set @ A)) @ (^ [X6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ X6) @ A2) & ((finite_finite2 @ A) @ X6))))))))))).
% 0.30/1.51 thf(fact_95_pair__fin__digraph__def, axiom, (! [A:$tType]: (((pair_p953238171igraph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_wf_digraph @ A) @ G) & ((pair_p1373739070axioms @ A) @ G)))))))).
% 0.30/1.51 thf(fact_96_pair__fin__digraph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_wf_digraph @ A) @ G2) => (((pair_p1373739070axioms @ A) @ G2) => ((pair_p953238171igraph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_97_pair__pre__digraph_Ounfold__congs_I3_J, axiom, (! [A:$tType,A7:$tType,R2:((pair_p1731315293ph_ext @ A7) @ A),R3:((pair_p1731315293ph_ext @ A7) @ A),V:A,F2:(A > A),F4:(A > A)]: (((R2 = R3) => (((((pair_pair_pre_more @ A7) @ A) @ R3) = V) => ((! [V2:A]: (((V2 = V) => ((F2 @ V2) = (F4 @ V2))))) => (((((pair_p1108258481update @ A) @ A7) @ F2) @ R2) = ((((pair_p1108258481update @ A) @ A7) @ F4) @ R3)))))))).
% 0.30/1.51 thf(fact_98_pair__pre__digraph_Ofold__congs_I3_J, axiom, (! [A:$tType,A7:$tType,R2:((pair_p1731315293ph_ext @ A7) @ A),R3:((pair_p1731315293ph_ext @ A7) @ A),V:A,F2:(A > A),F4:(A > A)]: (((R2 = R3) => (((((pair_pair_pre_more @ A7) @ A) @ R3) = V) => ((! [V2:A]: (((V = V2) => ((F2 @ V2) = (F4 @ V2))))) => (((((pair_p1108258481update @ A) @ A7) @ F2) @ R2) = ((((pair_p1108258481update @ A) @ A7) @ F4) @ R3)))))))).
% 0.30/1.51 thf(fact_99_finite__cartesian__product__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B2:(set @ B)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2)))) = ((A3 = (bot_bot @ (set @ A))) | ((B2 = (bot_bot @ (set @ B))) | (((finite_finite2 @ A) @ A3) & ((finite_finite2 @ B) @ B2)))))))).
% 0.30/1.51 thf(fact_100_bot__apply, axiom, (! [C:$tType,D2:$tType]: (((bot @ C) => ((bot_bot @ (D2 > C)) = (^ [X3:D2]: ((bot_bot @ C)))))))).
% 0.30/1.51 thf(fact_101_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_102_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_103_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 0.30/1.51 thf(fact_104_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 0.30/1.51 thf(fact_105_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 0.30/1.51 thf(fact_106_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_107_Sigma__empty1, axiom, (! [B:$tType,A:$tType,B2:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (bot_bot @ (set @ A))) @ B2) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 0.30/1.51 thf(fact_108_Times__empty, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B2:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2))) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = ((A3 = (bot_bot @ (set @ A))) | (B2 = (bot_bot @ (set @ B)))))))).
% 0.30/1.51 thf(fact_109_Sigma__empty2, axiom, (! [B:$tType,A:$tType,A3:(set @ A)]: ((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: ((bot_bot @ (set @ B))))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 0.30/1.51 thf(fact_110_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X3:A]: ($false))))))).
% 0.30/1.51 thf(fact_111_Sigma__empty__iff, axiom, (! [B:$tType,A:$tType,I:(set @ A),X5:(A > (set @ B))]: (((((((product_Sigma @ A) @ B) @ I) @ X5) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (! [X3:A]: (((((member @ A) @ X3) @ I) => ((X5 @ X3) = (bot_bot @ (set @ B)))))))))).
% 0.30/1.51 thf(fact_112_empty__in__Fpow, axiom, (! [A:$tType,A3:(set @ A)]: ((((member @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((finite_Fpow @ A) @ A3))))).
% 0.30/1.51 thf(fact_113_Fpow__not__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_Fpow @ A) @ A3) != (bot_bot @ (set @ (set @ A))))))).
% 0.30/1.51 thf(fact_114_emptyE, axiom, (! [A:$tType,A4:A]: ((~ ((((member @ A) @ A4) @ (bot_bot @ (set @ A)))))))).
% 0.30/1.51 thf(fact_115_equals0D, axiom, (! [A:$tType,A3:(set @ A),A4:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A4) @ A3))))))).
% 0.30/1.51 thf(fact_116_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_117_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_118_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 0.30/1.51 thf(fact_119_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ A4) @ (bot_bot @ A)) => (A4 = (bot_bot @ A))))))))).
% 0.30/1.51 thf(fact_120_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ A4) @ (bot_bot @ A)) = (A4 = (bot_bot @ A))))))))).
% 0.30/1.51 thf(fact_121_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A4))))))).
% 0.30/1.51 thf(fact_122_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 0.30/1.51 thf(fact_123_infinite__imp__nonempty, axiom, (! [A:$tType,S:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (S != (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_124_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))).
% 0.30/1.51 thf(fact_125_times__eq__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B2:(set @ B),C2:(set @ A),D:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2))) = ((((product_Sigma @ A) @ B) @ C2) @ (^ [Uu:A]: (D)))) = (((A3 = C2) & (B2 = D)) | (((A3 = (bot_bot @ (set @ A))) | (B2 = (bot_bot @ (set @ B)))) & ((C2 = (bot_bot @ (set @ A))) | (D = (bot_bot @ (set @ B)))))))))).
% 0.30/1.51 thf(fact_126_Fpow__mono, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A3)) @ ((finite_Fpow @ A) @ B2)))))).
% 0.30/1.51 thf(fact_127_pair__fin__digraph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => ((pair_pair_wf_digraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_128_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (? [X:A]: (((((member @ A) @ X) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X) @ Xa) => (X = Xa)))))))))))))))).
% 0.30/1.51 thf(fact_129_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (? [X:A]: (((((member @ A) @ X) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X) => (X = Xa)))))))))))))))).
% 0.30/1.51 thf(fact_130_pair__pre__digraph_Oupdate__convs_I3_J, axiom, (! [A:$tType,Z:$tType,More2:(Z > Z),Pverts:(set @ A),Parcs:(set @ ((product_prod @ A) @ A)),More:Z]: ((((((pair_p1108258481update @ Z) @ A) @ More2) @ (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts) @ Parcs) @ More)) = (((((pair_p1152105390ph_ext @ A) @ Z) @ Pverts) @ Parcs) @ (More2 @ More)))))).
% 0.30/1.51 thf(fact_131_times__subset__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),C2:(set @ B),B2:(set @ A),D:(set @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (C2)))) @ ((((product_Sigma @ A) @ B) @ B2) @ (^ [Uu:A]: (D)))) = ((A3 = (bot_bot @ (set @ A))) | ((C2 = (bot_bot @ (set @ B))) | ((((ord_less_eq @ (set @ A)) @ A3) @ B2) & (((ord_less_eq @ (set @ B)) @ C2) @ D)))))))).
% 0.30/1.51 thf(fact_132_finite__SigmaI2, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(A > (set @ B))]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A3) & ((B2 @ X3) != (bot_bot @ (set @ B)))))))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((finite_finite2 @ B) @ (B2 @ A5))))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2))))))).
% 0.30/1.51 thf(fact_133_finite__cartesian__productD1, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ B)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2)))) => ((B2 != (bot_bot @ (set @ B))) => ((finite_finite2 @ A) @ A3)))))).
% 0.30/1.51 thf(fact_134_finite__cartesian__productD2, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B2:(set @ B)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B2)))) => ((A3 != (bot_bot @ (set @ A))) => ((finite_finite2 @ B) @ B2)))))).
% 0.30/1.51 thf(fact_135_finite__transitivity__chain, axiom, (! [A:$tType,A3:(set @ A),R:(A > (A > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X:A]: ((~ (((R @ X) @ X))))) => ((! [X:A,Y4:A,Z5:A]: ((((R @ X) @ Y4) => (((R @ Y4) @ Z5) => ((R @ X) @ Z5))))) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (? [Y5:A]: (((((member @ A) @ Y5) @ A3) & ((R @ X) @ Y5))))))) => (A3 = (bot_bot @ (set @ A)))))))))).
% 0.30/1.51 thf(fact_136_pair__wf__digraph_Opair__wf__digraph__subdivide, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),E:((product_prod @ A) @ A),W:A]: ((((pair_pair_wf_digraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ E) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((~ ((((member @ A) @ W) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))) => ((pair_pair_wf_digraph @ A) @ ((((pair_subdivide @ A) @ G2) @ E) @ W)))))))).
% 0.30/1.51 thf(fact_137_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A2:(set @ A)]: ((A2 = (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_138_pair__wf__digraph_Oin__arcsD2, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),U:A,V3:A]: ((((pair_pair_wf_digraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V3)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => (((member @ A) @ V3) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2))))))).
% 0.30/1.51 thf(fact_139_pair__wf__digraph_Oin__arcsD1, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),U:A,V3:A]: ((((pair_pair_wf_digraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V3)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => (((member @ A) @ U) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2))))))).
% 0.30/1.51 thf(fact_140_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A4:A,B4:B,A8:A,B7:B]: (((((((product_Pair @ A) @ B) @ A4) @ B4) = ((((product_Pair @ A) @ B) @ A8) @ B7)) = ((A4 = A8) & (B4 = B7)))))).
% 0.30/1.51 thf(fact_141_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 0.30/1.51 thf(fact_142_SigmaI, axiom, (! [B:$tType,A:$tType,A4:A,A3:(set @ A),B4:B,B2:(A > (set @ B))]: (((((member @ A) @ A4) @ A3) => ((((member @ B) @ B4) @ (B2 @ A4)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2))))))).
% 0.30/1.51 thf(fact_143_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A4:A,B4:B,A3:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2)) = ((((member @ A) @ A4) @ A3) & (((member @ B) @ B4) @ (B2 @ A4))))))).
% 0.30/1.51 thf(fact_144_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 0.30/1.51 thf(fact_145_bot__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((bot_bot @ (A > (B > $o))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))))))))).
% 0.30/1.51 thf(fact_146_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 0.30/1.51 thf(fact_147_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B6:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B6)))) => (P @ Prod))))).
% 0.30/1.51 thf(fact_148_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y2:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B6:B]: ((Y2 != ((((product_Pair @ A) @ B) @ A5) @ B6))))))))).
% 0.30/1.51 thf(fact_149_prod__induct7, axiom, (! [G5:$tType,F5:$tType,E2:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5)))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5))))))]: (((! [A5:A,B6:B,C4:C,D3:D2,E3:E2,F:F5,G6:G5]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5))))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F5) @ G5)) @ E3) @ ((((product_Pair @ F5) @ G5) @ F) @ G6))))))))) => (P @ X4))))).
% 0.30/1.51 thf(fact_150_prod__induct6, axiom, (! [F5:$tType,E2:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5)))))]: (((! [A5:A,B6:B,C4:C,D3:D2,E3:E2,F:F5]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5)))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ F5)) @ D3) @ ((((product_Pair @ E2) @ F5) @ E3) @ F)))))))) => (P @ X4))))).
% 0.30/1.51 thf(fact_151_prod__induct5, axiom, (! [E2:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2)))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))))]: (((! [A5:A,B6:B,C4:C,D3:D2,E3:E2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E2)) @ C4) @ ((((product_Pair @ D2) @ E2) @ D3) @ E3))))))) => (P @ X4))))).
% 0.30/1.51 thf(fact_152_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A5:A,B6:B,C4:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B6) @ ((((product_Pair @ C) @ D2) @ C4) @ D3)))))) => (P @ X4))))).
% 0.30/1.51 thf(fact_153_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B6:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B6) @ C4))))) => (P @ X4))))).
% 0.30/1.51 thf(fact_154_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E2:$tType,F5:$tType,G5:$tType,Y2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5))))))]: ((~ ((! [A5:A,B6:B,C4:C,D3:D2,E3:E2,F:F5,G6:G5]: ((Y2 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5))))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5)))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ ((product_prod @ F5) @ G5))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F5) @ G5)) @ E3) @ ((((product_Pair @ F5) @ G5) @ F) @ G6)))))))))))))).
% 0.30/1.51 thf(fact_155_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E2:$tType,F5:$tType,Y2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5)))))]: ((~ ((! [A5:A,B6:B,C4:C,D3:D2,E3:E2,F:F5]: ((Y2 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5)))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E2) @ F5))) @ C4) @ ((((product_Pair @ D2) @ ((product_prod @ E2) @ F5)) @ D3) @ ((((product_Pair @ E2) @ F5) @ E3) @ F))))))))))))).
% 0.30/1.51 thf(fact_156_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E2:$tType,Y2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))))]: ((~ ((! [A5:A,B6:B,C4:C,D3:D2,E3:E2]: ((Y2 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E2))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E2)) @ C4) @ ((((product_Pair @ D2) @ E2) @ D3) @ E3)))))))))))).
% 0.30/1.51 thf(fact_157_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A5:A,B6:B,C4:C,D3:D2]: ((Y2 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B6) @ ((((product_Pair @ C) @ D2) @ C4) @ D3))))))))))).
% 0.30/1.51 thf(fact_158_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y2:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B6:B,C4:C]: ((Y2 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B6) @ C4)))))))))).
% 0.30/1.51 thf(fact_159_Pair__inject, axiom, (! [A:$tType,B:$tType,A4:A,B4:B,A8:A,B7:B]: (((((((product_Pair @ A) @ B) @ A4) @ B4) = ((((product_Pair @ A) @ B) @ A8) @ B7)) => (~ (((A4 = A8) => (B4 != B7)))))))).
% 0.30/1.51 thf(fact_160_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B6:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B6)))) => (P @ P2))))).
% 0.30/1.51 thf(fact_161_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X) @ Y4))))))).
% 0.30/1.51 thf(fact_162_subdivide_Oinduct, axiom, (! [A:$tType,P:(((pair_p1731315293ph_ext @ A) @ product_unit) > (((product_prod @ A) @ A) > (A > $o))),A0:((pair_p1731315293ph_ext @ A) @ product_unit),A1:((product_prod @ A) @ A),A22:A]: (((! [G7:((pair_p1731315293ph_ext @ A) @ product_unit),U2:A,V2:A,X_12:A]: ((((P @ G7) @ ((((product_Pair @ A) @ A) @ U2) @ V2)) @ X_12))) => (((P @ A0) @ A1) @ A22))))).
% 0.30/1.51 thf(fact_163_subdivide_Ocases, axiom, (! [A:$tType,X4:((product_prod @ ((pair_p1731315293ph_ext @ A) @ product_unit)) @ ((product_prod @ ((product_prod @ A) @ A)) @ A))]: ((~ ((! [G7:((pair_p1731315293ph_ext @ A) @ product_unit),U2:A,V2:A,W2:A]: ((X4 != ((((product_Pair @ ((pair_p1731315293ph_ext @ A) @ product_unit)) @ ((product_prod @ ((product_prod @ A) @ A)) @ A)) @ G7) @ ((((product_Pair @ ((product_prod @ A) @ A)) @ A) @ ((((product_Pair @ A) @ A) @ U2) @ V2)) @ W2)))))))))).
% 0.30/1.51 thf(fact_164_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S)))) = (R = S))))).
% 0.30/1.51 thf(fact_165_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S2:B,R:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S2)) @ R) => ((S3 = S2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S3)) @ R)))))).
% 0.30/1.51 thf(fact_166_SigmaE, axiom, (! [A:$tType,B:$tType,C3:((product_prod @ A) @ B),A3:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((product_Sigma @ A) @ B) @ A3) @ B2)) => (~ ((! [X:A]: (((((member @ A) @ X) @ A3) => (! [Y4:B]: (((((member @ B) @ Y4) @ (B2 @ X)) => (C3 != ((((product_Pair @ A) @ B) @ X) @ Y4)))))))))))))).
% 0.30/1.51 thf(fact_167_SigmaD1, axiom, (! [B:$tType,A:$tType,A4:A,B4:B,A3:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2)) => (((member @ A) @ A4) @ A3))))).
% 0.30/1.51 thf(fact_168_SigmaD2, axiom, (! [B:$tType,A:$tType,A4:A,B4:B,A3:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2)) => (((member @ B) @ B4) @ (B2 @ A4)))))).
% 0.30/1.51 thf(fact_169_SigmaE2, axiom, (! [B:$tType,A:$tType,A4:A,B4:B,A3:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ ((((product_Sigma @ A) @ B) @ A3) @ B2)) => (~ (((((member @ A) @ A4) @ A3) => (~ ((((member @ B) @ B4) @ (B2 @ A4))))))))))).
% 0.30/1.51 thf(fact_170_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S2))))).
% 0.30/1.51 thf(fact_171_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 0.30/1.51 thf(fact_172_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I2:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I2) @ J)) @ R4) => (((ord_less_eq @ B) @ (As @ I2)) @ (As @ J)))))))))))).
% 0.30/1.51 thf(fact_173_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A4:A,B4:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((F1 @ A4) @ B4))))).
% 0.30/1.51 thf(fact_174_less__by__empty, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A)),B2:(set @ ((product_prod @ A) @ A))]: (((A3 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A3) @ B2))))).
% 0.30/1.51 thf(fact_175_calculation, axiom, (((finite_finite2 @ a) @ (((pair_pair_pre_pverts @ a) @ product_unit) @ g)) & ((((finite_card @ a) @ (((pair_pair_pre_pverts @ a) @ product_unit) @ g)) = n) & ((((pair_pair_pre_parcs @ a) @ product_unit) @ g) = ((collect @ ((product_prod @ a) @ a)) @ ((((product_case_prod @ a) @ a) @ $o) @ (^ [U3:a,V4:a]: (((((member @ ((product_prod @ a) @ a)) @ ((((product_Pair @ a) @ a) @ U3) @ V4)) @ ((((product_Sigma @ a) @ a) @ (((pair_pair_pre_pverts @ a) @ product_unit) @ g)) @ (^ [Uu:a]: ((((pair_pair_pre_pverts @ a) @ product_unit) @ g))))) & (U3 != V4)))))))))).
% 0.30/1.51 thf(fact_176_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X:A,Y4:B]: ((((P @ X) @ Y4) => ((Q @ X) @ Y4)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 0.30/1.51 thf(fact_177_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B5:B]: ((P & ((Q @ A6) @ B5))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 0.30/1.51 thf(fact_178_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(B > (C > A)),A4:B,B4:C]: (((((((product_case_prod @ B) @ C) @ A) @ F2) @ ((((product_Pair @ B) @ C) @ A4) @ B4)) = ((F2 @ A4) @ B4))))).
% 0.30/1.51 thf(fact_179_case__prodI2, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),C3:(A > (B > $o))]: (((! [A5:A,B6:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B6)) => ((C3 @ A5) @ B6)))) => (((((product_case_prod @ A) @ B) @ $o) @ C3) @ P2))))).
% 0.30/1.51 thf(fact_180_case__prodI, axiom, (! [A:$tType,B:$tType,F2:(A > (B > $o)),A4:A,B4:B]: ((((F2 @ A4) @ B4) => (((((product_case_prod @ A) @ B) @ $o) @ F2) @ ((((product_Pair @ A) @ B) @ A4) @ B4)))))).
% 0.30/1.51 thf(fact_181_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B5:B]: (((P @ A6) & (Q @ B5)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [Uu:A]: (((collect @ B) @ Q)))))))).
% 0.30/1.51 thf(fact_182_bot2E, axiom, (! [A:$tType,B:$tType,X4:A,Y2:B]: ((~ ((((bot_bot @ (A > (B > $o))) @ X4) @ Y2)))))).
% 0.30/1.51 thf(fact_183_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A3:(A > (B > $o)),B2:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A3) @ B2) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A3))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B2))))))).
% 0.30/1.51 thf(fact_184_case__prodD, axiom, (! [A:$tType,B:$tType,F2:(A > (B > $o)),A4:A,B4:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F2) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) => ((F2 @ A4) @ B4))))).
% 0.30/1.51 thf(fact_185_case__prodE, axiom, (! [A:$tType,B:$tType,C3:(A > (B > $o)),P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C3) @ P2) => (~ ((! [X:A,Y4:B]: (((P2 = ((((product_Pair @ A) @ B) @ X) @ Y4)) => (~ (((C3 @ X) @ Y4)))))))))))).
% 0.30/1.51 thf(fact_186_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z3:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z3)) => (~ ((! [X:B,Y4:C]: (((Z3 = ((((product_Pair @ B) @ C) @ X) @ Y4)) => (~ ((Q @ ((P @ X) @ Y4))))))))))))).
% 0.30/1.51 thf(fact_187_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X3:A,Y3:B]: ((F2 @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) = F2)))).
% 0.30/1.51 thf(fact_188_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(A > (B > C)),G3:(((product_prod @ A) @ B) > C)]: (((! [X:A,Y4:B]: ((((F2 @ X) @ Y4) = (G3 @ ((((product_Pair @ A) @ B) @ X) @ Y4))))) => (((((product_case_prod @ A) @ B) @ C) @ F2) = G3))))).
% 0.30/1.51 thf(fact_189_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F2) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F2 @ X1) @ X22))))).
% 0.30/1.51 thf(fact_190_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F2:(A > (B > C)),G3:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X:A,Y4:B]: (((((((product_Pair @ A) @ B) @ X) @ Y4) = Q2) => (((F2 @ X) @ Y4) = ((G3 @ X) @ Y4))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F2) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 0.30/1.51 thf(fact_191_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X4:A,Y2:B,Q:(A > (B > $o))]: ((((P @ X4) @ Y2) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X4) @ Y2)))))).
% 0.30/1.51 thf(fact_192_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X4:A,Y2:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X4) @ Y2) => ((Q @ X4) @ Y2)))))).
% 0.30/1.51 thf(fact_193_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X3:A,Y3:B]: (((P @ X3) & ((Q @ X3) @ Y3)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [X3:A]: (((collect @ B) @ (Q @ X3))))))))).
% 0.30/1.51 thf(fact_194_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 0.30/1.51 thf(fact_195_prod_Ocase__distrib, axiom, (! [C:$tType,D2:$tType,B:$tType,A:$tType,H:(C > D2),F2:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F2) @ Prod)) = (((((product_case_prod @ A) @ B) @ D2) @ (^ [X12:A,X23:B]: ((H @ ((F2 @ X12) @ X23))))) @ Prod))))).
% 0.30/1.51 thf(fact_196_card__subset__eq, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((finite_card @ A) @ A3) = ((finite_card @ A) @ B2)) => (A3 = B2))))))).
% 0.30/1.51 thf(fact_197_infinite__arbitrarily__large, axiom, (! [A:$tType,A3:(set @ A),N:nat]: (((~ (((finite_finite2 @ A) @ A3))) => (? [B8:(set @ A)]: ((((finite_finite2 @ A) @ B8) & ((((finite_card @ A) @ B8) = N) & (((ord_less_eq @ (set @ A)) @ B8) @ A3))))))))).
% 0.30/1.51 thf(fact_198_finite__if__finite__subsets__card__bdd, axiom, (! [A:$tType,F6:(set @ A),C2:nat]: (((! [G7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ G7) @ F6) => (((finite_finite2 @ A) @ G7) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ G7)) @ C2))))) => (((finite_finite2 @ A) @ F6) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ F6)) @ C2)))))).
% 0.30/1.51 thf(fact_199_card__seteq, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B2)) @ ((finite_card @ A) @ A3)) => (A3 = B2))))))).
% 0.30/1.51 thf(fact_200_card__mono, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B2))))))).
% 0.30/1.51 thf(fact_201_obtain__subset__with__card__n, axiom, (! [A:$tType,N:nat,S:(set @ A)]: (((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ S)) => (~ ((! [T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T4) @ S) => ((((finite_card @ A) @ T4) = N) => (~ (((finite_finite2 @ A) @ T4))))))))))))).
% 0.30/1.51 thf(fact_202_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 0.30/1.51 thf(fact_203_card__Ex__subset, axiom, (! [A:$tType,K:nat,M:(set @ A)]: (((((ord_less_eq @ nat) @ K) @ ((finite_card @ A) @ M)) => (? [N2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ N2) @ M) & (((finite_card @ A) @ N2) = K)))))))).
% 0.30/1.51 thf(fact_204_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P2:((product_prod @ A) @ B),Z3:C,C3:(A > (B > (set @ C)))]: (((! [A5:A,B6:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B6)) => (((member @ C) @ Z3) @ ((C3 @ A5) @ B6))))) => (((member @ C) @ Z3) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C3) @ P2)))))).
% 0.30/1.51 thf(fact_205_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z3:A,C3:(B > (C > (set @ A))),A4:B,B4:C]: (((((member @ A) @ Z3) @ ((C3 @ A4) @ B4)) => (((member @ A) @ Z3) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ ((((product_Pair @ B) @ C) @ A4) @ B4))))))).
% 0.30/1.51 thf(fact_206_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P2:((product_prod @ A) @ B),C3:(A > (B > (C > $o))),X4:C]: (((! [A5:A,B6:B]: (((((((product_Pair @ A) @ B) @ A5) @ B6) = P2) => (((C3 @ A5) @ B6) @ X4)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P2) @ X4))))).
% 0.30/1.51 thf(fact_207_finite__Collect__le__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ K)))))))).
% 0.30/1.51 thf(fact_208_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R:(A > (B > (C > $o))),A4:A,B4:B,C3:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ C3) => (((R @ A4) @ B4) @ C3))))).
% 0.30/1.51 thf(fact_209_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(A > (B > (C > $o))),P2:((product_prod @ A) @ B),Z3:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P2) @ Z3) => (~ ((! [X:A,Y4:B]: (((P2 = ((((product_Pair @ A) @ B) @ X) @ Y4)) => (~ ((((C3 @ X) @ Y4) @ Z3)))))))))))).
% 0.30/1.51 thf(fact_210_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z3:A,C3:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z3) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P2)) => (~ ((! [X:B,Y4:C]: (((P2 = ((((product_Pair @ B) @ C) @ X) @ Y4)) => (~ ((((member @ A) @ Z3) @ ((C3 @ X) @ Y4))))))))))))).
% 0.30/1.51 thf(fact_211_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N4:(set @ nat)]: ((? [M2:nat]: ((! [X3:nat]: (((((member @ nat) @ X3) @ N4) => (((ord_less_eq @ nat) @ X3) @ M2)))))))))).
% 0.30/1.51 thf(fact_212_infinite__nat__iff__unbounded__le, axiom, (! [S:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S))) = (! [M2:nat]: ((? [N3:nat]: (((((ord_less_eq @ nat) @ M2) @ N3) & (((member @ nat) @ N3) @ S)))))))))).
% 0.30/1.51 thf(fact_213_bounded__Max__nat, axiom, (! [P:(nat > $o),X4:nat,M:nat]: (((P @ X4) => ((! [X:nat]: (((P @ X) => (((ord_less_eq @ nat) @ X) @ M)))) => (~ ((! [M3:nat]: (((P @ M3) => (~ ((! [X2:nat]: (((P @ X2) => (((ord_less_eq @ nat) @ X2) @ M3)))))))))))))))).
% 0.30/1.51 thf(fact_214_finite__less__ub, axiom, (! [F2:(nat > nat),U:nat]: (((! [N5:nat]: ((((ord_less_eq @ nat) @ N5) @ (F2 @ N5)))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ (F2 @ N3)) @ U))))))))).
% 0.30/1.51 thf(fact_215_card__le__if__inj__on__rel, axiom, (! [B:$tType,A:$tType,B2:(set @ A),A3:(set @ B),R2:(B > (A > $o))]: ((((finite_finite2 @ A) @ B2) => ((! [A5:B]: (((((member @ B) @ A5) @ A3) => (? [B9:A]: (((((member @ A) @ B9) @ B2) & ((R2 @ A5) @ B9))))))) => ((! [A12:B,A23:B,B6:A]: (((((member @ B) @ A12) @ A3) => ((((member @ B) @ A23) @ A3) => ((((member @ A) @ B6) @ B2) => (((R2 @ A12) @ B6) => (((R2 @ A23) @ B6) => (A12 = A23)))))))) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ A3)) @ ((finite_card @ A) @ B2)))))))).
% 0.30/1.51 thf(fact_216_pair__pseudo__graph_Opair__pseudo__graph__subdivide, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),E:((product_prod @ A) @ A),W:A]: ((((pair_p389585203_graph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ E) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((~ ((((member @ A) @ W) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))) => ((pair_p389585203_graph @ A) @ ((((pair_subdivide @ A) @ G2) @ E) @ W)))))))).
% 0.30/1.51 thf(fact_217_pair__graph_Opair__graph__subdivide, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),E:((product_prod @ A) @ A),W:A]: ((((pair_pair_graph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ E) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((~ ((((member @ A) @ W) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))) => ((pair_pair_graph @ A) @ ((((pair_subdivide @ A) @ G2) @ E) @ W)))))))).
% 0.30/1.51 thf(fact_218_pair__graph_Oaxioms_I2_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_graph @ A) @ G2) => ((pair_p389585203_graph @ A) @ G2))))).
% 0.30/1.51 thf(fact_219_pair__graph_Opair__graph, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_graph @ A) @ G2) => ((pair_pair_graph @ A) @ G2))))).
% 0.30/1.51 thf(fact_220_pair__pseudo__graph_Opair__pseudo__graph, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p389585203_graph @ A) @ G2) => ((pair_p389585203_graph @ A) @ G2))))).
% 0.30/1.51 thf(fact_221_pair__pseudo__graph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p389585203_graph @ A) @ G2) => ((pair_p953238171igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_222_case__prod__app, axiom, (! [A:$tType,D2:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D2 > A)) = (^ [F3:(B > (C > (D2 > A))),X3:((product_prod @ B) @ C),Y3:D2]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R4:C]: ((((F3 @ L) @ R4) @ Y3)))) @ X3))))))).
% 0.30/1.51 thf(fact_223_pair__bidirected__digraph_Opair__bidirected__digraph__subdivide, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),E:((product_prod @ A) @ A),W:A]: ((((pair_p1033978749igraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ E) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((~ ((((member @ A) @ W) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))) => ((pair_p1033978749igraph @ A) @ ((((pair_subdivide @ A) @ G2) @ E) @ W)))))))).
% 0.30/1.51 thf(fact_224_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),R:$o,X4:A,Y2:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) & R) => (R & (((P @ X4) @ Y2) => ((Q @ X4) @ Y2))))))).
% 0.30/1.51 thf(fact_225_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P2) = P2)))).
% 0.30/1.51 thf(fact_226_eq__subset, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((ord_less_eq @ (A > (A > $o))) @ (^ [Y:A,Z2:A]: ((Y = Z2)))) @ (^ [A6:A,B5:A]: ((((P @ A6) @ B5) | (A6 = B5)))))))).
% 0.30/1.51 thf(fact_227_pair__loopfree__digraph_Opair__loopfree__digraph__subdivide, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),E:((product_prod @ A) @ A),W:A]: ((((pair_p1282390076igraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ E) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((~ ((((member @ A) @ W) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))) => ((pair_p1282390076igraph @ A) @ ((((pair_subdivide @ A) @ G2) @ E) @ W)))))))).
% 0.30/1.51 thf(fact_228_pair__graph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_digraph @ A) @ G2) => (((pair_p389585203_graph @ A) @ G2) => ((pair_pair_graph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_229_pair__bidirected__digraph_Oaxioms_I2_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p1033978749igraph @ A) @ G2) => ((pair_p1282390076igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_230_pair__loopfree__digraph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p1282390076igraph @ A) @ G2) => ((pair_pair_wf_digraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_231_pair__digraph_Opair__wf__digraph, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_digraph @ A) @ G2) => ((pair_pair_wf_digraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_232_pair__digraph_Oaxioms_I2_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_digraph @ A) @ G2) => ((pair_p1282390076igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_233_pair__digraph_Opair__digraph, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_digraph @ A) @ G2) => ((pair_pair_digraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_234_pair__digraph__def, axiom, (! [A:$tType]: (((pair_pair_digraph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G) & ((pair_p1282390076igraph @ A) @ G)))))))).
% 0.30/1.51 thf(fact_235_pair__digraph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => (((pair_p1282390076igraph @ A) @ G2) => ((pair_pair_digraph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_236_pair__digraph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_digraph @ A) @ G2) => ((pair_p953238171igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_237_pair__graph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_graph @ A) @ G2) => ((pair_pair_digraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_238_pair__loopfree__digraph_Ono__loops_H, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),U:A,V3:A]: ((((pair_p1282390076igraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V3)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => (U != V3)))))).
% 0.30/1.51 thf(fact_239_pair__graph__def, axiom, (! [A:$tType]: (((pair_pair_graph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_digraph @ A) @ G) & ((pair_p389585203_graph @ A) @ G)))))))).
% 0.30/1.51 thf(fact_240_pair__loopfree__digraph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_wf_digraph @ A) @ G2) => (((pair_p814123999axioms @ A) @ G2) => ((pair_p1282390076igraph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_241_pair__loopfree__digraph__def, axiom, (! [A:$tType]: (((pair_p1282390076igraph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_wf_digraph @ A) @ G) & ((pair_p814123999axioms @ A) @ G)))))))).
% 0.30/1.51 thf(fact_242_pair__loopfree__digraph_Oaxioms_I2_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p1282390076igraph @ A) @ G2) => ((pair_p814123999axioms @ A) @ G2))))).
% 0.30/1.51 thf(fact_243_pair__sym__digraph_Opair__sym__digraph__subdivide, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),E:((product_prod @ A) @ A),W:A]: ((((pair_p460180511igraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ E) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => ((~ ((((member @ A) @ W) @ (((pair_pair_pre_pverts @ A) @ product_unit) @ G2)))) => ((pair_p460180511igraph @ A) @ ((((pair_subdivide @ A) @ G2) @ E) @ W)))))))).
% 0.30/1.51 thf(fact_244_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 0.30/1.51 thf(fact_245_pair__sym__digraph_Oarcs__symmetric, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit),A4:A,B4:A]: ((((pair_p460180511igraph @ A) @ G2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ A4)) @ (((pair_pair_pre_parcs @ A) @ product_unit) @ G2))))))).
% 0.30/1.51 thf(fact_246_pair__bidirected__digraph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p1033978749igraph @ A) @ G2) => ((pair_p460180511igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_247_pair__sym__digraph_Oaxioms_I1_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p460180511igraph @ A) @ G2) => ((pair_pair_wf_digraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_248_pair__pseudo__graph_Oaxioms_I2_J, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p389585203_graph @ A) @ G2) => ((pair_p460180511igraph @ A) @ G2))))).
% 0.30/1.51 thf(fact_249_pair__bidirected__digraph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p460180511igraph @ A) @ G2) => (((pair_p1282390076igraph @ A) @ G2) => ((pair_p1033978749igraph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_250_pair__bidirected__digraph__def, axiom, (! [A:$tType]: (((pair_p1033978749igraph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p460180511igraph @ A) @ G) & ((pair_p1282390076igraph @ A) @ G)))))))).
% 0.30/1.51 thf(fact_251_pair__pseudo__graph__def, axiom, (! [A:$tType]: (((pair_p389585203_graph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G) & ((pair_p460180511igraph @ A) @ G)))))))).
% 0.30/1.51 thf(fact_252_pair__pseudo__graph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_p953238171igraph @ A) @ G2) => (((pair_p460180511igraph @ A) @ G2) => ((pair_p389585203_graph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_253_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A4:B,B4:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A4) @ B4)) = ((C3 @ A4) @ B4))))).
% 0.30/1.51 thf(fact_254_pair__sym__digraph_Ointro, axiom, (! [A:$tType,G2:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_wf_digraph @ A) @ G2) => (((pair_p477239746axioms @ A) @ G2) => ((pair_p460180511igraph @ A) @ G2)))))).
% 0.30/1.51 thf(fact_255_pair__sym__digraph__def, axiom, (! [A:$tType]: (((pair_p460180511igraph @ A) = (^ [G:((pair_p1731315293ph_ext @ A) @ product_unit)]: ((((pair_pair_wf_digraph @ A) @ G) & ((pair_p477239746axioms @ A) @ G)))))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A7 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A7 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A7:$tType,A9:$tType]: ((((finite_finite @ A7) & (finite_finite @ A9)) => (finite_finite @ (A7 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A9:$tType]: (((order @ A9) => (order @ (A7 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A7 > A9)))))).
% 0.30/1.51 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A7 > A9)))))).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1, axiom, (order_bot @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, (preorder @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, (order @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, (ord @ nat)).
% 0.30/1.51 thf(tcon_Nat_Onat___Orderings_Obot_5, axiom, (bot @ nat)).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_6, axiom, (! [A7:$tType]: ((order_bot @ (set @ A7))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 0.30/1.51 thf(tcon_Set_Oset___Finite__Set_Ofinite_8, axiom, (! [A7:$tType]: (((finite_finite @ A7) => (finite_finite @ (set @ A7)))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Oord_10, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 0.30/1.51 thf(tcon_Set_Oset___Orderings_Obot_11, axiom, (! [A7:$tType]: ((bot @ (set @ A7))))).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_12, axiom, (order_bot @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_13, axiom, (preorder @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_14, axiom, (linorder @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Finite__Set_Ofinite_15, axiom, (finite_finite @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_16, axiom, (order @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Oord_17, axiom, (ord @ $o)).
% 0.30/1.51 thf(tcon_HOL_Obool___Orderings_Obot_18, axiom, (bot @ $o)).
% 0.30/1.51 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_19, axiom, (! [A7:$tType,A9:$tType]: ((((finite_finite @ A7) & (finite_finite @ A9)) => (finite_finite @ ((product_prod @ A7) @ A9)))))).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_20, axiom, (order_bot @ product_unit)).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_21, axiom, (preorder @ product_unit)).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_22, axiom, (linorder @ product_unit)).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_23, axiom, (finite_finite @ product_unit)).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Orderings_Oorder_24, axiom, (order @ product_unit)).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Orderings_Oord_25, axiom, (ord @ product_unit)).
% 0.30/1.51 thf(tcon_Product__Type_Ounit___Orderings_Obot_26, axiom, (bot @ product_unit)).
% 0.30/1.51 thf(conj_0, conjecture, ((finite_finite2 @ ((product_prod @ a) @ a)) @ (((pair_pair_pre_parcs @ a) @ product_unit) @ g))).
% 0.30/1.51 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.XcU2giCvCK/DTF2DTF14783.p
% 0.30/1.51 ------------------------
% 0.30/1.56 ---- Cleaned THF ---
% 0.30/1.56 thf(ty_t_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
% 0.30/1.56 pair_p1731315293ph_ext: $tType > $tType > $tType ).
% 0.30/1.56
% 0.30/1.56 thf(ty_t_Product__Type_Ounit,type,
% 0.30/1.56 product_unit: $tType ).
% 0.30/1.56
% 0.30/1.56 thf(ty_t_Product__Type_Oprod,type,
% 0.30/1.56 product_prod: $tType > $tType > $tType ).
% 0.30/1.56
% 0.30/1.56 thf(ty_t_Set_Oset,type,
% 0.30/1.56 set: $tType > $tType ).
% 0.30/1.56
% 0.30/1.56 thf(ty_t_Nat_Onat,type,
% 0.30/1.56 nat: $tType ).
% 0.30/1.56
% 0.30/1.56 thf(ty_tf_a,type,
% 0.30/1.56 a: $tType ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_HOL_Otype,type,
% 0.30/1.56 type:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Orderings_Obot,type,
% 0.30/1.56 bot:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Orderings_Oord,type,
% 0.30/1.56 ord:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Orderings_Oorder,type,
% 0.30/1.56 order:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.30/1.56 finite_finite:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/1.56 linorder:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/1.56 preorder:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/1.56 order_bot:
% 0.30/1.56 !>[A: $tType] : $o ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.30/1.56 bNF_Ca1785829860lChain:
% 0.30/1.56 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Finite__Set_OFpow,type,
% 0.30/1.56 finite_Fpow:
% 0.30/1.56 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Finite__Set_Ocard,type,
% 0.30/1.56 finite_card:
% 0.30/1.56 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Finite__Set_Ofinite,type,
% 0.30/1.56 finite_finite2:
% 0.30/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/1.56 bot_bot:
% 0.30/1.56 !>[A: $tType] : A ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/1.56 ord_less_eq:
% 0.30/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__bidirected__digraph,type,
% 0.30/1.56 pair_p1033978749igraph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__digraph,type,
% 0.30/1.56 pair_pair_digraph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__fin__digraph,type,
% 0.30/1.56 pair_p953238171igraph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__fin__digraph__axioms,type,
% 0.30/1.56 pair_p1373739070axioms:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__graph,type,
% 0.30/1.56 pair_pair_graph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__loopfree__digraph,type,
% 0.30/1.56 pair_p1282390076igraph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__loopfree__digraph__axioms,type,
% 0.30/1.56 pair_p814123999axioms:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore,type,
% 0.30/1.56 pair_pair_pre_more:
% 0.30/1.56 !>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > Z ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Omore__update,type,
% 0.30/1.56 pair_p1108258481update:
% 0.30/1.56 !>[Z: $tType,A: $tType] : ( ( Z > Z ) > ( pair_p1731315293ph_ext @ A @ Z ) > ( pair_p1731315293ph_ext @ A @ Z ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
% 0.30/1.56 pair_p1152105390ph_ext:
% 0.30/1.56 !>[A: $tType,Z: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > Z > ( pair_p1731315293ph_ext @ A @ Z ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Oparcs,type,
% 0.30/1.56 pair_pair_pre_parcs:
% 0.30/1.56 !>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__pre__digraph_Opverts,type,
% 0.30/1.56 pair_pair_pre_pverts:
% 0.30/1.56 !>[A: $tType,Z: $tType] : ( ( pair_p1731315293ph_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__pseudo__graph,type,
% 0.30/1.56 pair_p389585203_graph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__sym__digraph,type,
% 0.30/1.56 pair_p460180511igraph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__sym__digraph__axioms,type,
% 0.30/1.56 pair_p477239746axioms:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Opair__wf__digraph,type,
% 0.30/1.56 pair_pair_wf_digraph:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Pair__Digraph_Osubdivide,type,
% 0.30/1.56 pair_subdivide:
% 0.30/1.56 !>[A: $tType] : ( ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > ( pair_p1731315293ph_ext @ A @ product_unit ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Product__Type_OPair,type,
% 0.30/1.56 product_Pair:
% 0.30/1.56 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Product__Type_OSigma,type,
% 0.30/1.56 product_Sigma:
% 0.30/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/1.56 produc2004651681e_prod:
% 0.30/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/1.56 product_rec_prod:
% 0.30/1.56 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.30/1.56 product_case_prod:
% 0.30/1.56 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Product__Type_Oproduct,type,
% 0.30/1.56 product_product:
% 0.30/1.56 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Set_OCollect,type,
% 0.30/1.56 collect:
% 0.30/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_Set_Ois__empty,type,
% 0.30/1.56 is_empty:
% 0.30/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_c_member,type,
% 0.30/1.56 member:
% 0.30/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/1.56
% 0.30/1.56 thf(sy_v_G,type,
% 0.30/1.56 g: pair_p1731315293ph_ext @ a @ product_unit ).
% 0.30/1.56
% 0.30/1.56 thf(sy_v_n,type,
% 0.30/1.56 n: nat ).
% 0.30/1.56
% 0.30/1.56 thf(fact_0_finite__code,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( finite_finite @ A )
% 0.30/1.56 => ( ( finite_finite2 @ A )
% 0.30/1.56 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_1__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062,axiom,
% 0.30/1.56 ( finite_finite2 @ ( product_prod @ a @ a )
% 0.30/1.56 @ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
% 0.30/1.56 @ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_2__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062,axiom,
% 0.30/1.56 ( ord_less_eq @ ( set @ ( product_prod @ a @ a ) ) @ ( pair_pair_pre_parcs @ a @ product_unit @ g )
% 0.30/1.56 @ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
% 0.30/1.56 @ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_3_finite__set__choice,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.30/1.56 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ! [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ A3 )
% 0.30/1.56 => ? [X_1: B] : ( P @ X @ X_1 ) )
% 0.30/1.56 => ? [F: A > B] :
% 0.30/1.56 ! [X2: A] :
% 0.30/1.56 ( ( member @ A @ X2 @ A3 )
% 0.30/1.56 => ( P @ X2 @ ( F @ X2 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_4_finite,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( finite_finite @ A )
% 0.30/1.56 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_5_pair__fin__digraph__axioms__def,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( pair_p1373739070axioms @ A )
% 0.30/1.56 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.56 ( ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G ) )
% 0.30/1.56 & ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_6_pair__fin__digraph__axioms_Ointro,axiom,
% 0.30/1.56 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.56 ( ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.56 => ( ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.56 => ( pair_p1373739070axioms @ A @ G2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_7_pair__fin__digraph_Opair__finite__arcs,axiom,
% 0.30/1.56 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.56 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.56 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_8_finite__subset,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.56 => ( ( finite_finite2 @ A @ B2 )
% 0.30/1.56 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_9_infinite__super,axiom,
% 0.30/1.56 ! [A: $tType,S: set @ A,T2: set @ A] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
% 0.30/1.56 => ( ~ ( finite_finite2 @ A @ S )
% 0.30/1.56 => ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_10_rev__finite__subset,axiom,
% 0.30/1.56 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.30/1.56 ( ( finite_finite2 @ A @ B2 )
% 0.30/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.56 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_11_finite__has__maximal2,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [A3: set @ A,A4: A] :
% 0.30/1.56 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ( member @ A @ A4 @ A3 )
% 0.30/1.56 => ? [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ A3 )
% 0.30/1.56 & ( ord_less_eq @ A @ A4 @ X )
% 0.30/1.56 & ! [Xa: A] :
% 0.30/1.56 ( ( member @ A @ Xa @ A3 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ X @ Xa )
% 0.30/1.56 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_12_finite__Collect__conjI,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.56 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.30/1.56 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.30/1.56 => ( finite_finite2 @ A
% 0.30/1.56 @ ( collect @ A
% 0.30/1.56 @ ^ [X3: A] :
% 0.30/1.56 ( ( P @ X3 )
% 0.30/1.56 & ( Q @ X3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_13_finite__Collect__disjI,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.56 ( ( finite_finite2 @ A
% 0.30/1.56 @ ( collect @ A
% 0.30/1.56 @ ^ [X3: A] :
% 0.30/1.56 ( ( P @ X3 )
% 0.30/1.56 | ( Q @ X3 ) ) ) )
% 0.30/1.56 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.30/1.56 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_14_finite__Collect__subsets,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A] :
% 0.30/1.56 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( finite_finite2 @ ( set @ A )
% 0.30/1.56 @ ( collect @ ( set @ A )
% 0.30/1.56 @ ^ [B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_15_finite__SigmaI,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.56 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ! [A5: A] :
% 0.30/1.56 ( ( member @ A @ A5 @ A3 )
% 0.30/1.56 => ( finite_finite2 @ B @ ( B2 @ A5 ) ) )
% 0.30/1.56 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_16_pair__fin__digraph_Oaxioms_I2_J,axiom,
% 0.30/1.56 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.56 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.56 => ( pair_p1373739070axioms @ A @ G2 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_17_not__finite__existsD,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o] :
% 0.30/1.56 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.30/1.56 => ? [X_12: A] : ( P @ X_12 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_18_pigeonhole__infinite__rel,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,R: A > B > $o] :
% 0.30/1.56 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ( finite_finite2 @ B @ B2 )
% 0.30/1.56 => ( ! [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ A3 )
% 0.30/1.56 => ? [Xa: B] :
% 0.30/1.56 ( ( member @ B @ Xa @ B2 )
% 0.30/1.56 & ( R @ X @ Xa ) ) )
% 0.30/1.56 => ? [X: B] :
% 0.30/1.56 ( ( member @ B @ X @ B2 )
% 0.30/1.56 & ~ ( finite_finite2 @ A
% 0.30/1.56 @ ( collect @ A
% 0.30/1.56 @ ^ [A6: A] :
% 0.30/1.56 ( ( member @ A @ A6 @ A3 )
% 0.30/1.56 & ( R @ A6 @ X ) ) ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_19_pair__fin__digraph_Opair__fin__digraph,axiom,
% 0.30/1.56 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.56 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.56 => ( pair_p953238171igraph @ A @ G2 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_20_finite__cartesian__product,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
% 0.30/1.56 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ( finite_finite2 @ B @ B2 )
% 0.30/1.56 => ( finite_finite2 @ ( product_prod @ A @ B )
% 0.30/1.56 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.56 @ ^ [Uu: A] : B2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_21_pair__fin__digraph_Opair__finite__verts,axiom,
% 0.30/1.56 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.56 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.56 => ( finite_finite2 @ A @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_22_finite__has__minimal2,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [A3: set @ A,A4: A] :
% 0.30/1.56 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ( member @ A @ A4 @ A3 )
% 0.30/1.56 => ? [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ A3 )
% 0.30/1.56 & ( ord_less_eq @ A @ X @ A4 )
% 0.30/1.56 & ! [Xa: A] :
% 0.30/1.56 ( ( member @ A @ Xa @ A3 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ Xa @ X )
% 0.30/1.56 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_23_infinite__cartesian__product,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
% 0.30/1.56 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/1.56 => ( ~ ( finite_finite2 @ B @ B2 )
% 0.30/1.56 => ~ ( finite_finite2 @ ( product_prod @ A @ B )
% 0.30/1.56 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.56 @ ^ [Uu: A] : B2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_24_Times__subset__cancel2,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,X4: A,C2: set @ A,A3: set @ B,B2: set @ B] :
% 0.30/1.56 ( ( member @ A @ X4 @ C2 )
% 0.30/1.56 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.30/1.56 @ ( product_Sigma @ B @ A @ A3
% 0.30/1.56 @ ^ [Uu: B] : C2 )
% 0.30/1.56 @ ( product_Sigma @ B @ A @ B2
% 0.30/1.56 @ ^ [Uu: B] : C2 ) )
% 0.30/1.56 = ( ord_less_eq @ ( set @ B ) @ A3 @ B2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_25_Sigma__mono,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,C2: set @ A,B2: A > ( set @ B ),D: A > ( set @ B )] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C2 )
% 0.30/1.56 => ( ! [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ A3 )
% 0.30/1.56 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X ) @ ( D @ X ) ) )
% 0.30/1.56 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ C2 @ D ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_26_subsetI,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.56 ( ! [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ A3 )
% 0.30/1.56 => ( member @ A @ X @ B2 ) )
% 0.30/1.56 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_27_subset__antisym,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.30/1.56 => ( A3 = B2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_28_order__refl,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( preorder @ A )
% 0.30/1.56 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_29_pair__pre__digraph_Oequality,axiom,
% 0.30/1.56 ! [Z: $tType,A: $tType,R2: pair_p1731315293ph_ext @ A @ Z,R3: pair_p1731315293ph_ext @ A @ Z] :
% 0.30/1.56 ( ( ( pair_pair_pre_pverts @ A @ Z @ R2 )
% 0.30/1.56 = ( pair_pair_pre_pverts @ A @ Z @ R3 ) )
% 0.30/1.56 => ( ( ( pair_pair_pre_parcs @ A @ Z @ R2 )
% 0.30/1.56 = ( pair_pair_pre_parcs @ A @ Z @ R3 ) )
% 0.30/1.56 => ( ( ( pair_pair_pre_more @ A @ Z @ R2 )
% 0.30/1.56 = ( pair_pair_pre_more @ A @ Z @ R3 ) )
% 0.30/1.56 => ( R2 = R3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_30_Sigma__cong,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ A,C2: A > ( set @ B ),D: A > ( set @ B )] :
% 0.30/1.56 ( ( A3 = B2 )
% 0.30/1.56 => ( ! [X: A] :
% 0.30/1.56 ( ( member @ A @ X @ B2 )
% 0.30/1.56 => ( ( C2 @ X )
% 0.30/1.56 = ( D @ X ) ) )
% 0.30/1.56 => ( ( product_Sigma @ A @ B @ A3 @ C2 )
% 0.30/1.56 = ( product_Sigma @ A @ B @ B2 @ D ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_31_Times__eq__cancel2,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType,X4: A,C2: set @ A,A3: set @ B,B2: set @ B] :
% 0.30/1.56 ( ( member @ A @ X4 @ C2 )
% 0.30/1.56 => ( ( ( product_Sigma @ B @ A @ A3
% 0.30/1.56 @ ^ [Uu: B] : C2 )
% 0.30/1.56 = ( product_Sigma @ B @ A @ B2
% 0.30/1.56 @ ^ [Uu: B] : C2 ) )
% 0.30/1.56 = ( A3 = B2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_32_Collect__subset,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A,P: A > $o] :
% 0.30/1.56 ( ord_less_eq @ ( set @ A )
% 0.30/1.56 @ ( collect @ A
% 0.30/1.56 @ ^ [X3: A] :
% 0.30/1.56 ( ( member @ A @ X3 @ A3 )
% 0.30/1.56 & ( P @ X3 ) ) )
% 0.30/1.56 @ A3 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_33_less__eq__set__def,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.56 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.30/1.56 ( ord_less_eq @ ( A > $o )
% 0.30/1.56 @ ^ [X3: A] : ( member @ A @ X3 @ A2 )
% 0.30/1.56 @ ^ [X3: A] : ( member @ A @ X3 @ B3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_34_predicate1I,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.56 ( ! [X: A] :
% 0.30/1.56 ( ( P @ X )
% 0.30/1.56 => ( Q @ X ) )
% 0.30/1.56 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_35_predicate1D,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o,X4: A] :
% 0.30/1.56 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/1.56 => ( ( P @ X4 )
% 0.30/1.56 => ( Q @ X4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_36_rev__predicate1D,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,X4: A,Q: A > $o] :
% 0.30/1.56 ( ( P @ X4 )
% 0.30/1.56 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/1.56 => ( Q @ X4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_37_dual__order_Oantisym,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [B4: A,A4: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/1.56 => ( A4 = B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_38_dual__order_Oeq__iff,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/1.56 = ( ^ [A6: A,B5: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.30/1.56 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_39_dual__order_Otrans,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [B4: A,A4: A,C3: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ C3 @ B4 )
% 0.30/1.56 => ( ord_less_eq @ A @ C3 @ A4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_40_linorder__wlog,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( linorder @ A )
% 0.30/1.56 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.30/1.56 ( ! [A5: A,B6: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A5 @ B6 )
% 0.30/1.56 => ( P @ A5 @ B6 ) )
% 0.30/1.56 => ( ! [A5: A,B6: A] :
% 0.30/1.56 ( ( P @ B6 @ A5 )
% 0.30/1.56 => ( P @ A5 @ B6 ) )
% 0.30/1.56 => ( P @ A4 @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_41_dual__order_Orefl,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_42_order__trans,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( preorder @ A )
% 0.30/1.56 => ! [X4: A,Y2: A,Z3: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ Y2 @ Z3 )
% 0.30/1.56 => ( ord_less_eq @ A @ X4 @ Z3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_43_order__class_Oorder_Oantisym,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [A4: A,B4: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/1.56 => ( A4 = B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_44_ord__le__eq__trans,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( ord @ A )
% 0.30/1.56 => ! [A4: A,B4: A,C3: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/1.56 => ( ( B4 = C3 )
% 0.30/1.56 => ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/1.56 ! [A: $tType,A4: A,P: A > $o] :
% 0.30/1.56 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.30/1.56 = ( P @ A4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A] :
% 0.30/1.56 ( ( collect @ A
% 0.30/1.56 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.30/1.56 = A3 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_47_Collect__cong,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.56 ( ! [X: A] :
% 0.30/1.56 ( ( P @ X )
% 0.30/1.56 = ( Q @ X ) )
% 0.30/1.56 => ( ( collect @ A @ P )
% 0.30/1.56 = ( collect @ A @ Q ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_48_ext,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType,F2: A > B,G3: A > B] :
% 0.30/1.56 ( ! [X: A] :
% 0.30/1.56 ( ( F2 @ X )
% 0.30/1.56 = ( G3 @ X ) )
% 0.30/1.56 => ( F2 = G3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_49_ord__eq__le__trans,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( ord @ A )
% 0.30/1.56 => ! [A4: A,B4: A,C3: A] :
% 0.30/1.56 ( ( A4 = B4 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ B4 @ C3 )
% 0.30/1.56 => ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_50_order__class_Oorder_Oeq__iff,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/1.56 = ( ^ [A6: A,B5: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.30/1.56 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_51_antisym__conv,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [Y2: A,X4: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.30/1.56 = ( X4 = Y2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_52_le__cases3,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( linorder @ A )
% 0.30/1.56 => ! [X4: A,Y2: A,Z3: A] :
% 0.30/1.56 ( ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.30/1.56 => ~ ( ord_less_eq @ A @ Y2 @ Z3 ) )
% 0.30/1.56 => ( ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.30/1.56 => ~ ( ord_less_eq @ A @ X4 @ Z3 ) )
% 0.30/1.56 => ( ( ( ord_less_eq @ A @ X4 @ Z3 )
% 0.30/1.56 => ~ ( ord_less_eq @ A @ Z3 @ Y2 ) )
% 0.30/1.56 => ( ( ( ord_less_eq @ A @ Z3 @ Y2 )
% 0.30/1.56 => ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
% 0.30/1.56 => ( ( ( ord_less_eq @ A @ Y2 @ Z3 )
% 0.30/1.56 => ~ ( ord_less_eq @ A @ Z3 @ X4 ) )
% 0.30/1.56 => ~ ( ( ord_less_eq @ A @ Z3 @ X4 )
% 0.30/1.56 => ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_53_order_Otrans,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [A4: A,B4: A,C3: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ B4 @ C3 )
% 0.30/1.56 => ( ord_less_eq @ A @ A4 @ C3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_54_le__cases,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( linorder @ A )
% 0.30/1.56 => ! [X4: A,Y2: A] :
% 0.30/1.56 ( ~ ( ord_less_eq @ A @ X4 @ Y2 )
% 0.30/1.56 => ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_55_eq__refl,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( preorder @ A )
% 0.30/1.56 => ! [X4: A,Y2: A] :
% 0.30/1.56 ( ( X4 = Y2 )
% 0.30/1.56 => ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_56_linear,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( linorder @ A )
% 0.30/1.56 => ! [X4: A,Y2: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.30/1.56 | ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_57_antisym,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ! [X4: A,Y2: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.30/1.56 => ( ( ord_less_eq @ A @ Y2 @ X4 )
% 0.30/1.56 => ( X4 = Y2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_58_eq__iff,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( order @ A )
% 0.30/1.56 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/1.56 = ( ^ [X3: A,Y3: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/1.56 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_59_ord__le__eq__subst,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType] :
% 0.30/1.56 ( ( ( ord @ B )
% 0.30/1.56 & ( ord @ A ) )
% 0.30/1.56 => ! [A4: A,B4: A,F2: A > B,C3: B] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/1.56 => ( ( ( F2 @ B4 )
% 0.30/1.56 = C3 )
% 0.30/1.56 => ( ! [X: A,Y4: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/1.56 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.30/1.56 => ( ord_less_eq @ B @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_60_ord__eq__le__subst,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType] :
% 0.30/1.56 ( ( ( ord @ B )
% 0.30/1.56 & ( ord @ A ) )
% 0.30/1.56 => ! [A4: A,F2: B > A,B4: B,C3: B] :
% 0.30/1.56 ( ( A4
% 0.30/1.56 = ( F2 @ B4 ) )
% 0.30/1.56 => ( ( ord_less_eq @ B @ B4 @ C3 )
% 0.30/1.56 => ( ! [X: B,Y4: B] :
% 0.30/1.56 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.30/1.56 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.30/1.56 => ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_61_order__subst2,axiom,
% 0.30/1.56 ! [A: $tType,C: $tType] :
% 0.30/1.56 ( ( ( order @ C )
% 0.30/1.56 & ( order @ A ) )
% 0.30/1.56 => ! [A4: A,B4: A,F2: A > C,C3: C] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/1.56 => ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C3 )
% 0.30/1.56 => ( ! [X: A,Y4: A] :
% 0.30/1.56 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/1.56 => ( ord_less_eq @ C @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.30/1.56 => ( ord_less_eq @ C @ ( F2 @ A4 ) @ C3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_62_order__subst1,axiom,
% 0.30/1.56 ! [A: $tType,B: $tType] :
% 0.30/1.56 ( ( ( order @ B )
% 0.30/1.56 & ( order @ A ) )
% 0.30/1.56 => ! [A4: A,F2: B > A,B4: B,C3: B] :
% 0.30/1.56 ( ( ord_less_eq @ A @ A4 @ ( F2 @ B4 ) )
% 0.30/1.56 => ( ( ord_less_eq @ B @ B4 @ C3 )
% 0.30/1.56 => ( ! [X: B,Y4: B] :
% 0.30/1.56 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.30/1.56 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
% 0.30/1.56 => ( ord_less_eq @ A @ A4 @ ( F2 @ C3 ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_63_le__fun__def,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType] :
% 0.30/1.56 ( ( ord @ B )
% 0.30/1.56 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/1.56 = ( ^ [F3: A > B,G4: A > B] :
% 0.30/1.56 ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_64_le__funI,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType] :
% 0.30/1.56 ( ( ord @ B )
% 0.30/1.56 => ! [F2: A > B,G3: A > B] :
% 0.30/1.56 ( ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G3 @ X ) )
% 0.30/1.56 => ( ord_less_eq @ ( A > B ) @ F2 @ G3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_65_le__funE,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType] :
% 0.30/1.56 ( ( ord @ B )
% 0.30/1.56 => ! [F2: A > B,G3: A > B,X4: A] :
% 0.30/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G3 )
% 0.30/1.56 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G3 @ X4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_66_le__funD,axiom,
% 0.30/1.56 ! [B: $tType,A: $tType] :
% 0.30/1.56 ( ( ord @ B )
% 0.30/1.56 => ! [F2: A > B,G3: A > B,X4: A] :
% 0.30/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G3 )
% 0.30/1.56 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G3 @ X4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_67_Collect__mono__iff,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/1.56 = ( ! [X3: A] :
% 0.30/1.56 ( ( P @ X3 )
% 0.30/1.56 => ( Q @ X3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_68_set__eq__subset,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 ) )
% 0.30/1.56 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.30/1.56 & ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_69_subset__trans,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A,B2: set @ A,C2: set @ A] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.30/1.56 => ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_70_Collect__mono,axiom,
% 0.30/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/1.56 ( ! [X: A] :
% 0.30/1.56 ( ( P @ X )
% 0.30/1.56 => ( Q @ X ) )
% 0.30/1.56 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_71_subset__refl,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_72_subset__iff,axiom,
% 0.30/1.56 ! [A: $tType] :
% 0.30/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.56 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.30/1.56 ! [T3: A] :
% 0.30/1.56 ( ( member @ A @ T3 @ A2 )
% 0.30/1.56 => ( member @ A @ T3 @ B3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_73_equalityD2,axiom,
% 0.30/1.56 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.56 ( ( A3 = B2 )
% 0.30/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_74_equalityD1,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.57 ( ( A3 = B2 )
% 0.30/1.57 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_75_subset__eq,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/1.57 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.30/1.57 ! [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ A2 )
% 0.30/1.57 => ( member @ A @ X3 @ B3 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_76_equalityE,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.57 ( ( A3 = B2 )
% 0.30/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_77_subsetD,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,C3: A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ( ( member @ A @ C3 @ A3 )
% 0.30/1.57 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_78_in__mono,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ( ( member @ A @ X4 @ A3 )
% 0.30/1.57 => ( member @ A @ X4 @ B2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_79_pred__subset__eq,axiom,
% 0.30/1.57 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( A > $o )
% 0.30/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.30/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.30/1.57 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_80_pair__pre__digraph_Osurjective,axiom,
% 0.30/1.57 ! [Z: $tType,A: $tType,R2: pair_p1731315293ph_ext @ A @ Z] :
% 0.30/1.57 ( R2
% 0.30/1.57 = ( pair_p1152105390ph_ext @ A @ Z @ ( pair_pair_pre_pverts @ A @ Z @ R2 ) @ ( pair_pair_pre_parcs @ A @ Z @ R2 ) @ ( pair_pair_pre_more @ A @ Z @ R2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_81_member__product,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,X4: product_prod @ A @ B,A3: set @ A,B2: set @ B] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ X4 @ ( product_product @ A @ B @ A3 @ B2 ) )
% 0.30/1.57 = ( member @ ( product_prod @ A @ B ) @ X4
% 0.30/1.57 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : B2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_82_Product__Type_Oproduct__def,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType] :
% 0.30/1.57 ( ( product_product @ A @ B )
% 0.30/1.57 = ( ^ [A2: set @ A,B3: set @ B] :
% 0.30/1.57 ( product_Sigma @ A @ B @ A2
% 0.30/1.57 @ ^ [Uu: A] : B3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_83_subset__Collect__iff,axiom,
% 0.30/1.57 ! [A: $tType,B2: set @ A,A3: set @ A,P: A > $o] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.30/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ A3 )
% 0.30/1.57 & ( P @ X3 ) ) ) )
% 0.30/1.57 = ( ! [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ B2 )
% 0.30/1.57 => ( P @ X3 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_84_subset__CollectI,axiom,
% 0.30/1.57 ! [A: $tType,B2: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.30/1.57 => ( ! [X: A] :
% 0.30/1.57 ( ( member @ A @ X @ B2 )
% 0.30/1.57 => ( ( Q @ X )
% 0.30/1.57 => ( P @ X ) ) )
% 0.30/1.57 => ( ord_less_eq @ ( set @ A )
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ B2 )
% 0.30/1.57 & ( Q @ X3 ) ) )
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ A3 )
% 0.30/1.57 & ( P @ X3 ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_85_Collect__restrict,axiom,
% 0.30/1.57 ! [A: $tType,X5: set @ A,P: A > $o] :
% 0.30/1.57 ( ord_less_eq @ ( set @ A )
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ X5 )
% 0.30/1.57 & ( P @ X3 ) ) )
% 0.30/1.57 @ X5 ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_86_prop__restrict,axiom,
% 0.30/1.57 ! [A: $tType,X4: A,Z4: set @ A,X5: set @ A,P: A > $o] :
% 0.30/1.57 ( ( member @ A @ X4 @ Z4 )
% 0.30/1.57 => ( ( ord_less_eq @ ( set @ A ) @ Z4
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ X5 )
% 0.30/1.57 & ( P @ X3 ) ) ) )
% 0.30/1.57 => ( P @ X4 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_87_conj__subset__def,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( P @ X3 )
% 0.30/1.57 & ( Q @ X3 ) ) ) )
% 0.30/1.57 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
% 0.30/1.57 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_88_pair__pre__digraph_Oext__inject,axiom,
% 0.30/1.57 ! [A: $tType,Z: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z,Pverts2: set @ A,Parcs2: set @ ( product_prod @ A @ A ),More2: Z] :
% 0.30/1.57 ( ( ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More )
% 0.30/1.57 = ( pair_p1152105390ph_ext @ A @ Z @ Pverts2 @ Parcs2 @ More2 ) )
% 0.30/1.57 = ( ( Pverts = Pverts2 )
% 0.30/1.57 & ( Parcs = Parcs2 )
% 0.30/1.57 & ( More = More2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_89_pair__pre__digraph_Oinduct__scheme,axiom,
% 0.30/1.57 ! [Z: $tType,A: $tType,P: ( pair_p1731315293ph_ext @ A @ Z ) > $o,R2: pair_p1731315293ph_ext @ A @ Z] :
% 0.30/1.57 ( ! [Pverts3: set @ A,Parcs3: set @ ( product_prod @ A @ A ),More3: Z] : ( P @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts3 @ Parcs3 @ More3 ) )
% 0.30/1.57 => ( P @ R2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_90_pair__pre__digraph_Ocases__scheme,axiom,
% 0.30/1.57 ! [A: $tType,Z: $tType,R2: pair_p1731315293ph_ext @ A @ Z] :
% 0.30/1.57 ~ ! [Pverts3: set @ A,Parcs3: set @ ( product_prod @ A @ A ),More3: Z] :
% 0.30/1.57 ( R2
% 0.30/1.57 != ( pair_p1152105390ph_ext @ A @ Z @ Pverts3 @ Parcs3 @ More3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_91_pair__pre__digraph_Oselect__convs_I2_J,axiom,
% 0.30/1.57 ! [Z: $tType,A: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.30/1.57 ( ( pair_pair_pre_parcs @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.30/1.57 = Parcs ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_92_pair__pre__digraph_Oselect__convs_I1_J,axiom,
% 0.30/1.57 ! [Z: $tType,A: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.30/1.57 ( ( pair_pair_pre_pverts @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.30/1.57 = Pverts ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_93_pair__pre__digraph_Oselect__convs_I3_J,axiom,
% 0.30/1.57 ! [A: $tType,Z: $tType,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.30/1.57 ( ( pair_pair_pre_more @ A @ Z @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.30/1.57 = More ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_94_Fpow__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( finite_Fpow @ A )
% 0.30/1.57 = ( ^ [A2: set @ A] :
% 0.30/1.57 ( collect @ ( set @ A )
% 0.30/1.57 @ ^ [X6: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
% 0.30/1.57 & ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_95_pair__fin__digraph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_p953238171igraph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G )
% 0.30/1.57 & ( pair_p1373739070axioms @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_96_pair__fin__digraph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p1373739070axioms @ A @ G2 )
% 0.30/1.57 => ( pair_p953238171igraph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_97_pair__pre__digraph_Ounfold__congs_I3_J,axiom,
% 0.30/1.57 ! [A: $tType,A7: $tType,R2: pair_p1731315293ph_ext @ A7 @ A,R3: pair_p1731315293ph_ext @ A7 @ A,V: A,F2: A > A,F4: A > A] :
% 0.30/1.57 ( ( R2 = R3 )
% 0.30/1.57 => ( ( ( pair_pair_pre_more @ A7 @ A @ R3 )
% 0.30/1.57 = V )
% 0.30/1.57 => ( ! [V2: A] :
% 0.30/1.57 ( ( V2 = V )
% 0.30/1.57 => ( ( F2 @ V2 )
% 0.30/1.57 = ( F4 @ V2 ) ) )
% 0.30/1.57 => ( ( pair_p1108258481update @ A @ A7 @ F2 @ R2 )
% 0.30/1.57 = ( pair_p1108258481update @ A @ A7 @ F4 @ R3 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_98_pair__pre__digraph_Ofold__congs_I3_J,axiom,
% 0.30/1.57 ! [A: $tType,A7: $tType,R2: pair_p1731315293ph_ext @ A7 @ A,R3: pair_p1731315293ph_ext @ A7 @ A,V: A,F2: A > A,F4: A > A] :
% 0.30/1.57 ( ( R2 = R3 )
% 0.30/1.57 => ( ( ( pair_pair_pre_more @ A7 @ A @ R3 )
% 0.30/1.57 = V )
% 0.30/1.57 => ( ! [V2: A] :
% 0.30/1.57 ( ( V = V2 )
% 0.30/1.57 => ( ( F2 @ V2 )
% 0.30/1.57 = ( F4 @ V2 ) ) )
% 0.30/1.57 => ( ( pair_p1108258481update @ A @ A7 @ F2 @ R2 )
% 0.30/1.57 = ( pair_p1108258481update @ A @ A7 @ F4 @ R3 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_99_finite__cartesian__product__iff,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
% 0.30/1.57 ( ( finite_finite2 @ ( product_prod @ A @ B )
% 0.30/1.57 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : B2 ) )
% 0.30/1.57 = ( ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 | ( B2
% 0.30/1.57 = ( bot_bot @ ( set @ B ) ) )
% 0.30/1.57 | ( ( finite_finite2 @ A @ A3 )
% 0.30/1.57 & ( finite_finite2 @ B @ B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_100_bot__apply,axiom,
% 0.30/1.57 ! [C: $tType,D2: $tType] :
% 0.30/1.57 ( ( bot @ C )
% 0.30/1.57 => ( ( bot_bot @ ( D2 > C ) )
% 0.30/1.57 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_101_empty__iff,axiom,
% 0.30/1.57 ! [A: $tType,C3: A] :
% 0.30/1.57 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_102_all__not__in__conv,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] :
% 0.30/1.57 ( ( ! [X3: A] :
% 0.30/1.57 ~ ( member @ A @ X3 @ A3 ) )
% 0.30/1.57 = ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_103_Collect__empty__eq,axiom,
% 0.30/1.57 ! [A: $tType,P: A > $o] :
% 0.30/1.57 ( ( ( collect @ A @ P )
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 = ( ! [X3: A] :
% 0.30/1.57 ~ ( P @ X3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_104_empty__Collect__eq,axiom,
% 0.30/1.57 ! [A: $tType,P: A > $o] :
% 0.30/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/1.57 = ( collect @ A @ P ) )
% 0.30/1.57 = ( ! [X3: A] :
% 0.30/1.57 ~ ( P @ X3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_105_empty__subsetI,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_106_subset__empty,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 = ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_107_Sigma__empty1,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.30/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_108_Times__empty,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
% 0.30/1.57 ( ( ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : B2 )
% 0.30/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/1.57 = ( ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 | ( B2
% 0.30/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_109_Sigma__empty2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A3: set @ A] :
% 0.30/1.57 ( ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 0.30/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_110_empty__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( bot_bot @ ( set @ A ) )
% 0.30/1.57 = ( collect @ A
% 0.30/1.57 @ ^ [X3: A] : $false ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_111_Sigma__empty__iff,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,I: set @ A,X5: A > ( set @ B )] :
% 0.30/1.57 ( ( ( product_Sigma @ A @ B @ I @ X5 )
% 0.30/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/1.57 = ( ! [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ I )
% 0.30/1.57 => ( ( X5 @ X3 )
% 0.30/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_112_empty__in__Fpow,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_113_Fpow__not__empty,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] :
% 0.30/1.57 ( ( finite_Fpow @ A @ A3 )
% 0.30/1.57 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_114_emptyE,axiom,
% 0.30/1.57 ! [A: $tType,A4: A] :
% 0.30/1.57 ~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_115_equals0D,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,A4: A] :
% 0.30/1.57 ( ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 => ~ ( member @ A @ A4 @ A3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_116_equals0I,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] :
% 0.30/1.57 ( ! [Y4: A] :
% 0.30/1.57 ~ ( member @ A @ Y4 @ A3 )
% 0.30/1.57 => ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_117_ex__in__conv,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] :
% 0.30/1.57 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.30/1.57 = ( A3
% 0.30/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_118_bot__fun__def,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType] :
% 0.30/1.57 ( ( bot @ B )
% 0.30/1.57 => ( ( bot_bot @ ( A > B ) )
% 0.30/1.57 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_119_bot_Oextremum__uniqueI,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( order_bot @ A )
% 0.30/1.57 => ! [A4: A] :
% 0.30/1.57 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 0.30/1.57 => ( A4
% 0.30/1.57 = ( bot_bot @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_120_bot_Oextremum__unique,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( order_bot @ A )
% 0.30/1.57 => ! [A4: A] :
% 0.30/1.57 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 0.30/1.57 = ( A4
% 0.30/1.57 = ( bot_bot @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_121_bot_Oextremum,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( order_bot @ A )
% 0.30/1.57 => ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_122_finite_OemptyI,axiom,
% 0.30/1.57 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_123_infinite__imp__nonempty,axiom,
% 0.30/1.57 ! [A: $tType,S: set @ A] :
% 0.30/1.57 ( ~ ( finite_finite2 @ A @ S )
% 0.30/1.57 => ( S
% 0.30/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_124_subset__emptyI,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A] :
% 0.30/1.57 ( ! [X: A] :
% 0.30/1.57 ~ ( member @ A @ X @ A3 )
% 0.30/1.57 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_125_times__eq__iff,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B,C2: set @ A,D: set @ B] :
% 0.30/1.57 ( ( ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : B2 )
% 0.30/1.57 = ( product_Sigma @ A @ B @ C2
% 0.30/1.57 @ ^ [Uu: A] : D ) )
% 0.30/1.57 = ( ( ( A3 = C2 )
% 0.30/1.57 & ( B2 = D ) )
% 0.30/1.57 | ( ( ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 | ( B2
% 0.30/1.57 = ( bot_bot @ ( set @ B ) ) ) )
% 0.30/1.57 & ( ( C2
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 | ( D
% 0.30/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_126_Fpow__mono,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_127_pair__fin__digraph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_128_finite__has__maximal,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( order @ A )
% 0.30/1.57 => ! [A3: set @ A] :
% 0.30/1.57 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.57 => ( ( A3
% 0.30/1.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 => ? [X: A] :
% 0.30/1.57 ( ( member @ A @ X @ A3 )
% 0.30/1.57 & ! [Xa: A] :
% 0.30/1.57 ( ( member @ A @ Xa @ A3 )
% 0.30/1.57 => ( ( ord_less_eq @ A @ X @ Xa )
% 0.30/1.57 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_129_finite__has__minimal,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( order @ A )
% 0.30/1.57 => ! [A3: set @ A] :
% 0.30/1.57 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.57 => ( ( A3
% 0.30/1.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 => ? [X: A] :
% 0.30/1.57 ( ( member @ A @ X @ A3 )
% 0.30/1.57 & ! [Xa: A] :
% 0.30/1.57 ( ( member @ A @ Xa @ A3 )
% 0.30/1.57 => ( ( ord_less_eq @ A @ Xa @ X )
% 0.30/1.57 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_130_pair__pre__digraph_Oupdate__convs_I3_J,axiom,
% 0.30/1.57 ! [A: $tType,Z: $tType,More2: Z > Z,Pverts: set @ A,Parcs: set @ ( product_prod @ A @ A ),More: Z] :
% 0.30/1.57 ( ( pair_p1108258481update @ Z @ A @ More2 @ ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ More ) )
% 0.30/1.57 = ( pair_p1152105390ph_ext @ A @ Z @ Pverts @ Parcs @ ( More2 @ More ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_131_times__subset__iff,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A3: set @ A,C2: set @ B,B2: set @ A,D: set @ B] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.30/1.57 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : C2 )
% 0.30/1.57 @ ( product_Sigma @ A @ B @ B2
% 0.30/1.57 @ ^ [Uu: A] : D ) )
% 0.30/1.57 = ( ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 | ( C2
% 0.30/1.57 = ( bot_bot @ ( set @ B ) ) )
% 0.30/1.57 | ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 & ( ord_less_eq @ ( set @ B ) @ C2 @ D ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_132_finite__SigmaI2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( finite_finite2 @ A
% 0.30/1.57 @ ( collect @ A
% 0.30/1.57 @ ^ [X3: A] :
% 0.30/1.57 ( ( member @ A @ X3 @ A3 )
% 0.30/1.57 & ( ( B2 @ X3 )
% 0.30/1.57 != ( bot_bot @ ( set @ B ) ) ) ) ) )
% 0.30/1.57 => ( ! [A5: A] :
% 0.30/1.57 ( ( member @ A @ A5 @ A3 )
% 0.30/1.57 => ( finite_finite2 @ B @ ( B2 @ A5 ) ) )
% 0.30/1.57 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_133_finite__cartesian__productD1,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
% 0.30/1.57 ( ( finite_finite2 @ ( product_prod @ A @ B )
% 0.30/1.57 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : B2 ) )
% 0.30/1.57 => ( ( B2
% 0.30/1.57 != ( bot_bot @ ( set @ B ) ) )
% 0.30/1.57 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_134_finite__cartesian__productD2,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
% 0.30/1.57 ( ( finite_finite2 @ ( product_prod @ A @ B )
% 0.30/1.57 @ ( product_Sigma @ A @ B @ A3
% 0.30/1.57 @ ^ [Uu: A] : B2 ) )
% 0.30/1.57 => ( ( A3
% 0.30/1.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 => ( finite_finite2 @ B @ B2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_135_finite__transitivity__chain,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,R: A > A > $o] :
% 0.30/1.57 ( ( finite_finite2 @ A @ A3 )
% 0.30/1.57 => ( ! [X: A] :
% 0.30/1.57 ~ ( R @ X @ X )
% 0.30/1.57 => ( ! [X: A,Y4: A,Z5: A] :
% 0.30/1.57 ( ( R @ X @ Y4 )
% 0.30/1.57 => ( ( R @ Y4 @ Z5 )
% 0.30/1.57 => ( R @ X @ Z5 ) ) )
% 0.30/1.57 => ( ! [X: A] :
% 0.30/1.57 ( ( member @ A @ X @ A3 )
% 0.30/1.57 => ? [Y5: A] :
% 0.30/1.57 ( ( member @ A @ Y5 @ A3 )
% 0.30/1.57 & ( R @ X @ Y5 ) ) )
% 0.30/1.57 => ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_136_pair__wf__digraph_Opair__wf__digraph__subdivide,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( pair_pair_wf_digraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_137_Set_Ois__empty__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( is_empty @ A )
% 0.30/1.57 = ( ^ [A2: set @ A] :
% 0.30/1.57 ( A2
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_138_pair__wf__digraph_Oin__arcsD2,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( member @ A @ V3 @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_139_pair__wf__digraph_Oin__arcsD1,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( member @ A @ U @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_140_old_Oprod_Oinject,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A4: A,B4: B,A8: A,B7: B] :
% 0.30/1.57 ( ( ( product_Pair @ A @ B @ A4 @ B4 )
% 0.30/1.57 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 0.30/1.57 = ( ( A4 = A8 )
% 0.30/1.57 & ( B4 = B7 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_141_prod_Oinject,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.30/1.57 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.30/1.57 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.30/1.57 = ( ( X1 = Y1 )
% 0.30/1.57 & ( X22 = Y22 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_142_SigmaI,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A4: A,A3: set @ A,B4: B,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( member @ A @ A4 @ A3 )
% 0.30/1.57 => ( ( member @ B @ B4 @ ( B2 @ A4 ) )
% 0.30/1.57 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_143_mem__Sigma__iff,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.30/1.57 = ( ( member @ A @ A4 @ A3 )
% 0.30/1.57 & ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_144_bot__empty__eq,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( bot_bot @ ( A > $o ) )
% 0.30/1.57 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_145_bot__empty__eq2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType] :
% 0.30/1.57 ( ( bot_bot @ ( A > B > $o ) )
% 0.30/1.57 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_146_bot__set__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( bot_bot @ ( set @ A ) )
% 0.30/1.57 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_147_old_Oprod_Oinducts,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/1.57 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.30/1.57 => ( P @ Prod ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_148_old_Oprod_Oexhaust,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
% 0.30/1.57 ~ ! [A5: A,B6: B] :
% 0.30/1.57 ( Y2
% 0.30/1.57 != ( product_Pair @ A @ B @ A5 @ B6 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_149_prod__induct7,axiom,
% 0.30/1.57 ! [G5: $tType,F5: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) )] :
% 0.30/1.57 ( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5,G6: G5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G5 ) @ E3 @ ( product_Pair @ F5 @ G5 @ F @ G6 ) ) ) ) ) ) )
% 0.30/1.57 => ( P @ X4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_150_prod__induct6,axiom,
% 0.30/1.57 ! [F5: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) )] :
% 0.30/1.57 ( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F5 ) @ D3 @ ( product_Pair @ E2 @ F5 @ E3 @ F ) ) ) ) ) )
% 0.30/1.57 => ( P @ X4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_151_prod__induct5,axiom,
% 0.30/1.57 ! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 0.30/1.57 ( ! [A5: A,B6: B,C4: C,D3: D2,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) )
% 0.30/1.57 => ( P @ X4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_152_prod__induct4,axiom,
% 0.30/1.57 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/1.57 ( ! [A5: A,B6: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B6 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
% 0.30/1.57 => ( P @ X4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_153_prod__induct3,axiom,
% 0.30/1.57 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/1.57 ( ! [A5: A,B6: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) )
% 0.30/1.57 => ( P @ X4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_154_prod__cases7,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F5: $tType,G5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) )] :
% 0.30/1.57 ~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5,G6: G5] :
% 0.30/1.57 ( Y2
% 0.30/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F5 @ G5 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G5 ) @ E3 @ ( product_Pair @ F5 @ G5 @ F @ G6 ) ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_155_prod__cases6,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) )] :
% 0.30/1.57 ~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2,F: F5] :
% 0.30/1.57 ( Y2
% 0.30/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F5 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F5 ) @ D3 @ ( product_Pair @ E2 @ F5 @ E3 @ F ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_156_prod__cases5,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
% 0.30/1.57 ~ ! [A5: A,B6: B,C4: C,D3: D2,E3: E2] :
% 0.30/1.57 ( Y2
% 0.30/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_157_prod__cases4,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/1.57 ~ ! [A5: A,B6: B,C4: C,D3: D2] :
% 0.30/1.57 ( Y2
% 0.30/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B6 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_158_prod__cases3,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/1.57 ~ ! [A5: A,B6: B,C4: C] :
% 0.30/1.57 ( Y2
% 0.30/1.57 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_159_Pair__inject,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,A4: A,B4: B,A8: A,B7: B] :
% 0.30/1.57 ( ( ( product_Pair @ A @ B @ A4 @ B4 )
% 0.30/1.57 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 0.30/1.57 => ~ ( ( A4 = A8 )
% 0.30/1.57 => ( B4 != B7 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_160_prod__cases,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/1.57 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.30/1.57 => ( P @ P2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_161_surj__pair,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/1.57 ? [X: A,Y4: B] :
% 0.30/1.57 ( P2
% 0.30/1.57 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_162_subdivide_Oinduct,axiom,
% 0.30/1.57 ! [A: $tType,P: ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > $o,A0: pair_p1731315293ph_ext @ A @ product_unit,A1: product_prod @ A @ A,A22: A] :
% 0.30/1.57 ( ! [G7: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V2: A,X_12: A] : ( P @ G7 @ ( product_Pair @ A @ A @ U2 @ V2 ) @ X_12 )
% 0.30/1.57 => ( P @ A0 @ A1 @ A22 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_163_subdivide_Ocases,axiom,
% 0.30/1.57 ! [A: $tType,X4: product_prod @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A )] :
% 0.30/1.57 ~ ! [G7: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V2: A,W2: A] :
% 0.30/1.57 ( X4
% 0.30/1.57 != ( product_Pair @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A ) @ G7 @ ( product_Pair @ ( product_prod @ A @ A ) @ A @ ( product_Pair @ A @ A @ U2 @ V2 ) @ W2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_164_pred__equals__eq2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/1.57 ( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.30/1.57 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) ) )
% 0.30/1.57 = ( R = S ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_165_ssubst__Pair__rhs,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
% 0.30/1.57 => ( ( S3 = S2 )
% 0.30/1.57 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_166_SigmaE,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.30/1.57 => ~ ! [X: A] :
% 0.30/1.57 ( ( member @ A @ X @ A3 )
% 0.30/1.57 => ! [Y4: B] :
% 0.30/1.57 ( ( member @ B @ Y4 @ ( B2 @ X ) )
% 0.30/1.57 => ( C3
% 0.30/1.57 != ( product_Pair @ A @ B @ X @ Y4 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_167_SigmaD1,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.30/1.57 => ( member @ A @ A4 @ A3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_168_SigmaD2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.30/1.57 => ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_169_SigmaE2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A4: A,B4: B,A3: set @ A,B2: A > ( set @ B )] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
% 0.30/1.57 => ~ ( ( member @ A @ A4 @ A3 )
% 0.30/1.57 => ~ ( member @ B @ B4 @ ( B2 @ A4 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_170_subrelI,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.30/1.57 ( ! [X: A,Y4: B] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 )
% 0.30/1.57 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
% 0.30/1.57 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_171_pred__subset__eq2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/1.57 ( ( ord_less_eq @ ( A > B > $o )
% 0.30/1.57 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 0.30/1.57 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 0.30/1.57 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_172_relChain__def,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType] :
% 0.30/1.57 ( ( ord @ B )
% 0.30/1.57 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.30/1.57 = ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/1.57 ! [I2: A,J: A] :
% 0.30/1.57 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R4 )
% 0.30/1.57 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_173_old_Oprod_Orec,axiom,
% 0.30/1.57 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B4: B] :
% 0.30/1.57 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/1.57 = ( F1 @ A4 @ B4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_174_less__by__empty,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
% 0.30/1.57 ( ( A3
% 0.30/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/1.57 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_175_calculation,axiom,
% 0.30/1.57 ( ( finite_finite2 @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g ) )
% 0.30/1.57 & ( ( finite_card @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g ) )
% 0.30/1.57 = n )
% 0.30/1.57 & ( ( pair_pair_pre_parcs @ a @ product_unit @ g )
% 0.30/1.57 = ( collect @ ( product_prod @ a @ a )
% 0.30/1.57 @ ( product_case_prod @ a @ a @ $o
% 0.30/1.57 @ ^ [U3: a,V4: a] :
% 0.30/1.57 ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ U3 @ V4 )
% 0.30/1.57 @ ( product_Sigma @ a @ a @ ( pair_pair_pre_pverts @ a @ product_unit @ g )
% 0.30/1.57 @ ^ [Uu: a] : ( pair_pair_pre_pverts @ a @ product_unit @ g ) ) )
% 0.30/1.57 & ( U3 != V4 ) ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_176_predicate2I,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.30/1.57 ( ! [X: A,Y4: B] :
% 0.30/1.57 ( ( P @ X @ Y4 )
% 0.30/1.57 => ( Q @ X @ Y4 ) )
% 0.30/1.57 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_177_split__part,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ $o
% 0.30/1.57 @ ^ [A6: A,B5: B] :
% 0.30/1.57 ( P
% 0.30/1.57 & ( Q @ A6 @ B5 ) ) )
% 0.30/1.57 = ( ^ [Ab: product_prod @ A @ B] :
% 0.30/1.57 ( P
% 0.30/1.57 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_178_case__prod__conv,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A4: B,B4: C] :
% 0.30/1.57 ( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A4 @ B4 ) )
% 0.30/1.57 = ( F2 @ A4 @ B4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_179_case__prodI2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
% 0.30/1.57 ( ! [A5: A,B6: B] :
% 0.30/1.57 ( ( P2
% 0.30/1.57 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.30/1.57 => ( C3 @ A5 @ B6 ) )
% 0.30/1.57 => ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_180_case__prodI,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B4: B] :
% 0.30/1.57 ( ( F2 @ A4 @ B4 )
% 0.30/1.57 => ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_181_Collect__case__prod,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 0.30/1.57 ( ( collect @ ( product_prod @ A @ B )
% 0.30/1.57 @ ( product_case_prod @ A @ B @ $o
% 0.30/1.57 @ ^ [A6: A,B5: B] :
% 0.30/1.57 ( ( P @ A6 )
% 0.30/1.57 & ( Q @ B5 ) ) ) )
% 0.30/1.57 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.30/1.57 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_182_bot2E,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,X4: A,Y2: B] :
% 0.30/1.57 ~ ( bot_bot @ ( A > B > $o ) @ X4 @ Y2 ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_183_Collect__case__prod__mono,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,A3: A > B > $o,B2: A > B > $o] :
% 0.30/1.57 ( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B2 )
% 0.30/1.57 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_184_case__prodD,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,F2: A > B > $o,A4: A,B4: B] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/1.57 => ( F2 @ A4 @ B4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_185_case__prodE,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
% 0.30/1.57 => ~ ! [X: A,Y4: B] :
% 0.30/1.57 ( ( P2
% 0.30/1.57 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.30/1.57 => ~ ( C3 @ X @ Y4 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_186_case__prodE2,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z3: product_prod @ B @ C] :
% 0.30/1.57 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z3 ) )
% 0.30/1.57 => ~ ! [X: B,Y4: C] :
% 0.30/1.57 ( ( Z3
% 0.30/1.57 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 0.30/1.57 => ~ ( Q @ ( P @ X @ Y4 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_187_case__prod__eta,axiom,
% 0.30/1.57 ! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ C
% 0.30/1.57 @ ^ [X3: A,Y3: B] : ( F2 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.30/1.57 = F2 ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_188_cond__case__prod__eta,axiom,
% 0.30/1.57 ! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 0.30/1.57 ( ! [X: A,Y4: B] :
% 0.30/1.57 ( ( F2 @ X @ Y4 )
% 0.30/1.57 = ( G3 @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
% 0.30/1.57 => ( ( product_case_prod @ A @ B @ C @ F2 )
% 0.30/1.57 = G3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_189_old_Oprod_Ocase,axiom,
% 0.30/1.57 ! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.30/1.57 = ( F2 @ X1 @ X22 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_190_split__cong,axiom,
% 0.30/1.57 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F2: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.30/1.57 ( ! [X: A,Y4: B] :
% 0.30/1.57 ( ( ( product_Pair @ A @ B @ X @ Y4 )
% 0.30/1.57 = Q2 )
% 0.30/1.57 => ( ( F2 @ X @ Y4 )
% 0.30/1.57 = ( G3 @ X @ Y4 ) ) )
% 0.30/1.57 => ( ( P2 = Q2 )
% 0.30/1.57 => ( ( product_case_prod @ A @ B @ C @ F2 @ P2 )
% 0.30/1.57 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_191_rev__predicate2D,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,P: A > B > $o,X4: A,Y2: B,Q: A > B > $o] :
% 0.30/1.57 ( ( P @ X4 @ Y2 )
% 0.30/1.57 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/1.57 => ( Q @ X4 @ Y2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_192_predicate2D,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X4: A,Y2: B] :
% 0.30/1.57 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/1.57 => ( ( P @ X4 @ Y2 )
% 0.30/1.57 => ( Q @ X4 @ Y2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_193_Collect__case__prod__Sigma,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 0.30/1.57 ( ( collect @ ( product_prod @ A @ B )
% 0.30/1.57 @ ( product_case_prod @ A @ B @ $o
% 0.30/1.57 @ ^ [X3: A,Y3: B] :
% 0.30/1.57 ( ( P @ X3 )
% 0.30/1.57 & ( Q @ X3 @ Y3 ) ) ) )
% 0.30/1.57 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.30/1.57 @ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_194_prod_Odisc__eq__case,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/1.57 ( product_case_prod @ A @ B @ $o
% 0.30/1.57 @ ^ [Uu: A,Uv: B] : $true
% 0.30/1.57 @ Prod ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_195_prod_Ocase__distrib,axiom,
% 0.30/1.57 ! [C: $tType,D2: $tType,B: $tType,A: $tType,H: C > D2,F2: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/1.57 ( ( H @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
% 0.30/1.57 = ( product_case_prod @ A @ B @ D2
% 0.30/1.57 @ ^ [X12: A,X23: B] : ( H @ ( F2 @ X12 @ X23 ) )
% 0.30/1.57 @ Prod ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_196_card__subset__eq,axiom,
% 0.30/1.57 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.30/1.57 ( ( finite_finite2 @ A @ B2 )
% 0.30/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ( ( ( finite_card @ A @ A3 )
% 0.30/1.57 = ( finite_card @ A @ B2 ) )
% 0.30/1.57 => ( A3 = B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_197_infinite__arbitrarily__large,axiom,
% 0.30/1.57 ! [A: $tType,A3: set @ A,N: nat] :
% 0.30/1.57 ( ~ ( finite_finite2 @ A @ A3 )
% 0.30/1.57 => ? [B8: set @ A] :
% 0.30/1.57 ( ( finite_finite2 @ A @ B8 )
% 0.30/1.57 & ( ( finite_card @ A @ B8 )
% 0.30/1.57 = N )
% 0.30/1.57 & ( ord_less_eq @ ( set @ A ) @ B8 @ A3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_198_finite__if__finite__subsets__card__bdd,axiom,
% 0.30/1.57 ! [A: $tType,F6: set @ A,C2: nat] :
% 0.30/1.57 ( ! [G7: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ G7 @ F6 )
% 0.30/1.57 => ( ( finite_finite2 @ A @ G7 )
% 0.30/1.57 => ( ord_less_eq @ nat @ ( finite_card @ A @ G7 ) @ C2 ) ) )
% 0.30/1.57 => ( ( finite_finite2 @ A @ F6 )
% 0.30/1.57 & ( ord_less_eq @ nat @ ( finite_card @ A @ F6 ) @ C2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_199_card__seteq,axiom,
% 0.30/1.57 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.30/1.57 ( ( finite_finite2 @ A @ B2 )
% 0.30/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A3 ) )
% 0.30/1.57 => ( A3 = B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_200_card__mono,axiom,
% 0.30/1.57 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.30/1.57 ( ( finite_finite2 @ A @ B2 )
% 0.30/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.30/1.57 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_201_obtain__subset__with__card__n,axiom,
% 0.30/1.57 ! [A: $tType,N: nat,S: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
% 0.30/1.57 => ~ ! [T4: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
% 0.30/1.57 => ( ( ( finite_card @ A @ T4 )
% 0.30/1.57 = N )
% 0.30/1.57 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_202_Collect__empty__eq__bot,axiom,
% 0.30/1.57 ! [A: $tType,P: A > $o] :
% 0.30/1.57 ( ( ( collect @ A @ P )
% 0.30/1.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/1.57 = ( P
% 0.30/1.57 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_203_card__Ex__subset,axiom,
% 0.30/1.57 ! [A: $tType,K: nat,M: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ M ) )
% 0.30/1.57 => ? [N2: set @ A] :
% 0.30/1.57 ( ( ord_less_eq @ ( set @ A ) @ N2 @ M )
% 0.30/1.57 & ( ( finite_card @ A @ N2 )
% 0.30/1.57 = K ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_204_mem__case__prodI2,axiom,
% 0.30/1.57 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z3: C,C3: A > B > ( set @ C )] :
% 0.30/1.57 ( ! [A5: A,B6: B] :
% 0.30/1.57 ( ( P2
% 0.30/1.57 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.30/1.57 => ( member @ C @ Z3 @ ( C3 @ A5 @ B6 ) ) )
% 0.30/1.57 => ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_205_mem__case__prodI,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),A4: B,B4: C] :
% 0.30/1.57 ( ( member @ A @ Z3 @ ( C3 @ A4 @ B4 ) )
% 0.30/1.57 => ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_206_case__prodI2_H,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X4: C] :
% 0.30/1.57 ( ! [A5: A,B6: B] :
% 0.30/1.57 ( ( ( product_Pair @ A @ B @ A5 @ B6 )
% 0.30/1.57 = P2 )
% 0.30/1.57 => ( C3 @ A5 @ B6 @ X4 ) )
% 0.30/1.57 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_207_finite__Collect__le__nat,axiom,
% 0.30/1.57 ! [K: nat] :
% 0.30/1.57 ( finite_finite2 @ nat
% 0.30/1.57 @ ( collect @ nat
% 0.30/1.57 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_208_case__prodD_H,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A4: A,B4: B,C3: C] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A4 @ B4 ) @ C3 )
% 0.30/1.57 => ( R @ A4 @ B4 @ C3 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_209_case__prodE_H,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z3: C] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z3 )
% 0.30/1.57 => ~ ! [X: A,Y4: B] :
% 0.30/1.57 ( ( P2
% 0.30/1.57 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.30/1.57 => ~ ( C3 @ X @ Y4 @ Z3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_210_mem__case__prodE,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.30/1.57 ( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 0.30/1.57 => ~ ! [X: B,Y4: C] :
% 0.30/1.57 ( ( P2
% 0.30/1.57 = ( product_Pair @ B @ C @ X @ Y4 ) )
% 0.30/1.57 => ~ ( member @ A @ Z3 @ ( C3 @ X @ Y4 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_211_finite__nat__set__iff__bounded__le,axiom,
% 0.30/1.57 ( ( finite_finite2 @ nat )
% 0.30/1.57 = ( ^ [N4: set @ nat] :
% 0.30/1.57 ? [M2: nat] :
% 0.30/1.57 ! [X3: nat] :
% 0.30/1.57 ( ( member @ nat @ X3 @ N4 )
% 0.30/1.57 => ( ord_less_eq @ nat @ X3 @ M2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_212_infinite__nat__iff__unbounded__le,axiom,
% 0.30/1.57 ! [S: set @ nat] :
% 0.30/1.57 ( ( ~ ( finite_finite2 @ nat @ S ) )
% 0.30/1.57 = ( ! [M2: nat] :
% 0.30/1.57 ? [N3: nat] :
% 0.30/1.57 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 0.30/1.57 & ( member @ nat @ N3 @ S ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_213_bounded__Max__nat,axiom,
% 0.30/1.57 ! [P: nat > $o,X4: nat,M: nat] :
% 0.30/1.57 ( ( P @ X4 )
% 0.30/1.57 => ( ! [X: nat] :
% 0.30/1.57 ( ( P @ X )
% 0.30/1.57 => ( ord_less_eq @ nat @ X @ M ) )
% 0.30/1.57 => ~ ! [M3: nat] :
% 0.30/1.57 ( ( P @ M3 )
% 0.30/1.57 => ~ ! [X2: nat] :
% 0.30/1.57 ( ( P @ X2 )
% 0.30/1.57 => ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_214_finite__less__ub,axiom,
% 0.30/1.57 ! [F2: nat > nat,U: nat] :
% 0.30/1.57 ( ! [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( F2 @ N5 ) )
% 0.30/1.57 => ( finite_finite2 @ nat
% 0.30/1.57 @ ( collect @ nat
% 0.30/1.57 @ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F2 @ N3 ) @ U ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_215_card__le__if__inj__on__rel,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,B2: set @ A,A3: set @ B,R2: B > A > $o] :
% 0.30/1.57 ( ( finite_finite2 @ A @ B2 )
% 0.30/1.57 => ( ! [A5: B] :
% 0.30/1.57 ( ( member @ B @ A5 @ A3 )
% 0.30/1.57 => ? [B9: A] :
% 0.30/1.57 ( ( member @ A @ B9 @ B2 )
% 0.30/1.57 & ( R2 @ A5 @ B9 ) ) )
% 0.30/1.57 => ( ! [A12: B,A23: B,B6: A] :
% 0.30/1.57 ( ( member @ B @ A12 @ A3 )
% 0.30/1.57 => ( ( member @ B @ A23 @ A3 )
% 0.30/1.57 => ( ( member @ A @ B6 @ B2 )
% 0.30/1.57 => ( ( R2 @ A12 @ B6 )
% 0.30/1.57 => ( ( R2 @ A23 @ B6 )
% 0.30/1.57 => ( A12 = A23 ) ) ) ) ) )
% 0.30/1.57 => ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_216_pair__pseudo__graph_Opair__pseudo__graph__subdivide,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.30/1.57 ( ( pair_p389585203_graph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( pair_p389585203_graph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_217_pair__graph_Opair__graph__subdivide,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.30/1.57 ( ( pair_pair_graph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( pair_pair_graph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_218_pair__graph_Oaxioms_I2_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_graph @ A @ G2 )
% 0.30/1.57 => ( pair_p389585203_graph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_219_pair__graph_Opair__graph,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_graph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_graph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_220_pair__pseudo__graph_Opair__pseudo__graph,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p389585203_graph @ A @ G2 )
% 0.30/1.57 => ( pair_p389585203_graph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_221_pair__pseudo__graph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p389585203_graph @ A @ G2 )
% 0.30/1.57 => ( pair_p953238171igraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_222_case__prod__app,axiom,
% 0.30/1.57 ! [A: $tType,D2: $tType,C: $tType,B: $tType] :
% 0.30/1.57 ( ( product_case_prod @ B @ C @ ( D2 > A ) )
% 0.30/1.57 = ( ^ [F3: B > C > D2 > A,X3: product_prod @ B @ C,Y3: D2] :
% 0.30/1.57 ( product_case_prod @ B @ C @ A
% 0.30/1.57 @ ^ [L: B,R4: C] : ( F3 @ L @ R4 @ Y3 )
% 0.30/1.57 @ X3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_223_pair__bidirected__digraph_Opair__bidirected__digraph__subdivide,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.30/1.57 ( ( pair_p1033978749igraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( pair_p1033978749igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_224_predicate2D__conj,axiom,
% 0.30/1.57 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X4: A,Y2: B] :
% 0.30/1.57 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/1.57 & R )
% 0.30/1.57 => ( R
% 0.30/1.57 & ( ( P @ X4 @ Y2 )
% 0.30/1.57 => ( Q @ X4 @ Y2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_225_case__prod__Pair__iden,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.30/1.57 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 0.30/1.57 = P2 ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_226_eq__subset,axiom,
% 0.30/1.57 ! [A: $tType,P: A > A > $o] :
% 0.30/1.57 ( ord_less_eq @ ( A > A > $o )
% 0.30/1.57 @ ^ [Y: A,Z2: A] : ( Y = Z2 )
% 0.30/1.57 @ ^ [A6: A,B5: A] :
% 0.30/1.57 ( ( P @ A6 @ B5 )
% 0.30/1.57 | ( A6 = B5 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_227_pair__loopfree__digraph_Opair__loopfree__digraph__subdivide,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.30/1.57 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( pair_p1282390076igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_228_pair__graph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_digraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p389585203_graph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_graph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_229_pair__bidirected__digraph_Oaxioms_I2_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p1033978749igraph @ A @ G2 )
% 0.30/1.57 => ( pair_p1282390076igraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_230_pair__loopfree__digraph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_231_pair__digraph_Opair__wf__digraph,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_digraph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_232_pair__digraph_Oaxioms_I2_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_digraph @ A @ G2 )
% 0.30/1.57 => ( pair_p1282390076igraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_233_pair__digraph_Opair__digraph,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_digraph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_digraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_234_pair__digraph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_pair_digraph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p953238171igraph @ A @ G )
% 0.30/1.57 & ( pair_p1282390076igraph @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_235_pair__digraph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p1282390076igraph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_digraph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_236_pair__digraph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_digraph @ A @ G2 )
% 0.30/1.57 => ( pair_p953238171igraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_237_pair__graph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_graph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_digraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_238_pair__loopfree__digraph_Ono__loops_H,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,U: A,V3: A] :
% 0.30/1.57 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V3 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( U != V3 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_239_pair__graph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_pair_graph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_digraph @ A @ G )
% 0.30/1.57 & ( pair_p389585203_graph @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_240_pair__loopfree__digraph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p814123999axioms @ A @ G2 )
% 0.30/1.57 => ( pair_p1282390076igraph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_241_pair__loopfree__digraph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_p1282390076igraph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G )
% 0.30/1.57 & ( pair_p814123999axioms @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_242_pair__loopfree__digraph_Oaxioms_I2_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p1282390076igraph @ A @ G2 )
% 0.30/1.57 => ( pair_p814123999axioms @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_243_pair__sym__digraph_Opair__sym__digraph__subdivide,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,E: product_prod @ A @ A,W: A] :
% 0.30/1.57 ( ( pair_p460180511igraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ E @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( ~ ( member @ A @ W @ ( pair_pair_pre_pverts @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( pair_p460180511igraph @ A @ ( pair_subdivide @ A @ G2 @ E @ W ) ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_244_internal__case__prod__def,axiom,
% 0.30/1.57 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/1.57 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.30/1.57 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_245_pair__sym__digraph_Oarcs__symmetric,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit,A4: A,B4: A] :
% 0.30/1.57 ( ( pair_p460180511igraph @ A @ G2 )
% 0.30/1.57 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) )
% 0.30/1.57 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ ( pair_pair_pre_parcs @ A @ product_unit @ G2 ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_246_pair__bidirected__digraph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p1033978749igraph @ A @ G2 )
% 0.30/1.57 => ( pair_p460180511igraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_247_pair__sym__digraph_Oaxioms_I1_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p460180511igraph @ A @ G2 )
% 0.30/1.57 => ( pair_pair_wf_digraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_248_pair__pseudo__graph_Oaxioms_I2_J,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p389585203_graph @ A @ G2 )
% 0.30/1.57 => ( pair_p460180511igraph @ A @ G2 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_249_pair__bidirected__digraph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p460180511igraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p1282390076igraph @ A @ G2 )
% 0.30/1.57 => ( pair_p1033978749igraph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_250_pair__bidirected__digraph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_p1033978749igraph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p460180511igraph @ A @ G )
% 0.30/1.57 & ( pair_p1282390076igraph @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_251_pair__pseudo__graph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_p389585203_graph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p953238171igraph @ A @ G )
% 0.30/1.57 & ( pair_p460180511igraph @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_252_pair__pseudo__graph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_p953238171igraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p460180511igraph @ A @ G2 )
% 0.30/1.57 => ( pair_p389585203_graph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_253_internal__case__prod__conv,axiom,
% 0.30/1.57 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A4: B,B4: C] :
% 0.30/1.57 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A4 @ B4 ) )
% 0.30/1.57 = ( C3 @ A4 @ B4 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_254_pair__sym__digraph_Ointro,axiom,
% 0.30/1.57 ! [A: $tType,G2: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G2 )
% 0.30/1.57 => ( ( pair_p477239746axioms @ A @ G2 )
% 0.30/1.57 => ( pair_p460180511igraph @ A @ G2 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(fact_255_pair__sym__digraph__def,axiom,
% 0.30/1.57 ! [A: $tType] :
% 0.30/1.57 ( ( pair_p460180511igraph @ A )
% 0.30/1.57 = ( ^ [G: pair_p1731315293ph_ext @ A @ product_unit] :
% 0.30/1.57 ( ( pair_pair_wf_digraph @ A @ G )
% 0.30/1.57 & ( pair_p477239746axioms @ A @ G ) ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( order_bot @ A9 )
% 0.30/1.57 => ( order_bot @ ( A7 > A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( preorder @ A9 )
% 0.30/1.57 => ( preorder @ ( A7 > A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( ( finite_finite @ A7 )
% 0.30/1.57 & ( finite_finite @ A9 ) )
% 0.30/1.57 => ( finite_finite @ ( A7 > A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( order @ A9 )
% 0.30/1.57 => ( order @ ( A7 > A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( ord @ A9 )
% 0.30/1.57 => ( ord @ ( A7 > A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( bot @ A9 )
% 0.30/1.57 => ( bot @ ( A7 > A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.30/1.57 order_bot @ nat ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.30/1.57 preorder @ nat ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/1.57 linorder @ nat ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.30/1.57 order @ nat ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.30/1.57 ord @ nat ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.30/1.57 bot @ nat ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.30/1.57 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.30/1.57 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 0.30/1.57 ! [A7: $tType] :
% 0.30/1.57 ( ( finite_finite @ A7 )
% 0.30/1.57 => ( finite_finite @ ( set @ A7 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.30/1.57 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.30/1.57 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.30/1.57 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
% 0.30/1.57 order_bot @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
% 0.30/1.57 preorder @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
% 0.30/1.57 linorder @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
% 0.30/1.57 finite_finite @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.30/1.57 order @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
% 0.30/1.57 ord @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
% 0.30/1.57 bot @ $o ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_19,axiom,
% 0.30/1.57 ! [A7: $tType,A9: $tType] :
% 0.30/1.57 ( ( ( finite_finite @ A7 )
% 0.30/1.57 & ( finite_finite @ A9 ) )
% 0.30/1.57 => ( finite_finite @ ( product_prod @ A7 @ A9 ) ) ) ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_20,axiom,
% 0.30/1.57 order_bot @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_21,axiom,
% 0.30/1.57 preorder @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_22,axiom,
% 0.30/1.57 linorder @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_23,axiom,
% 0.30/1.57 finite_finite @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder_24,axiom,
% 0.30/1.57 order @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oord_25,axiom,
% 0.30/1.57 ord @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(tcon_Product__Type_Ounit___Orderings_Obot_26,axiom,
% 0.30/1.57 bot @ product_unit ).
% 0.30/1.57
% 0.30/1.57 thf(conj_0,conjecture,
% 0.30/1.57 finite_finite2 @ ( product_prod @ a @ a ) @ ( pair_pair_pre_parcs @ a @ product_unit @ g ) ).
% 0.30/1.57 ------------------------
% 2.94/1.60 % (14976)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/2Mi)
% 2.94/1.60 % (14978)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_14783 for (2999ds/275Mi)
% 2.94/1.60 % (14979)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/18Mi)
% 2.94/1.60 % (14976)Instruction limit reached!
% 2.94/1.60 % (14976)------------------------------
% 2.94/1.60 % (14976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (14976)Termination reason: Unknown
% 2.94/1.60 % (14976)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (14976)Memory used [KB]: 1407
% 2.94/1.60 % (14976)Time elapsed: 0.003 s
% 2.94/1.60 % (14976)Instructions burned: 3 (million)
% 2.94/1.60 % (14976)------------------------------
% 2.94/1.60 % (14976)------------------------------
% 2.94/1.60 % (14974)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_14783 for (2999ds/4Mi)
% 2.94/1.60 % (14979)Instruction limit reached!
% 2.94/1.60 % (14979)------------------------------
% 2.94/1.60 % (14979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (14979)Termination reason: Unknown
% 2.94/1.60 % (14973)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/183Mi)
% 2.94/1.60 % (14979)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (14979)Memory used [KB]: 1791
% 2.94/1.60 % (14979)Time elapsed: 0.008 s
% 2.94/1.60 % (14979)Instructions burned: 19 (million)
% 2.94/1.60 % (14979)------------------------------
% 2.94/1.60 % (14979)------------------------------
% 2.94/1.60 % (14975)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_14783 for (2999ds/27Mi)
% 2.94/1.61 % (14977)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_14783 for (2999ds/2Mi)
% 2.94/1.61 % (14977)Instruction limit reached!
% 2.94/1.61 % (14977)------------------------------
% 2.94/1.61 % (14977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (14977)Termination reason: Unknown
% 2.94/1.61 % (14977)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (14977)Memory used [KB]: 1407
% 2.94/1.61 % (14977)Time elapsed: 0.004 s
% 2.94/1.61 % (14977)Instructions burned: 3 (million)
% 2.94/1.61 % (14977)------------------------------
% 2.94/1.61 % (14977)------------------------------
% 2.94/1.61 % (14974)Instruction limit reached!
% 2.94/1.61 % (14974)------------------------------
% 2.94/1.61 % (14974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (14974)Termination reason: Unknown
% 2.94/1.61 % (14974)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (14974)Memory used [KB]: 1407
% 2.94/1.61 % (14974)Time elapsed: 0.005 s
% 2.94/1.61 % (14974)Instructions burned: 5 (million)
% 2.94/1.61 % (14974)------------------------------
% 2.94/1.61 % (14974)------------------------------
% 2.94/1.61 % (14980)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/3Mi)
% 2.94/1.61 % (14980)Instruction limit reached!
% 2.94/1.61 % (14980)------------------------------
% 2.94/1.61 % (14980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (14980)Termination reason: Unknown
% 2.94/1.61 % (14980)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (14980)Memory used [KB]: 1407
% 2.94/1.61 % (14980)Time elapsed: 0.002 s
% 2.94/1.61 % (14980)Instructions burned: 3 (million)
% 2.94/1.61 % (14980)------------------------------
% 2.94/1.61 % (14980)------------------------------
% 2.94/1.62 % (14981)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_14783 for (2999ds/37Mi)
% 2.94/1.62 % (14978)Refutation not found, incomplete strategy
% 2.94/1.62 % (14978)------------------------------
% 2.94/1.62 % (14978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (14978)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.62
% 2.94/1.62
% 2.94/1.62 % (14978)Memory used [KB]: 6652
% 2.94/1.62 % (14978)Time elapsed: 0.021 s
% 2.94/1.62 % (14978)Instructions burned: 55 (million)
% 2.94/1.62 % (14978)------------------------------
% 2.94/1.62 % (14978)------------------------------
% 2.94/1.62 % (14984)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_14783 for (2999ds/1041Mi)
% 2.94/1.62 % (14975)Instruction limit reached!
% 2.94/1.62 % (14975)------------------------------
% 2.94/1.62 % (14975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (14975)Termination reason: Unknown
% 2.94/1.62 % (14975)Termination phase: Property scanning
% 2.94/1.62
% 2.94/1.62 % (14975)Memory used [KB]: 1918
% 2.94/1.62 % (14975)Time elapsed: 0.019 s
% 2.94/1.62 % (14975)Instructions burned: 27 (million)
% 2.94/1.62 % (14975)------------------------------
% 2.94/1.62 % (14975)------------------------------
% 2.94/1.62 % (14982)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_14783 for (2999ds/15Mi)
% 2.94/1.62 % (14983)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_14783 for (2999ds/3Mi)
% 2.94/1.63 % (14983)Instruction limit reached!
% 2.94/1.63 % (14983)------------------------------
% 2.94/1.63 % (14983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (14983)Termination reason: Unknown
% 2.94/1.63 % (14983)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (14983)Memory used [KB]: 1407
% 2.94/1.63 % (14983)Time elapsed: 0.004 s
% 2.94/1.63 % (14983)Instructions burned: 3 (million)
% 2.94/1.63 % (14983)------------------------------
% 2.94/1.63 % (14983)------------------------------
% 2.94/1.63 % (14981)Instruction limit reached!
% 2.94/1.63 % (14981)------------------------------
% 2.94/1.63 % (14981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (14981)Termination reason: Unknown
% 2.94/1.63 % (14981)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (14981)Memory used [KB]: 2302
% 2.94/1.63 % (14981)Time elapsed: 0.014 s
% 2.94/1.63 % (14981)Instructions burned: 40 (million)
% 2.94/1.63 % (14981)------------------------------
% 2.94/1.63 % (14981)------------------------------
% 2.94/1.63 % (14985)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/7Mi)
% 2.94/1.63 % (14985)Instruction limit reached!
% 2.94/1.63 % (14985)------------------------------
% 2.94/1.63 % (14985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (14985)Termination reason: Unknown
% 2.94/1.63 % (14985)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (14985)Memory used [KB]: 1535
% 2.94/1.63 % (14985)Time elapsed: 0.005 s
% 2.94/1.63 % (14985)Instructions burned: 9 (million)
% 2.94/1.63 % (14985)------------------------------
% 2.94/1.63 % (14985)------------------------------
% 2.94/1.63 % (14982)Instruction limit reached!
% 2.94/1.63 % (14982)------------------------------
% 2.94/1.63 % (14982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (14982)Termination reason: Unknown
% 2.94/1.63 % (14982)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (14982)Memory used [KB]: 1663
% 2.94/1.63 % (14982)Time elapsed: 0.011 s
% 2.94/1.63 % (14982)Instructions burned: 15 (million)
% 2.94/1.63 % (14982)------------------------------
% 2.94/1.63 % (14982)------------------------------
% 2.94/1.64 % (14988)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_14783 for (2999ds/3Mi)
% 2.94/1.64 % (14986)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/16Mi)
% 2.94/1.64 % (14988)Instruction limit reached!
% 2.94/1.64 % (14988)------------------------------
% 2.94/1.64 % (14988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (14988)Termination reason: Unknown
% 2.94/1.64 % (14988)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (14988)Memory used [KB]: 1407
% 2.94/1.64 % (14988)Time elapsed: 0.003 s
% 2.94/1.64 % (14988)Instructions burned: 4 (million)
% 2.94/1.64 % (14988)------------------------------
% 2.94/1.64 % (14988)------------------------------
% 2.94/1.64 % (14989)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/7Mi)
% 2.94/1.64 % (14987)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_14783 for (2999ds/3Mi)
% 2.94/1.65 % (14989)Instruction limit reached!
% 2.94/1.65 % (14989)------------------------------
% 2.94/1.65 % (14989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (14989)Termination reason: Unknown
% 2.94/1.65 % (14989)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (14989)Memory used [KB]: 1535
% 2.94/1.65 % (14989)Time elapsed: 0.005 s
% 2.94/1.65 % (14989)Instructions burned: 9 (million)
% 2.94/1.65 % (14989)------------------------------
% 2.94/1.65 % (14989)------------------------------
% 2.94/1.65 % (14987)Instruction limit reached!
% 2.94/1.65 % (14987)------------------------------
% 2.94/1.65 % (14987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (14987)Termination reason: Unknown
% 2.94/1.65 % (14987)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (14987)Memory used [KB]: 1407
% 2.94/1.65 % (14987)Time elapsed: 0.004 s
% 2.94/1.65 % (14987)Instructions burned: 3 (million)
% 2.94/1.65 % (14987)------------------------------
% 2.94/1.65 % (14987)------------------------------
% 2.94/1.65 % (14986)Instruction limit reached!
% 2.94/1.65 % (14986)------------------------------
% 2.94/1.65 % (14986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (14986)Termination reason: Unknown
% 2.94/1.65 % (14986)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (14986)Memory used [KB]: 1791
% 2.94/1.65 % (14986)Time elapsed: 0.011 s
% 2.94/1.65 % (14986)Instructions burned: 16 (million)
% 2.94/1.65 % (14986)------------------------------
% 2.94/1.65 % (14986)------------------------------
% 2.94/1.65 % (14990)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/3Mi)
% 2.94/1.65 % (14991)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/4Mi)
% 2.94/1.65 % (14990)Instruction limit reached!
% 2.94/1.65 % (14990)------------------------------
% 2.94/1.65 % (14990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (14990)Termination reason: Unknown
% 2.94/1.65 % (14990)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (14990)Memory used [KB]: 1407
% 2.94/1.65 % (14990)Time elapsed: 0.003 s
% 2.94/1.65 % (14990)Instructions burned: 3 (million)
% 2.94/1.65 % (14990)------------------------------
% 2.94/1.65 % (14990)------------------------------
% 2.94/1.65 % (14991)Instruction limit reached!
% 2.94/1.65 % (14991)------------------------------
% 2.94/1.65 % (14991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (14991)Termination reason: Unknown
% 2.94/1.65 % (14991)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (14991)Memory used [KB]: 1407
% 2.94/1.65 % (14991)Time elapsed: 0.003 s
% 2.94/1.65 % (14991)Instructions burned: 4 (million)
% 2.94/1.65 % (14991)------------------------------
% 2.94/1.65 % (14991)------------------------------
% 2.94/1.66 % (14992)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/18Mi)
% 2.94/1.66 % (14994)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/6Mi)
% 2.94/1.66 % (14996)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/21Mi)
% 2.94/1.66 % (14994)Instruction limit reached!
% 2.94/1.66 % (14994)------------------------------
% 2.94/1.66 % (14994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (14994)Termination reason: Unknown
% 2.94/1.66 % (14994)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (14994)Memory used [KB]: 1535
% 2.94/1.66 % (14994)Time elapsed: 0.004 s
% 2.94/1.66 % (14994)Instructions burned: 8 (million)
% 2.94/1.66 % (14994)------------------------------
% 2.94/1.66 % (14994)------------------------------
% 2.94/1.66 % (14992)Instruction limit reached!
% 2.94/1.66 % (14992)------------------------------
% 2.94/1.66 % (14992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (14992)Termination reason: Unknown
% 2.94/1.66 % (14992)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (14992)Memory used [KB]: 1791
% 2.94/1.66 % (14992)Time elapsed: 0.008 s
% 2.94/1.66 % (14992)Instructions burned: 18 (million)
% 2.94/1.66 % (14992)------------------------------
% 2.94/1.66 % (14992)------------------------------
% 2.94/1.66 % (14993)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/710Mi)
% 2.94/1.67 % (14995)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/902Mi)
% 2.94/1.67 % (14996)Instruction limit reached!
% 2.94/1.67 % (14996)------------------------------
% 2.94/1.67 % (14996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (14996)Termination reason: Unknown
% 2.94/1.67 % (14996)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (14996)Memory used [KB]: 1918
% 2.94/1.67 % (14996)Time elapsed: 0.010 s
% 2.94/1.67 % (14996)Instructions burned: 23 (million)
% 2.94/1.67 % (14996)------------------------------
% 2.94/1.67 % (14996)------------------------------
% 2.94/1.67 % (14997)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/5Mi)
% 2.94/1.67 % (14998)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/6Mi)
% 2.94/1.67 % (14997)Instruction limit reached!
% 2.94/1.67 % (14997)------------------------------
% 2.94/1.67 % (14997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (14997)Termination reason: Unknown
% 2.94/1.67 % (14997)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (14997)Memory used [KB]: 1535
% 2.94/1.67 % (14997)Time elapsed: 0.004 s
% 2.94/1.67 % (14997)Instructions burned: 7 (million)
% 2.94/1.67 % (14997)------------------------------
% 2.94/1.67 % (14997)------------------------------
% 2.94/1.67 % (14998)Instruction limit reached!
% 2.94/1.67 % (14998)------------------------------
% 2.94/1.67 % (14998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (14998)Termination reason: Unknown
% 2.94/1.67 % (14998)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (14998)Memory used [KB]: 1535
% 2.94/1.67 % (14998)Time elapsed: 0.004 s
% 2.94/1.67 % (14998)Instructions burned: 7 (million)
% 2.94/1.67 % (14998)------------------------------
% 2.94/1.67 % (14998)------------------------------
% 2.94/1.68 % (14999)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/377Mi)
% 2.94/1.68 % (15000)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/779Mi)
% 2.94/1.68 % (15001)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_14783 for (2999ds/19Mi)
% 2.94/1.69 % (15001)Instruction limit reached!
% 2.94/1.69 % (15001)------------------------------
% 2.94/1.69 % (15001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.69 % (15001)Termination reason: Unknown
% 2.94/1.69 % (15001)Termination phase: shuffling
% 2.94/1.69
% 2.94/1.69 % (15001)Memory used [KB]: 1791
% 2.94/1.69 % (15001)Time elapsed: 0.008 s
% 2.94/1.69 % (15001)Instructions burned: 20 (million)
% 2.94/1.69 % (15001)------------------------------
% 2.94/1.69 % (15001)------------------------------
% 2.94/1.70 % (15002)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_14783 for (2998ds/879Mi)
% 2.94/1.70 % (14993)First to succeed.
% 2.94/1.71 % (14993)Refutation found. Thanks to Tanya!
% 2.94/1.71 % SZS status Theorem for DTF2THF_14783
% 2.94/1.71 % SZS output start Proof for DTF2THF_14783
% 2.94/1.71 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.94/1.71 thf(type_def_6, type, set: ($tType) > $tType).
% 2.94/1.71 thf(type_def_8, type, nat: $tType).
% 2.94/1.71 thf(type_def_9, type, product_unit: $tType).
% 2.94/1.71 thf(type_def_10, type, pair_p1731315293ph_ext: ($tType * $tType) > $tType).
% 2.94/1.71 thf(type_def_11, type, a: $tType).
% 2.94/1.71 thf(func_def_0, type, pair_p1731315293ph_ext: ($tType * $tType) > $tType).
% 2.94/1.71 thf(func_def_1, type, product_unit: $tType).
% 2.94/1.71 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 2.94/1.71 thf(func_def_3, type, set: ($tType) > $tType).
% 2.94/1.71 thf(func_def_4, type, nat: $tType).
% 2.94/1.71 thf(func_def_5, type, a: $tType).
% 2.94/1.71 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_10, type, finite_finite: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_13, type, order_bot: !>[X0: $tType]:($o)).
% 2.94/1.71 thf(func_def_14, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.94/1.71 thf(func_def_15, type, finite_Fpow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.94/1.71 thf(func_def_16, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 2.94/1.71 thf(func_def_17, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.94/1.71 thf(func_def_18, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.94/1.71 thf(func_def_19, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.94/1.71 thf(func_def_20, type, pair_p1033978749igraph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_21, type, pair_pair_digraph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_22, type, pair_p953238171igraph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_23, type, pair_p1373739070axioms: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_24, type, pair_pair_graph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_25, type, pair_p1282390076igraph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_26, type, pair_p814123999axioms: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_27, type, pair_pair_pre_more: !>[X0: $tType, X1: $tType]:(pair_p1731315293ph_ext @ X0 @ X1 > X1)).
% 2.94/1.71 thf(func_def_28, type, pair_p1108258481update: !>[X0: $tType, X1: $tType]:((X0 > X0) > pair_p1731315293ph_ext @ X1 @ X0 > pair_p1731315293ph_ext @ X1 @ X0)).
% 2.94/1.71 thf(func_def_29, type, pair_p1152105390ph_ext: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > X1 > pair_p1731315293ph_ext @ X0 @ X1)).
% 2.94/1.71 thf(func_def_30, type, pair_pair_pre_parcs: !>[X0: $tType, X1: $tType]:(pair_p1731315293ph_ext @ X0 @ X1 > set @ product_prod @ X0 @ X0)).
% 2.94/1.71 thf(func_def_31, type, pair_pair_pre_pverts: !>[X0: $tType, X1: $tType]:(pair_p1731315293ph_ext @ X0 @ X1 > set @ X0)).
% 2.94/1.71 thf(func_def_32, type, pair_p389585203_graph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_33, type, pair_p460180511igraph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_34, type, pair_p477239746axioms: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_35, type, pair_pair_wf_digraph: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > $o)).
% 2.94/1.71 thf(func_def_36, type, pair_subdivide: !>[X0: $tType]:(pair_p1731315293ph_ext @ X0 @ product_unit > product_prod @ X0 @ X0 > X0 > pair_p1731315293ph_ext @ X0 @ product_unit)).
% 2.94/1.71 thf(func_def_37, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.94/1.71 thf(func_def_38, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 2.94/1.71 thf(func_def_39, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.94/1.71 thf(func_def_40, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.94/1.71 thf(func_def_41, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.94/1.71 thf(func_def_42, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 2.94/1.71 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.94/1.71 thf(func_def_44, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.94/1.71 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.94/1.71 thf(func_def_46, type, g: pair_p1731315293ph_ext @ a @ product_unit).
% 2.94/1.71 thf(func_def_47, type, n: nat).
% 2.94/1.71 thf(func_def_64, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.94/1.71 thf(func_def_65, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 2.94/1.71 thf(func_def_66, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 2.94/1.71 thf(func_def_68, type, ph4: !>[X0: $tType]:(X0)).
% 2.94/1.71 thf(f959,plain,(
% 2.94/1.71 $false),
% 2.94/1.71 inference(subsumption_resolution,[status(thm)],[f958,f833])).
% 2.94/1.71 thf(f833,plain,(
% 2.94/1.71 ($true = (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g)))))),
% 2.94/1.71 inference(cnf_transformation,[status(thm)],[f450])).
% 2.94/1.71 thf(f450,plain,(
% 2.94/1.71 ($true = (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g)))))),
% 2.94/1.71 inference(fool_elimination,[status(thm)],[f449])).
% 2.94/1.71 thf(f449,plain,(
% 2.94/1.71 (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[X0 : a] : (pair_pair_pre_pverts @ a @ product_unit @ g))))),
% 2.94/1.71 inference(rectify,[status(thm)],[f2])).
% 2.94/1.71 thf(f2,axiom,(
% 2.94/1.71 (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[X6 : a] : (pair_pair_pre_pverts @ a @ product_unit @ g))))),
% 2.94/1.71 file('/export/starexec/sandbox2/tmp/tmp.XcU2giCvCK/DTF2THF_14783.p',fact_1__092_060open_062finite_A_Ipverts_AG_A_092_060times_062_Apverts_AG_J_092_060close_062)).
% 2.94/1.71 thf(f958,plain,(
% 2.94/1.71 ($true != (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g)))))),
% 2.94/1.71 inference(subsumption_resolution,[status(thm)],[f946,f826])).
% 2.94/1.71 thf(f826,plain,(
% 2.94/1.71 ((finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g)) != $true)),
% 2.94/1.71 inference(cnf_transformation,[status(thm)],[f809])).
% 2.94/1.71 thf(f809,plain,(
% 2.94/1.71 ((finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g)) != $true)),
% 2.94/1.71 inference(flattening,[status(thm)],[f744])).
% 2.94/1.71 thf(f744,plain,(
% 2.94/1.71 ~((finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g)) = $true)),
% 2.94/1.71 inference(fool_elimination,[status(thm)],[f743])).
% 2.94/1.71 thf(f743,plain,(
% 2.94/1.71 ~(finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g))),
% 2.94/1.71 inference(rectify,[status(thm)],[f291])).
% 2.94/1.71 thf(f291,negated_conjecture,(
% 2.94/1.71 ~(finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g))),
% 2.94/1.71 inference(negated_conjecture,[status(cth)],[f290])).
% 2.94/1.71 thf(f290,conjecture,(
% 2.94/1.71 (finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g))),
% 2.94/1.71 file('/export/starexec/sandbox2/tmp/tmp.XcU2giCvCK/DTF2THF_14783.p',conj_0)).
% 2.94/1.71 thf(f946,plain,(
% 2.94/1.71 ((finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g)) = $true) | ($true != (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g)))))),
% 2.94/1.71 inference(trivial_inequality_removal,[status(thm)],[f934])).
% 2.94/1.71 thf(f934,plain,(
% 2.94/1.71 ($true != (finite_finite2 @ product_prod @ a @ a @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g))))) | ((finite_finite2 @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g)) = $true) | ($true != $true)),
% 2.94/1.71 inference(superposition,[status(thm)],[f829,f830])).
% 2.94/1.71 thf(f830,plain,(
% 2.94/1.71 ($true = (ord_less_eq @ set @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g) @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g)))))),
% 2.94/1.71 inference(cnf_transformation,[status(thm)],[f512])).
% 2.94/1.71 thf(f512,plain,(
% 2.94/1.71 ($true = (ord_less_eq @ set @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g) @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[Y0 : a]: (pair_pair_pre_pverts @ a @ product_unit @ g)))))),
% 2.94/1.71 inference(fool_elimination,[status(thm)],[f511])).
% 2.94/1.71 thf(f511,plain,(
% 2.94/1.71 (ord_less_eq @ set @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g) @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[X0 : a] : (pair_pair_pre_pverts @ a @ product_unit @ g))))),
% 2.94/1.71 inference(rectify,[status(thm)],[f3])).
% 2.94/1.71 thf(f3,axiom,(
% 2.94/1.71 (ord_less_eq @ set @ product_prod @ a @ a @ (pair_pair_pre_parcs @ a @ product_unit @ g) @ (product_Sigma @ a @ a @ (pair_pair_pre_pverts @ a @ product_unit @ g) @ (^[X6 : a] : (pair_pair_pre_pverts @ a @ product_unit @ g))))),
% 2.94/1.71 file('/export/starexec/sandbox2/tmp/tmp.XcU2giCvCK/DTF2THF_14783.p',fact_2__092_060open_062parcs_AG_A_092_060subseteq_062_Apverts_AG_A_092_060times_062_Apverts_AG_092_060close_062)).
% 2.94/1.71 thf(f829,plain,(
% 2.94/1.71 ( ! [X0 : $tType,X2 : set @ X0,X1 : set @ X0] : (($true != (ord_less_eq @ set @ X0 @ X1 @ X2)) | ($true != (finite_finite2 @ X0 @ X2)) | ($true = (finite_finite2 @ X0 @ X1))) )),
% 2.94/1.71 inference(cnf_transformation,[status(thm)],[f816])).
% 2.94/1.71 thf(f816,plain,(
% 2.94/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : (($true = (finite_finite2 @ X0 @ X1)) | ($true != (ord_less_eq @ set @ X0 @ X1 @ X2)) | ($true != (finite_finite2 @ X0 @ X2)))),
% 2.94/1.71 inference(flattening,[status(thm)],[f815])).
% 2.94/1.71 thf(f815,plain,(
% 2.94/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((($true != (finite_finite2 @ X0 @ X2)) | ($true = (finite_finite2 @ X0 @ X1))) | ($true != (ord_less_eq @ set @ X0 @ X1 @ X2)))),
% 2.94/1.71 inference(ennf_transformation,[status(thm)],[f810])).
% 2.94/1.71 thf(f810,plain,(
% 2.94/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : (($true = (ord_less_eq @ set @ X0 @ X1 @ X2)) => (($true != (finite_finite2 @ X0 @ X1)) => ($true != (finite_finite2 @ X0 @ X2))))),
% 2.94/1.71 inference(flattening,[status(thm)],[f350])).
% 2.94/1.71 thf(f350,plain,(
% 2.94/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : (($true = (ord_less_eq @ set @ X0 @ X1 @ X2)) => (~($true = (finite_finite2 @ X0 @ X1)) => ~($true = (finite_finite2 @ X0 @ X2))))),
% 2.94/1.71 inference(fool_elimination,[status(thm)],[f349])).
% 2.94/1.71 thf(f349,plain,(
% 2.94/1.71 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((ord_less_eq @ set @ X0 @ X1 @ X2) => (~(finite_finite2 @ X0 @ X1) => ~(finite_finite2 @ X0 @ X2)))),
% 2.94/1.71 inference(rectify,[status(thm)],[f10])).
% 2.94/1.71 thf(f10,axiom,(
% 2.94/1.71 ! [X0 : $tType,X16 : set @ X0,X17 : set @ X0] : ((ord_less_eq @ set @ X0 @ X16 @ X17) => (~(finite_finite2 @ X0 @ X16) => ~(finite_finite2 @ X0 @ X17)))),
% 2.94/1.71 file('/export/starexec/sandbox2/tmp/tmp.XcU2giCvCK/DTF2THF_14783.p',fact_9_infinite__super)).
% 2.94/1.71 % SZS output end Proof for DTF2THF_14783
% 2.94/1.71 % (14993)------------------------------
% 2.94/1.71 % (14993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.71 % (14993)Termination reason: Refutation
% 2.94/1.71
% 2.94/1.71 % (14993)Memory used [KB]: 6908
% 2.94/1.71 % (14993)Time elapsed: 0.042 s
% 2.94/1.71 % (14993)Instructions burned: 106 (million)
% 2.94/1.71 % (14993)------------------------------
% 2.94/1.71 % (14993)------------------------------
% 2.94/1.71 % (14972)Success in time 0.125 s
%------------------------------------------------------------------------------