%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP138^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n028.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 08:55:51 AM UTC 2026
% Result : Theorem 2.99s 1.70s
% Output : Refutation 2.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP138^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n028.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:27:26 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP138^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Paraconsistency problem prob_525__3267638_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Paraconsistency/prob_525__3267638_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.00 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 337 ( 141 unt; 48 typ; 0 def)
% 0.30/0.46 % Number of atoms : 835 ( 272 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 16 ( 2 avg)
% 0.30/0.46 % Number of connectives : 4144 ( 79 ~; 1 |; 69 &;3603 @)
% 0.30/0.46 % ( 0 <=>; 392 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.30/0.46 % Number of types : 5 ( 4 usr)
% 0.30/0.46 % Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 47 ( 44 usr; 4 con; 0-6 aty)
% 0.30/0.46 % Number of variables : 1245 ( 109 ^;1073 !; 30 ?;1245 :)
% 0.30/0.46 % ( 33 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:21:25.150
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (7)
% 0.30/0.46 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 0.30/0.46 paraco415392788lle_tv: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 0.30/0.46 paraco414474393lle_fm: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.46 product_prod: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_String_Ochar,type,
% 0.30/0.46 char: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_List_Olist,type,
% 0.30/0.46 list: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Nat_Onat,type,
% 0.30/0.46 nat: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (41)
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.46 ord:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.30/0.46 top:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.46 no_bot:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Enum_Ofinite__lattice,type,
% 0.30/0.46 finite_lattice:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.30/0.46 order_top:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Complete__Lattices_OSup,type,
% 0.30/0.46 complete_Sup:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.30/0.46 comple187826305attice:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.30/0.46 comple1035589618norder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 0.30/0.46 condit378418413attice:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.30/0.46 condit1037483654norder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.30/0.46 complete_Sup_Sup:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_OFpow,type,
% 0.30/0.46 finite_Fpow:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Oswap,type,
% 0.30/0.46 swap:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.46 ord_less:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.30/0.46 order_Greatest:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.30/0.46 order_strict_mono:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.30/0.46 top_top:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Odomain,type,
% 0.30/0.46 paraco506338565domain: ( set @ nat ) > ( set @ paraco415392788lle_tv ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 0.30/0.46 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 0.30/0.46 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 0.30/0.46 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 0.30/0.46 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 0.30/0.46 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 0.30/0.46 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 0.30/0.46 paraco251304083_Truth: paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 0.30/0.46 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 0.30/0.46 paraco152590079rec_tv:
% 0.30/0.46 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 0.30/0.46 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
% 0.30/0.46 paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.46 product_Pair:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Relation_OPowp,type,
% 0.30/0.46 powp:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OPow,type,
% 0.30/0.46 pow:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Oimage,type,
% 0.30/0.46 image:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_p,type,
% 0.30/0.46 p: paraco414474393lle_fm ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (256)
% 0.30/0.46 thf(fact_0_universal__domain,axiom,
% 0.30/0.46 ( ( paraco506338565domain
% 0.30/0.46 @ ( collect @ nat
% 0.30/0.46 @ ^ [N: nat] : $true ) )
% 0.30/0.46 = ( collect @ paraco415392788lle_tv
% 0.30/0.46 @ ^ [X: paraco415392788lle_tv] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % universal_domain
% 0.30/0.46 thf(fact_1_tv_Oinject_I1_J,axiom,
% 0.30/0.46 ! [X1: $o,Y1: $o] :
% 0.30/0.46 ( ( ( paraco2040174112le_Det @ X1 )
% 0.30/0.46 = ( paraco2040174112le_Det @ Y1 ) )
% 0.30/0.46 = ( X1 = Y1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % tv.inject(1)
% 0.30/0.46 thf(fact_2_valid__in__def,axiom,
% 0.30/0.46 ( paraco2086025920lid_in
% 0.30/0.46 = ( ^ [U: set @ nat,P: paraco414474393lle_fm] :
% 0.30/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ paraco415392788lle_tv ) @ ( image @ ( list @ char ) @ paraco415392788lle_tv @ I @ ( top_top @ ( set @ ( list @ char ) ) ) ) @ ( paraco506338565domain @ U ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I @ P )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % valid_in_def
% 0.30/0.46 thf(fact_3_image__ident,axiom,
% 0.30/0.46 ! [A: $tType,Y: set @ A] :
% 0.30/0.46 ( ( image @ A @ A
% 0.30/0.46 @ ^ [X: A] : X
% 0.30/0.46 @ Y )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 % image_ident
% 0.30/0.46 thf(fact_4_UNIV__I,axiom,
% 0.30/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_I
% 0.30/0.46 thf(fact_5_iso__tuple__UNIV__I,axiom,
% 0.30/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % iso_tuple_UNIV_I
% 0.30/0.46 thf(fact_6_top__apply,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType] :
% 0.30/0.46 ( ( top @ C )
% 0.30/0.46 => ( ( top_top @ ( D > C ) )
% 0.30/0.46 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_apply
% 0.30/0.46 thf(fact_7_subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( member @ A @ X3 @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetI
% 0.30/0.46 thf(fact_8_subset__antisym,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_antisym
% 0.30/0.46 thf(fact_9_image__eqI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B,A2: set @ B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F @ X2 ) )
% 0.30/0.46 => ( ( member @ B @ X2 @ A2 )
% 0.30/0.46 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_eqI
% 0.30/0.46 thf(fact_10_order__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_11_rangeE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( B3
% 0.30/0.46 != ( F @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rangeE
% 0.30/0.46 thf(fact_12_range__composition,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
% 0.30/0.46 ( ( image @ B @ A
% 0.30/0.46 @ ^ [X: B] : ( F @ ( G @ X ) )
% 0.30/0.46 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_composition
% 0.30/0.46 thf(fact_13_image__Collect__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P2: A > $o,F: A > B,B2: set @ B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P2 @ X3 )
% 0.30/0.46 => ( member @ B @ ( F @ X3 ) @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_Collect_subsetI
% 0.30/0.46 thf(fact_14_top__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_set_def
% 0.30/0.46 thf(fact_15_less__eq__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A3: set @ A,B4: set @ A] :
% 0.30/0.46 ( ord_less_eq @ ( A > $o )
% 0.30/0.46 @ ^ [X: A] : ( member @ A @ X @ A3 )
% 0.30/0.46 @ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_set_def
% 0.30/0.46 thf(fact_16_dual__order_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [B3: A,A4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( A4 = B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.antisym
% 0.30/0.46 thf(fact_17_dual__order_Oeq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.30/0.46 = ( ^ [A5: A,B5: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.30/0.46 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.eq_iff
% 0.30/0.46 thf(fact_18_dual__order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [B3: A,A4: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/0.46 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.trans
% 0.30/0.46 thf(fact_19_linorder__wlog,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [P2: A > A > $o,A4: A,B3: A] :
% 0.30/0.46 ( ! [A6: A,B6: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.30/0.46 => ( P2 @ A6 @ B6 ) )
% 0.30/0.46 => ( ! [A6: A,B6: A] :
% 0.30/0.46 ( ( P2 @ B6 @ A6 )
% 0.30/0.46 => ( P2 @ A6 @ B6 ) )
% 0.30/0.46 => ( P2 @ A4 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linorder_wlog
% 0.30/0.46 thf(fact_20_dual__order_Orefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.refl
% 0.30/0.46 thf(fact_21_order__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X2: A,Y3: A,Z2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.46 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_trans
% 0.30/0.46 thf(fact_22_order__class_Oorder_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [A4: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.30/0.46 => ( A4 = B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.antisym
% 0.30/0.46 thf(fact_23_ord__le__eq__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [A4: A,B3: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( B3 = C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_trans
% 0.30/0.46 thf(fact_24_ord__eq__le__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [A4: A,B3: A,C2: A] :
% 0.30/0.46 ( ( A4 = B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_trans
% 0.30/0.46 thf(fact_25_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.30/0.46 = ( ^ [A5: A,B5: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.30/0.46 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.eq_iff
% 0.30/0.46 thf(fact_26_antisym__conv,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [Y3: A,X2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ Y3 @ X2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.46 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_27_le__cases3,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [X2: A,Y3: A,Z2: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ X2 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.30/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases3
% 0.30/0.46 thf(fact_28_order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [A4: A,B3: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_29_le__cases,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [X2: A,Y3: A] :
% 0.30/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases
% 0.30/0.46 thf(fact_30_eq__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X2: A,Y3: A] :
% 0.30/0.46 ( ( X2 = Y3 )
% 0.30/0.46 => ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_31_linear,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ! [X2: A,Y3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.46 | ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linear
% 0.30/0.46 thf(fact_32_antisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [X2: A,Y3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y3 @ X2 )
% 0.30/0.46 => ( X2 = Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_33_eq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.30/0.46 = ( ^ [X: A,Y4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.46 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_34_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B )
% 0.30/0.46 & ( ord @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,F: A > B,C2: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ( F @ B3 )
% 0.30/0.46 = C2 )
% 0.30/0.46 => ( ! [X3: A,Y5: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_35_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B )
% 0.30/0.46 & ( ord @ A ) )
% 0.30/0.46 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 0.30/0.46 ( ( A4
% 0.30/0.46 = ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/0.46 => ( ! [X3: B,Y5: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_36_order__subst2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( ( order @ C )
% 0.30/0.46 & ( order @ A ) )
% 0.30/0.46 => ! [A4: A,B3: A,F: A > C,C2: C] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.30/0.46 => ( ! [X3: A,Y5: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst2
% 0.30/0.46 thf(fact_37_order__subst1,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( order @ B )
% 0.30/0.46 & ( order @ A ) )
% 0.30/0.46 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/0.46 => ( ! [X3: B,Y5: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst1
% 0.30/0.46 thf(fact_38_le__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.46 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_fun_def
% 0.30/0.46 thf(fact_39_le__funI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ! [F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funI
% 0.30/0.46 thf(fact_40_le__funE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funE
% 0.30/0.46 thf(fact_41_le__funD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B )
% 0.30/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funD
% 0.30/0.46 thf(fact_42_rev__image__eqI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X2: A,A2: set @ A,B3: B,F: A > B] :
% 0.30/0.46 ( ( member @ A @ X2 @ A2 )
% 0.30/0.46 => ( ( B3
% 0.30/0.46 = ( F @ X2 ) )
% 0.30/0.46 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_image_eqI
% 0.30/0.46 thf(fact_43_ball__imageD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 => ( P2 @ X3 ) )
% 0.30/0.46 => ! [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ A2 )
% 0.30/0.46 => ( P2 @ ( F @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_imageD
% 0.30/0.46 thf(fact_44_image__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 0.30/0.46 ( ( M = N2 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ N2 )
% 0.30/0.46 => ( ( F @ X3 )
% 0.30/0.46 = ( G @ X3 ) ) )
% 0.30/0.46 => ( ( image @ A @ B @ F @ M )
% 0.30/0.46 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_cong
% 0.30/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A4: A,P2: A > $o] :
% 0.30/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P2 ) )
% 0.30/0.46 = ( P2 @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_47_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P2 @ X3 )
% 0.30/0.46 = ( Q @ X3 ) )
% 0.30/0.46 => ( ( collect @ A @ P2 )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_48_ext,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( F @ X3 )
% 0.30/0.46 = ( G @ X3 ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % ext
% 0.30/0.46 thf(fact_49_bex__imageD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 0.30/0.46 ( ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 & ( P2 @ X4 ) )
% 0.30/0.46 => ? [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A2 )
% 0.30/0.46 & ( P2 @ ( F @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex_imageD
% 0.30/0.46 thf(fact_50_image__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
% 0.30/0.46 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 = ( ? [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 & ( Z2
% 0.30/0.46 = ( F @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_iff
% 0.30/0.46 thf(fact_51_imageI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X2: A,A2: set @ A,F: A > B] :
% 0.30/0.46 ( ( member @ A @ X2 @ A2 )
% 0.30/0.46 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % imageI
% 0.30/0.46 thf(fact_52_Collect__mono__iff,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.30/0.46 = ( ! [X: A] :
% 0.30/0.46 ( ( P2 @ X )
% 0.30/0.46 => ( Q @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono_iff
% 0.30/0.46 thf(fact_53_set__eq__subset,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 0.30/0.46 = ( ^ [A3: set @ A,B4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_eq_subset
% 0.30/0.46 thf(fact_54_subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_trans
% 0.30/0.46 thf(fact_55_Collect__mono,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P2 @ X3 )
% 0.30/0.46 => ( Q @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono
% 0.30/0.46 thf(fact_56_subset__refl,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % subset_refl
% 0.30/0.46 thf(fact_57_subset__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A3: set @ A,B4: set @ A] :
% 0.30/0.46 ! [T: A] :
% 0.30/0.46 ( ( member @ A @ T @ A3 )
% 0.30/0.46 => ( member @ A @ T @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff
% 0.30/0.46 thf(fact_58_equalityD2,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD2
% 0.30/0.46 thf(fact_59_equalityD1,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD1
% 0.30/0.46 thf(fact_60_subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A3: set @ A,B4: set @ A] :
% 0.30/0.46 ! [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ A3 )
% 0.30/0.46 => ( member @ A @ X @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_eq
% 0.30/0.46 thf(fact_61_equalityE,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityE
% 0.30/0.46 thf(fact_62_subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.30/0.46 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetD
% 0.30/0.46 thf(fact_63_in__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ( member @ A @ X2 @ A2 )
% 0.30/0.46 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_mono
% 0.30/0.46 thf(fact_64_UNIV__witness,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_witness
% 0.30/0.46 thf(fact_65_UNIV__eq__I,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_eq_I
% 0.30/0.46 thf(fact_66_Compr__image__eq,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 & ( P2 @ X ) ) )
% 0.30/0.46 = ( image @ B @ A @ F
% 0.30/0.46 @ ( collect @ B
% 0.30/0.46 @ ^ [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 & ( P2 @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compr_image_eq
% 0.30/0.46 thf(fact_67_image__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A2: set @ C] :
% 0.30/0.46 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A2 ) )
% 0.30/0.46 = ( image @ C @ A
% 0.30/0.46 @ ^ [X: C] : ( F @ ( G @ X ) )
% 0.30/0.46 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_image
% 0.30/0.46 thf(fact_68_imageE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F @ X3 ) )
% 0.30/0.46 => ~ ( member @ B @ X3 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % imageE
% 0.30/0.46 thf(fact_69_Collect__restrict,axiom,
% 0.30/0.46 ! [A: $tType,X5: set @ A,P2: A > $o] :
% 0.30/0.46 ( ord_less_eq @ ( set @ A )
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ X5 )
% 0.30/0.46 & ( P2 @ X ) ) )
% 0.30/0.46 @ X5 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_restrict
% 0.30/0.46 thf(fact_70_prop__restrict,axiom,
% 0.30/0.46 ! [A: $tType,X2: A,Z3: set @ A,X5: set @ A,P2: A > $o] :
% 0.30/0.46 ( ( member @ A @ X2 @ Z3 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ X5 )
% 0.30/0.46 & ( P2 @ X ) ) ) )
% 0.30/0.46 => ( P2 @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prop_restrict
% 0.30/0.46 thf(fact_71_Collect__subset,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,P2: A > $o] :
% 0.30/0.46 ( ord_less_eq @ ( set @ A )
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 & ( P2 @ X ) ) )
% 0.30/0.46 @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_subset
% 0.30/0.46 thf(fact_72_UNIV__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A
% 0.30/0.46 @ ^ [X: A] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_def
% 0.30/0.46 thf(fact_73_top_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 0.30/0.46 => ( A4
% 0.30/0.46 = ( top_top @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top.extremum_uniqueI
% 0.30/0.46 thf(fact_74_top_Oextremum__unique,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 0.30/0.46 = ( A4
% 0.30/0.46 = ( top_top @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top.extremum_unique
% 0.30/0.46 thf(fact_75_top__greatest,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A )
% 0.30/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_greatest
% 0.30/0.46 thf(fact_76_subset__image__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 = ( ? [AA: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 0.30/0.46 & ( B2
% 0.30/0.46 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_image_iff
% 0.30/0.46 thf(fact_77_image__subset__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.30/0.46 = ( ! [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 => ( member @ A @ ( F @ X ) @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_subset_iff
% 0.30/0.46 thf(fact_78_subset__imageE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 => ~ ! [C4: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ C4 @ A2 )
% 0.30/0.46 => ( B2
% 0.30/0.46 != ( image @ B @ A @ F @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_imageE
% 0.30/0.46 thf(fact_79_image__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( member @ B @ ( F @ X3 ) @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_subsetI
% 0.30/0.46 thf(fact_80_image__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_mono
% 0.30/0.46 thf(fact_81_range__eqI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F @ X2 ) )
% 0.30/0.46 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_eqI
% 0.30/0.46 thf(fact_82_rangeI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rangeI
% 0.30/0.46 thf(fact_83_subset__UNIV,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_UNIV
% 0.30/0.46 thf(fact_84_surj__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [Y4: A] :
% 0.30/0.46 ? [X: B] :
% 0.30/0.46 ( Y4
% 0.30/0.46 = ( F @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_def
% 0.30/0.46 thf(fact_85_surjI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( G @ ( F @ X3 ) )
% 0.30/0.46 = X3 )
% 0.30/0.46 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjI
% 0.30/0.46 thf(fact_86_surjE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( F @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjE
% 0.30/0.46 thf(fact_87_surjD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ? [X3: B] :
% 0.30/0.46 ( Y3
% 0.30/0.46 = ( F @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjD
% 0.30/0.46 thf(fact_88_all__subset__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: ( set @ A ) > $o] :
% 0.30/0.46 ( ( ! [B4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 => ( P2 @ B4 ) ) )
% 0.30/0.46 = ( ! [B4: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ B4 @ A2 )
% 0.30/0.46 => ( P2 @ ( image @ B @ A @ F @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % all_subset_image
% 0.30/0.46 thf(fact_89_valid__def,axiom,
% 0.30/0.46 ( paraco769098683_valid
% 0.30/0.46 = ( ^ [P: paraco414474393lle_fm] :
% 0.30/0.46 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.46 ( ( paraco876059933e_eval @ I @ P )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % valid_def
% 0.30/0.46 thf(fact_90_pred__subset__eq,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > $o )
% 0.30/0.46 @ ^ [X: A] : ( member @ A @ X @ R )
% 0.30/0.46 @ ^ [X: A] : ( member @ A @ X @ S ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.30/0.46
% 0.30/0.46 % pred_subset_eq
% 0.30/0.46 thf(fact_91_subset__Collect__iff,axiom,
% 0.30/0.46 ! [A: $tType,B2: set @ A,A2: set @ A,P2: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 & ( P2 @ X ) ) ) )
% 0.30/0.46 = ( ! [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ B2 )
% 0.30/0.46 => ( P2 @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Collect_iff
% 0.30/0.46 thf(fact_92_subset__CollectI,axiom,
% 0.30/0.46 ! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P2: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ B2 )
% 0.30/0.46 => ( ( Q @ X3 )
% 0.30/0.46 => ( P2 @ X3 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A )
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ B2 )
% 0.30/0.46 & ( Q @ X ) ) )
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 & ( P2 @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_CollectI
% 0.30/0.46 thf(fact_93_conj__subset__def,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,P2: A > $o,Q: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2
% 0.30/0.46 @ ( collect @ A
% 0.30/0.46 @ ^ [X: A] :
% 0.30/0.46 ( ( P2 @ X )
% 0.30/0.46 & ( Q @ X ) ) ) )
% 0.30/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P2 ) )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conj_subset_def
% 0.30/0.46 thf(fact_94_Sup_OSUP__identity__eq,axiom,
% 0.30/0.46 ! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
% 0.30/0.46 ( ( Sup
% 0.30/0.46 @ ( image @ A @ A
% 0.30/0.46 @ ^ [X: A] : X
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( Sup @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup.SUP_identity_eq
% 0.30/0.46 thf(fact_95_predicate1I,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P2 @ X3 )
% 0.30/0.46 => ( Q @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% 0.30/0.46
% 0.30/0.46 % predicate1I
% 0.30/0.46 thf(fact_96_top1I,axiom,
% 0.30/0.46 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.30/0.46
% 0.30/0.46 % top1I
% 0.30/0.46 thf(fact_97_predicate1D,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,Q: A > $o,X2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 0.30/0.46 => ( ( P2 @ X2 )
% 0.30/0.46 => ( Q @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % predicate1D
% 0.30/0.46 thf(fact_98_rev__predicate1D,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,X2: A,Q: A > $o] :
% 0.30/0.46 ( ( P2 @ X2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 0.30/0.46 => ( Q @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_predicate1D
% 0.30/0.46 thf(fact_99_Inf_OINF__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D2: B > A,Inf: ( set @ A ) > A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ B2 )
% 0.30/0.46 => ( ( C3 @ X3 )
% 0.30/0.46 = ( D2 @ X3 ) ) )
% 0.30/0.46 => ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
% 0.30/0.46 = ( Inf @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inf.INF_cong
% 0.30/0.46 thf(fact_100_Sup_OSUP__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D2: B > A,Sup: ( set @ A ) > A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ B2 )
% 0.30/0.46 => ( ( C3 @ X3 )
% 0.30/0.46 = ( D2 @ X3 ) ) )
% 0.30/0.46 => ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
% 0.30/0.46 = ( Sup @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup.SUP_cong
% 0.30/0.46 thf(fact_101_top__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( A > $o ) )
% 0.30/0.46 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_empty_eq
% 0.30/0.46 thf(fact_102_Inf_OINF__identity__eq,axiom,
% 0.30/0.46 ! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
% 0.30/0.46 ( ( Inf
% 0.30/0.46 @ ( image @ A @ A
% 0.30/0.46 @ ^ [X: A] : X
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( Inf @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inf.INF_identity_eq
% 0.30/0.46 thf(fact_103_top__conj_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,P2: $o,X2: A] :
% 0.30/0.46 ( ( P2
% 0.30/0.46 & ( top_top @ ( A > $o ) @ X2 ) )
% 0.30/0.46 = P2 ) ).
% 0.30/0.46
% 0.30/0.46 % top_conj(2)
% 0.30/0.46 thf(fact_104_top__conj_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X2: A,P2: $o] :
% 0.30/0.46 ( ( ( top_top @ ( A > $o ) @ X2 )
% 0.30/0.46 & P2 )
% 0.30/0.46 = P2 ) ).
% 0.30/0.46
% 0.30/0.46 % top_conj(1)
% 0.30/0.46 thf(fact_105_eval_Osimps_I2_J,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.46 ( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval.simps(2)
% 0.30/0.46 thf(fact_106_image__Fpow__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A2 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_Fpow_mono
% 0.30/0.46 thf(fact_107_eval_Osimps_I5_J,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ Q2 ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco876059933e_eval @ I2 @ Q2 ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval.simps(5)
% 0.30/0.46 thf(fact_108_fm_Oinject_I4_J,axiom,
% 0.30/0.46 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.30/0.46 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 0.30/0.46 = ( ( X51 = Y51 )
% 0.30/0.46 & ( X52 = Y52 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.inject(4)
% 0.30/0.46 thf(fact_109_fm_Odistinct_I15_J,axiom,
% 0.30/0.46 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.46 ( paraco251304083_Truth
% 0.30/0.46 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(15)
% 0.30/0.46 thf(fact_110_Fpow__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Fpow_mono
% 0.30/0.46 thf(fact_111_eval__equality__simplify,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco2040174112le_Det
% 0.30/0.46 @ ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ Q2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval_equality_simplify
% 0.30/0.46 thf(fact_112_image__Pow__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_Pow_mono
% 0.30/0.46 thf(fact_113_eval_Osimps_I4_J,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ Q2 ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ P3 ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco876059933e_eval @ I2 @ Q2 ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ Q2 ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ P3 ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval.simps(4)
% 0.30/0.46 thf(fact_114_Powp__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: A > $o,B2: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > $o ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A2 ) @ ( powp @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Powp_mono
% 0.30/0.46 thf(fact_115_Greatest__equality,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [P2: A > $o,X2: A] :
% 0.30/0.46 ( ( P2 @ X2 )
% 0.30/0.46 => ( ! [Y5: A] :
% 0.30/0.46 ( ( P2 @ Y5 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 0.30/0.46 => ( ( order_Greatest @ A @ P2 )
% 0.30/0.46 = X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Greatest_equality
% 0.30/0.46 thf(fact_116_GreatestI2__order,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [P2: A > $o,X2: A,Q: A > $o] :
% 0.30/0.46 ( ( P2 @ X2 )
% 0.30/0.46 => ( ! [Y5: A] :
% 0.30/0.46 ( ( P2 @ Y5 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( P2 @ X3 )
% 0.30/0.46 => ( ! [Y6: A] :
% 0.30/0.46 ( ( P2 @ Y6 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y6 @ X3 ) )
% 0.30/0.46 => ( Q @ X3 ) ) )
% 0.30/0.46 => ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % GreatestI2_order
% 0.30/0.46 thf(fact_117_fm_Oinject_I3_J,axiom,
% 0.30/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.30/0.46 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 0.30/0.46 = ( ( X41 = Y41 )
% 0.30/0.46 & ( X42 = Y42 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.inject(3)
% 0.30/0.46 thf(fact_118_PowI,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % PowI
% 0.30/0.46 thf(fact_119_Pow__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pow_iff
% 0.30/0.46 thf(fact_120_Pow__UNIV,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pow_UNIV
% 0.30/0.46 thf(fact_121_Pow__top,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pow_top
% 0.30/0.46 thf(fact_122_Powp__Pow__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( powp @ A
% 0.30/0.46 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.30/0.46 = ( ^ [X: set @ A] : ( member @ ( set @ A ) @ X @ ( pow @ A @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Powp_Pow_eq
% 0.30/0.46 thf(fact_123_PowD,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % PowD
% 0.30/0.46 thf(fact_124_Cantors__paradox,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ~ ? [F3: A > ( set @ A )] :
% 0.30/0.46 ( ( image @ A @ ( set @ A ) @ F3 @ A2 )
% 0.30/0.46 = ( pow @ A @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Cantors_paradox
% 0.30/0.46 thf(fact_125_Pow__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( pow @ A )
% 0.30/0.46 = ( ^ [A3: set @ A] :
% 0.30/0.46 ( collect @ ( set @ A )
% 0.30/0.46 @ ^ [B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pow_def
% 0.30/0.46 thf(fact_126_fm_Odistinct_I25_J,axiom,
% 0.30/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.30/0.46 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(25)
% 0.30/0.46 thf(fact_127_fm_Odistinct_I13_J,axiom,
% 0.30/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.46 ( paraco251304083_Truth
% 0.30/0.46 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(13)
% 0.30/0.46 thf(fact_128_Pow__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pow_mono
% 0.30/0.46 thf(fact_129_image__Pow__surj,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ A2 )
% 0.30/0.46 = B2 )
% 0.30/0.46 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
% 0.30/0.46 = ( pow @ A @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_Pow_surj
% 0.30/0.46 thf(fact_130_conjunction,axiom,
% 0.30/0.46 ! [P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 0.30/0.46 = ( ( paraco769098683_valid @ P3 )
% 0.30/0.46 & ( paraco769098683_valid @ Q2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conjunction
% 0.30/0.46 thf(fact_131_Fpow__subset__Pow,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( pow @ A @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Fpow_subset_Pow
% 0.30/0.46 thf(fact_132_tv_Osimps_I7_J,axiom,
% 0.30/0.46 ! [A: $tType,F1: $o > A,F22: nat > A,X1: $o] :
% 0.30/0.46 ( ( paraco152590079rec_tv @ A @ F1 @ F22 @ ( paraco2040174112le_Det @ X1 ) )
% 0.30/0.46 = ( F1 @ X1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % tv.simps(7)
% 0.30/0.46 thf(fact_133_strict__mono__inv,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( linorder @ B )
% 0.30/0.46 & ( linorder @ A ) )
% 0.30/0.46 => ! [F: A > B,G: B > A] :
% 0.30/0.46 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.46 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ ( set @ B ) ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( G @ ( F @ X3 ) )
% 0.30/0.46 = X3 )
% 0.30/0.46 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % strict_mono_inv
% 0.30/0.46 thf(fact_134_eval__negation,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ P3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval_negation
% 0.30/0.46 thf(fact_135_surj__swap__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: B,B3: B,F: B > A] :
% 0.30/0.46 ( ( ( image @ B @ A @ ( swap @ B @ A @ A4 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_swap_iff
% 0.30/0.46 thf(fact_136_swap__nilpotent,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: A,B3: A,F: A > B] :
% 0.30/0.46 ( ( swap @ A @ B @ A4 @ B3 @ ( swap @ A @ B @ A4 @ B3 @ F ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % swap_nilpotent
% 0.30/0.46 thf(fact_137_swap__self,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: A,F: A > B] :
% 0.30/0.46 ( ( swap @ A @ B @ A4 @ A4 @ F )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % swap_self
% 0.30/0.46 thf(fact_138_swap__apply_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A4: B,B3: B,F: B > A] :
% 0.30/0.46 ( ( swap @ B @ A @ A4 @ B3 @ F @ A4 )
% 0.30/0.46 = ( F @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_apply(1)
% 0.30/0.46 thf(fact_139_swap__apply_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A4: B,B3: B,F: B > A] :
% 0.30/0.46 ( ( swap @ B @ A @ A4 @ B3 @ F @ B3 )
% 0.30/0.46 = ( F @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_apply(2)
% 0.30/0.46 thf(fact_140_swap__apply_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C2: B,A4: B,B3: B,F: B > A] :
% 0.30/0.46 ( ( C2 != A4 )
% 0.30/0.46 => ( ( C2 != B3 )
% 0.30/0.46 => ( ( swap @ B @ A @ A4 @ B3 @ F @ C2 )
% 0.30/0.46 = ( F @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_apply(3)
% 0.30/0.46 thf(fact_141_fm_Oinject_I2_J,axiom,
% 0.30/0.46 ! [X32: paraco414474393lle_fm,Y32: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( paraco329115265le_Neg @ X32 )
% 0.30/0.46 = ( paraco329115265le_Neg @ Y32 ) )
% 0.30/0.46 = ( X32 = Y32 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.inject(2)
% 0.30/0.46 thf(fact_142_swap__image__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: A,F: A > B] :
% 0.30/0.46 ( ( member @ A @ A4 @ A2 )
% 0.30/0.46 => ( ( member @ A @ B3 @ A2 )
% 0.30/0.46 => ( ( image @ A @ B @ ( swap @ A @ B @ A4 @ B3 @ F ) @ A2 )
% 0.30/0.46 = ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_image_eq
% 0.30/0.46 thf(fact_143_swap__commute,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( swap @ A @ B )
% 0.30/0.46 = ( ^ [A5: A,B5: A] : ( swap @ A @ B @ B5 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_commute
% 0.30/0.46 thf(fact_144_swap__triple,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: A,C2: A,B3: A,F: A > B] :
% 0.30/0.46 ( ( A4 != C2 )
% 0.30/0.46 => ( ( B3 != C2 )
% 0.30/0.46 => ( ( swap @ A @ B @ A4 @ B3 @ ( swap @ A @ B @ B3 @ C2 @ ( swap @ A @ B @ A4 @ B3 @ F ) ) )
% 0.30/0.46 = ( swap @ A @ B @ A4 @ C2 @ F ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_triple
% 0.30/0.46 thf(fact_145_strict__mono__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( linorder @ A )
% 0.30/0.46 & ( order @ B ) )
% 0.30/0.46 => ! [F: A > B,X2: A,Y3: A] :
% 0.30/0.46 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.46 => ( ( ( F @ X2 )
% 0.30/0.46 = ( F @ Y3 ) )
% 0.30/0.46 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % strict_mono_eq
% 0.30/0.46 thf(fact_146_double__negation,axiom,
% 0.30/0.46 ( paraco876059933e_eval
% 0.30/0.46 = ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % double_negation
% 0.30/0.46 thf(fact_147_strict__mono__less__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( linorder @ A )
% 0.30/0.46 & ( order @ B ) )
% 0.30/0.46 => ! [F: A > B,X2: A,Y3: A] :
% 0.30/0.46 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.46 => ( ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % strict_mono_less_eq
% 0.30/0.46 thf(fact_148_fm_Odistinct_I21_J,axiom,
% 0.30/0.46 ! [X32: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco329115265le_Neg @ X32 )
% 0.30/0.46 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(21)
% 0.30/0.46 thf(fact_149_fm_Odistinct_I19_J,axiom,
% 0.30/0.46 ! [X32: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco329115265le_Neg @ X32 )
% 0.30/0.46 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(19)
% 0.30/0.46 thf(fact_150_fm_Odistinct_I11_J,axiom,
% 0.30/0.46 ! [X32: paraco414474393lle_fm] :
% 0.30/0.46 ( paraco251304083_Truth
% 0.30/0.46 != ( paraco329115265le_Neg @ X32 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(11)
% 0.30/0.46 thf(fact_151_surj__imp__surj__swap,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A,A4: B,B3: B] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ( ( image @ B @ A @ ( swap @ B @ A @ A4 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_imp_surj_swap
% 0.30/0.46 thf(fact_152_eval__equality,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ Q2 ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco876059933e_eval @ I2 @ Q2 ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ Q2 ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ P3 ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $true ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q2 ) ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.46 => ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) ) ) )
% 0.30/0.46 & ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 0.30/0.46 != ( paraco2040174112le_Det @ $false ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 0.30/0.46 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval_equality
% 0.30/0.46 thf(fact_153_fm_Oexhaust,axiom,
% 0.30/0.46 ! [Y3: paraco414474393lle_fm] :
% 0.30/0.46 ( ! [X12: list @ char] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( paraco27778325le_Pro @ X12 ) )
% 0.30/0.46 => ( ( Y3 != paraco251304083_Truth )
% 0.30/0.46 => ( ! [X33: paraco414474393lle_fm] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( paraco329115265le_Neg @ X33 ) )
% 0.30/0.46 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 0.30/0.46 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 0.30/0.46 => ~ ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 0.30/0.46 ( Y3
% 0.30/0.46 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.exhaust
% 0.30/0.46 thf(fact_154_fm_Oinduct,axiom,
% 0.30/0.46 ! [P2: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 0.30/0.46 ( ! [X3: list @ char] : ( P2 @ ( paraco27778325le_Pro @ X3 ) )
% 0.30/0.46 => ( ( P2 @ paraco251304083_Truth )
% 0.30/0.46 => ( ! [X3: paraco414474393lle_fm] :
% 0.30/0.46 ( ( P2 @ X3 )
% 0.30/0.46 => ( P2 @ ( paraco329115265le_Neg @ X3 ) ) )
% 0.30/0.46 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.30/0.46 ( ( P2 @ X1a )
% 0.30/0.46 => ( ( P2 @ X22 )
% 0.30/0.46 => ( P2 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 0.30/0.46 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.30/0.46 ( ( P2 @ X1a )
% 0.30/0.46 => ( ( P2 @ X22 )
% 0.30/0.46 => ( P2 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 0.30/0.46 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 0.30/0.46 ( ( P2 @ X1a )
% 0.30/0.46 => ( ( P2 @ X22 )
% 0.30/0.46 => ( P2 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 0.30/0.46 => ( P2 @ Fm ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.induct
% 0.30/0.46 thf(fact_155_SUP__subset__mono,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ B,B2: set @ B,F: B > A,G: B > A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_subset_mono
% 0.30/0.46 thf(fact_156_UN__ball__bex__simps_I4_J,axiom,
% 0.30/0.46 ! [F4: $tType,E: $tType,B2: E > ( set @ F4 ),A2: set @ E,P2: F4 > $o] :
% 0.30/0.46 ( ( ? [X: F4] :
% 0.30/0.46 ( ( member @ F4 @ X @ ( complete_Sup_Sup @ ( set @ F4 ) @ ( image @ E @ ( set @ F4 ) @ B2 @ A2 ) ) )
% 0.30/0.46 & ( P2 @ X ) ) )
% 0.30/0.46 = ( ? [X: E] :
% 0.30/0.46 ( ( member @ E @ X @ A2 )
% 0.30/0.46 & ? [Y4: F4] :
% 0.30/0.46 ( ( member @ F4 @ Y4 @ ( B2 @ X ) )
% 0.30/0.46 & ( P2 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_ball_bex_simps(4)
% 0.30/0.46 thf(fact_157_UN__ball__bex__simps_I2_J,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P2: C > $o] :
% 0.30/0.46 ( ( ! [X: C] :
% 0.30/0.46 ( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
% 0.30/0.46 => ( P2 @ X ) ) )
% 0.30/0.46 = ( ! [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 => ! [Y4: C] :
% 0.30/0.46 ( ( member @ C @ Y4 @ ( B2 @ X ) )
% 0.30/0.46 => ( P2 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_ball_bex_simps(2)
% 0.30/0.46 thf(fact_158_bex__UN,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
% 0.30/0.46 ( ( ? [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 0.30/0.46 & ( P2 @ X ) ) )
% 0.30/0.46 = ( ? [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 & ? [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 0.30/0.46 & ( P2 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex_UN
% 0.30/0.46 thf(fact_159_ball__UN,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
% 0.30/0.46 ( ( ! [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 0.30/0.46 => ( P2 @ X ) ) )
% 0.30/0.46 = ( ! [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 => ! [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 0.30/0.46 => ( P2 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_UN
% 0.30/0.46 thf(fact_160_Union__Pow__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Union_Pow_eq
% 0.30/0.46 thf(fact_161_fm_Oinject_I5_J,axiom,
% 0.30/0.46 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 0.30/0.46 ( ( ( paraco1628874225le_Eql @ X612 @ X622 )
% 0.30/0.46 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 0.30/0.46 = ( ( X612 = Y61 )
% 0.30/0.46 & ( X622 = Y62 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.inject(5)
% 0.30/0.46 thf(fact_162_fm_Oinject_I1_J,axiom,
% 0.30/0.46 ! [X1: list @ char,Y1: list @ char] :
% 0.30/0.46 ( ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.46 = ( paraco27778325le_Pro @ Y1 ) )
% 0.30/0.46 = ( X1 = Y1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.inject(1)
% 0.30/0.46 thf(fact_163_Sup__apply,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( complete_Sup @ B )
% 0.30/0.46 => ( ( complete_Sup_Sup @ ( A > B ) )
% 0.30/0.46 = ( ^ [A3: set @ ( A > B ),X: A] :
% 0.30/0.46 ( complete_Sup_Sup @ B
% 0.30/0.46 @ ( image @ ( A > B ) @ B
% 0.30/0.46 @ ^ [F2: A > B] : ( F2 @ X )
% 0.30/0.46 @ A3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_apply
% 0.30/0.46 thf(fact_164_UN__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 0.30/0.46 = ( ? [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 & ( member @ A @ B3 @ ( B2 @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_iff
% 0.30/0.46 thf(fact_165_UN__I,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ A @ A4 @ A2 )
% 0.30/0.46 => ( ( member @ B @ B3 @ ( B2 @ A4 ) )
% 0.30/0.46 => ( member @ B @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_I
% 0.30/0.46 thf(fact_166_SUP__apply,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.46 ( ( complete_Sup @ A )
% 0.30/0.46 => ! [F: C > B > A,A2: set @ C,X2: B] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F @ A2 ) @ X2 )
% 0.30/0.46 = ( complete_Sup_Sup @ A
% 0.30/0.46 @ ( image @ C @ A
% 0.30/0.46 @ ^ [Y4: C] : ( F @ Y4 @ X2 )
% 0.30/0.46 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_apply
% 0.30/0.46 thf(fact_167_SUP__identity__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( complete_Sup @ A )
% 0.30/0.46 => ! [A2: set @ A] :
% 0.30/0.46 ( ( complete_Sup_Sup @ A
% 0.30/0.46 @ ( image @ A @ A
% 0.30/0.46 @ ^ [X: A] : X
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_identity_eq
% 0.30/0.46 thf(fact_168_Sup__UNIV,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_UNIV
% 0.30/0.46 thf(fact_169_SUP__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( complete_Sup @ A )
% 0.30/0.46 => ! [A2: set @ B,B2: set @ B,C3: B > A,D2: B > A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ B2 )
% 0.30/0.46 => ( ( C3 @ X3 )
% 0.30/0.46 = ( D2 @ X3 ) ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C3 @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_cong
% 0.30/0.46 thf(fact_170_SUP__UNION,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [F: B > A,G: C > ( set @ B ),A2: set @ C] :
% 0.30/0.46 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G @ A2 ) ) ) )
% 0.30/0.46 = ( complete_Sup_Sup @ A
% 0.30/0.46 @ ( image @ C @ A
% 0.30/0.46 @ ^ [Y4: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G @ Y4 ) ) )
% 0.30/0.46 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_UNION
% 0.30/0.46 thf(fact_171_Sup__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( complete_Sup @ B )
% 0.30/0.46 => ( ( complete_Sup_Sup @ ( A > B ) )
% 0.30/0.46 = ( ^ [A3: set @ ( A > B ),X: A] :
% 0.30/0.46 ( complete_Sup_Sup @ B
% 0.30/0.46 @ ( image @ ( A > B ) @ B
% 0.30/0.46 @ ^ [F2: A > B] : ( F2 @ X )
% 0.30/0.46 @ A3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_fun_def
% 0.30/0.46 thf(fact_172_SUP__UN__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > $o )
% 0.30/0.46 @ ( image @ B @ ( A > $o )
% 0.30/0.46 @ ^ [I: B,X: A] : ( member @ A @ X @ ( R2 @ I ) )
% 0.30/0.46 @ S ) )
% 0.30/0.46 = ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_UN_eq
% 0.30/0.46 thf(fact_173_SUP__commute,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [F: B > C > A,B2: set @ C,A2: set @ B] :
% 0.30/0.46 ( ( complete_Sup_Sup @ A
% 0.30/0.46 @ ( image @ B @ A
% 0.30/0.46 @ ^ [I: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F @ I ) @ B2 ) )
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ A
% 0.30/0.46 @ ( image @ C @ A
% 0.30/0.46 @ ^ [J: C] :
% 0.30/0.46 ( complete_Sup_Sup @ A
% 0.30/0.46 @ ( image @ B @ A
% 0.30/0.46 @ ^ [I: B] : ( F @ I @ J )
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_commute
% 0.30/0.46 thf(fact_174_fm_Odistinct_I9_J,axiom,
% 0.30/0.46 ! [X1: list @ char,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.46 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(9)
% 0.30/0.46 thf(fact_175_Union__least,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
% 0.30/0.46 ( ! [X6: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ X6 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ X6 @ C3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Union_least
% 0.30/0.46 thf(fact_176_Union__upper,axiom,
% 0.30/0.46 ! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ B2 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Union_upper
% 0.30/0.46 thf(fact_177_Union__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 0.30/0.46 ( ! [X3: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ X3 @ A2 )
% 0.30/0.46 => ? [Y6: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ Y6 @ B2 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ X3 @ Y6 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Union_subsetI
% 0.30/0.46 thf(fact_178_Sup__upper2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [U2: A,A2: set @ A,V: A] :
% 0.30/0.46 ( ( member @ A @ U2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ V @ U2 )
% 0.30/0.46 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_upper2
% 0.30/0.46 thf(fact_179_Sup__le__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
% 0.30/0.46 = ( ! [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_le_iff
% 0.30/0.46 thf(fact_180_Sup__upper,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [X2: A,A2: set @ A] :
% 0.30/0.46 ( ( member @ A @ X2 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_upper
% 0.30/0.46 thf(fact_181_Sup__least,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ A,Z2: A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_least
% 0.30/0.46 thf(fact_182_Sup__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ! [A6: A] :
% 0.30/0.46 ( ( member @ A @ A6 @ A2 )
% 0.30/0.46 => ? [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ B2 )
% 0.30/0.46 & ( ord_less_eq @ A @ A6 @ X4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_mono
% 0.30/0.46 thf(fact_183_Sup__eqI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ A,X2: A] :
% 0.30/0.46 ( ! [Y5: A] :
% 0.30/0.46 ( ( member @ A @ Y5 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 0.30/0.46 => ( ! [Y5: A] :
% 0.30/0.46 ( ! [Z4: A] :
% 0.30/0.46 ( ( member @ A @ Z4 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ Z4 @ Y5 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ A2 )
% 0.30/0.46 = X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_eqI
% 0.30/0.46 thf(fact_184_UN__UN__flatten,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,C3: B > ( set @ A ),B2: C > ( set @ B ),A2: set @ C] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) ) )
% 0.30/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.30/0.46 @ ( image @ C @ ( set @ A )
% 0.30/0.46 @ ^ [Y4: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( B2 @ Y4 ) ) )
% 0.30/0.46 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_UN_flatten
% 0.30/0.46 thf(fact_185_UN__E,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A2 )
% 0.30/0.46 => ~ ( member @ A @ B3 @ ( B2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_E
% 0.30/0.46 thf(fact_186_UN__extend__simps_I9_J,axiom,
% 0.30/0.46 ! [S2: $tType,R3: $tType,Q3: $tType,C3: R3 > ( set @ S2 ),B2: Q3 > ( set @ R3 ),A2: set @ Q3] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ S2 )
% 0.30/0.46 @ ( image @ Q3 @ ( set @ S2 )
% 0.30/0.46 @ ^ [X: Q3] : ( complete_Sup_Sup @ ( set @ S2 ) @ ( image @ R3 @ ( set @ S2 ) @ C3 @ ( B2 @ X ) ) )
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ ( set @ S2 ) @ ( image @ R3 @ ( set @ S2 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R3 ) @ ( image @ Q3 @ ( set @ R3 ) @ B2 @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_extend_simps(9)
% 0.30/0.46 thf(fact_187_UN__Pow__subset,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
% 0.30/0.46 ( ord_less_eq @ ( set @ ( set @ A ) )
% 0.30/0.46 @ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
% 0.30/0.46 @ ( image @ B @ ( set @ ( set @ A ) )
% 0.30/0.46 @ ^ [X: B] : ( pow @ A @ ( B2 @ X ) )
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_Pow_subset
% 0.30/0.46 thf(fact_188_SUP__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
% 0.30/0.46 ( ! [I3: B] :
% 0.30/0.46 ( ( member @ B @ I3 @ A2 )
% 0.30/0.46 => ? [X4: C] :
% 0.30/0.46 ( ( member @ C @ X4 @ B2 )
% 0.30/0.46 & ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
% 0.30/0.46 => ( ! [J2: C] :
% 0.30/0.46 ( ( member @ C @ J2 @ B2 )
% 0.30/0.46 => ? [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ A2 )
% 0.30/0.46 & ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_eq
% 0.30/0.46 thf(fact_189_Sup__subset__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_subset_mono
% 0.30/0.46 thf(fact_190_eval_Osimps_I1_J,axiom,
% 0.30/0.46 ! [I2: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
% 0.30/0.46 ( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S3 ) )
% 0.30/0.46 = ( I2 @ S3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval.simps(1)
% 0.30/0.46 thf(fact_191_fm_Odistinct_I23_J,axiom,
% 0.30/0.46 ! [X32: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco329115265le_Neg @ X32 )
% 0.30/0.46 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(23)
% 0.30/0.46 thf(fact_192_fm_Odistinct_I3_J,axiom,
% 0.30/0.46 ! [X1: list @ char,X32: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.46 != ( paraco329115265le_Neg @ X32 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(3)
% 0.30/0.46 thf(fact_193_fm_Odistinct_I27_J,axiom,
% 0.30/0.46 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 0.30/0.46 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(27)
% 0.30/0.46 thf(fact_194_fm_Odistinct_I29_J,axiom,
% 0.30/0.46 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 0.30/0.46 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(29)
% 0.30/0.46 thf(fact_195_Union__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Union_mono
% 0.30/0.46 thf(fact_196_fm_Odistinct_I5_J,axiom,
% 0.30/0.46 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.46 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(5)
% 0.30/0.46 thf(fact_197_fm_Odistinct_I7_J,axiom,
% 0.30/0.46 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 0.30/0.46 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.46 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(7)
% 0.30/0.46 thf(fact_198_fm_Odistinct_I17_J,axiom,
% 0.30/0.46 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 0.30/0.46 ( paraco251304083_Truth
% 0.30/0.46 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(17)
% 0.30/0.46 thf(fact_199_SUP__eqI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ B,F: B > A,X2: A] :
% 0.30/0.46 ( ! [I3: B] :
% 0.30/0.46 ( ( member @ B @ I3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ I3 ) @ X2 ) )
% 0.30/0.46 => ( ! [Y5: A] :
% 0.30/0.46 ( ! [I4: B] :
% 0.30/0.46 ( ( member @ B @ I4 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ I4 ) @ Y5 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 = X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_eqI
% 0.30/0.46 thf(fact_200_SUP__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
% 0.30/0.46 ( ! [N3: B] :
% 0.30/0.46 ( ( member @ B @ N3 @ A2 )
% 0.30/0.46 => ? [X4: C] :
% 0.30/0.46 ( ( member @ C @ X4 @ B2 )
% 0.30/0.46 & ( ord_less_eq @ A @ ( F @ N3 ) @ ( G @ X4 ) ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_mono
% 0.30/0.46 thf(fact_201_SUP__least,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [A2: set @ B,F: B > A,U2: A] :
% 0.30/0.46 ( ! [I3: B] :
% 0.30/0.46 ( ( member @ B @ I3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ I3 ) @ U2 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_least
% 0.30/0.46 thf(fact_202_SUP__mono_H,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [F: B > A,G: B > A,A2: set @ B] :
% 0.30/0.46 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_mono'
% 0.30/0.46 thf(fact_203_SUP__upper,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [I2: B,A2: set @ B,F: B > A] :
% 0.30/0.46 ( ( member @ B @ I2 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ I2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_upper
% 0.30/0.46 thf(fact_204_SUP__le__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [F: B > A,A2: set @ B,U2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U2 )
% 0.30/0.46 = ( ! [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X ) @ U2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_le_iff
% 0.30/0.46 thf(fact_205_SUP__upper2,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [I2: B,A2: set @ B,U2: A,F: B > A] :
% 0.30/0.46 ( ( member @ B @ I2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ U2 @ ( F @ I2 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_upper2
% 0.30/0.46 thf(fact_206_fm_Odistinct_I1_J,axiom,
% 0.30/0.46 ! [X1: list @ char] :
% 0.30/0.46 ( ( paraco27778325le_Pro @ X1 )
% 0.30/0.46 != paraco251304083_Truth ) ).
% 0.30/0.46
% 0.30/0.46 % fm.distinct(1)
% 0.30/0.46 thf(fact_207_subset__Pow__Union,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Pow_Union
% 0.30/0.46 thf(fact_208_UN__extend__simps_I10_J,axiom,
% 0.30/0.46 ! [V2: $tType,U3: $tType,T2: $tType,B2: U3 > ( set @ V2 ),F: T2 > U3,A2: set @ T2] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ V2 )
% 0.30/0.46 @ ( image @ T2 @ ( set @ V2 )
% 0.30/0.46 @ ^ [A5: T2] : ( B2 @ ( F @ A5 ) )
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ ( set @ V2 ) @ ( image @ U3 @ ( set @ V2 ) @ B2 @ ( image @ T2 @ U3 @ F @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_extend_simps(10)
% 0.30/0.46 thf(fact_209_image__UN,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F: B > A,B2: C > ( set @ B ),A2: set @ C] :
% 0.30/0.46 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) )
% 0.30/0.46 = ( complete_Sup_Sup @ ( set @ A )
% 0.30/0.46 @ ( image @ C @ ( set @ A )
% 0.30/0.46 @ ^ [X: C] : ( image @ B @ A @ F @ ( B2 @ X ) )
% 0.30/0.46 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_UN
% 0.30/0.46 thf(fact_210_UN__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F @ A2 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_mono
% 0.30/0.46 thf(fact_211_UN__least,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ A,B2: A > ( set @ B ),C3: set @ B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ C3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) @ C3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_least
% 0.30/0.46 thf(fact_212_UN__upper,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A4: A,A2: set @ A,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ A @ A4 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ A4 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_upper
% 0.30/0.46 thf(fact_213_UN__image__subset,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,F: B > ( set @ A ),G: C > ( set @ B ),X2: C,X5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ ( G @ X2 ) ) ) @ X5 )
% 0.30/0.46 = ( ord_less_eq @ ( set @ B ) @ ( G @ X2 )
% 0.30/0.46 @ ( collect @ B
% 0.30/0.46 @ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F @ X ) @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_image_subset
% 0.30/0.46 thf(fact_214_UN__subset__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: B > ( set @ A ),I5: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A2 @ I5 ) ) @ B2 )
% 0.30/0.46 = ( ! [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ I5 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( A2 @ X ) @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_subset_iff
% 0.30/0.46 thf(fact_215_Union__UNIV,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Union_UNIV
% 0.30/0.46 thf(fact_216_image__Union,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,S: set @ ( set @ B )] :
% 0.30/0.46 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
% 0.30/0.46 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_Union
% 0.30/0.46 thf(fact_217_UN__extend__simps_I8_J,axiom,
% 0.30/0.46 ! [P4: $tType,O: $tType,B2: O > ( set @ P4 ),A2: set @ ( set @ O )] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ P4 )
% 0.30/0.46 @ ( image @ ( set @ O ) @ ( set @ P4 )
% 0.30/0.46 @ ^ [Y4: set @ O] : ( complete_Sup_Sup @ ( set @ P4 ) @ ( image @ O @ ( set @ P4 ) @ B2 @ Y4 ) )
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( complete_Sup_Sup @ ( set @ P4 ) @ ( image @ O @ ( set @ P4 ) @ B2 @ ( complete_Sup_Sup @ ( set @ O ) @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_extend_simps(8)
% 0.30/0.46 thf(fact_218_top__finite__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_lattice @ A )
% 0.30/0.46 => ( ( top_top @ A )
% 0.30/0.46 = ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_finite_def
% 0.30/0.46 thf(fact_219_eval_Ocases,axiom,
% 0.30/0.46 ! [X2: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 0.30/0.46 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S4: list @ char] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S4 ) ) )
% 0.30/0.46 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 0.30/0.46 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P5 ) ) )
% 0.30/0.46 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P5 @ Q4 ) ) )
% 0.30/0.46 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P5 @ Q4 ) ) )
% 0.30/0.46 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
% 0.30/0.46 ( X2
% 0.30/0.46 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P5 @ Q4 ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eval.cases
% 0.30/0.46 thf(fact_220_cSup__eq__maximum,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( condit378418413attice @ A )
% 0.30/0.46 => ! [Z2: A,X5: set @ A] :
% 0.30/0.46 ( ( member @ A @ Z2 @ X5 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ X5 )
% 0.30/0.46 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ X5 )
% 0.30/0.46 = Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cSup_eq_maximum
% 0.30/0.46 thf(fact_221_UN__ball__bex__simps_I3_J,axiom,
% 0.30/0.46 ! [D: $tType,A2: set @ ( set @ D ),P2: D > $o] :
% 0.30/0.46 ( ( ? [X: D] :
% 0.30/0.46 ( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
% 0.30/0.46 & ( P2 @ X ) ) )
% 0.30/0.46 = ( ? [X: set @ D] :
% 0.30/0.46 ( ( member @ ( set @ D ) @ X @ A2 )
% 0.30/0.46 & ? [Y4: D] :
% 0.30/0.46 ( ( member @ D @ Y4 @ X )
% 0.30/0.46 & ( P2 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_ball_bex_simps(3)
% 0.30/0.46 thf(fact_222_UN__ball__bex__simps_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ ( set @ A ),P2: A > $o] :
% 0.30/0.46 ( ( ! [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 0.30/0.46 => ( P2 @ X ) ) )
% 0.30/0.46 = ( ! [X: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ X @ A2 )
% 0.30/0.46 => ! [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ X )
% 0.30/0.46 => ( P2 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_ball_bex_simps(1)
% 0.30/0.46 thf(fact_223_UnionI,axiom,
% 0.30/0.46 ! [A: $tType,X5: set @ A,C3: set @ ( set @ A ),A2: A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ X5 @ C3 )
% 0.30/0.46 => ( ( member @ A @ A2 @ X5 )
% 0.30/0.46 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UnionI
% 0.30/0.46 thf(fact_224_Union__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 0.30/0.46 = ( ? [X: set @ A] :
% 0.30/0.46 ( ( member @ ( set @ A ) @ X @ C3 )
% 0.30/0.46 & ( member @ A @ A2 @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Union_iff
% 0.30/0.46 thf(fact_225_Sup1__I,axiom,
% 0.30/0.46 ! [A: $tType,P2: A > $o,A2: set @ ( A > $o ),A4: A] :
% 0.30/0.46 ( ( member @ ( A > $o ) @ P2 @ A2 )
% 0.30/0.46 => ( ( P2 @ A4 )
% 0.30/0.46 => ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup1_I
% 0.30/0.46 thf(fact_226_SUP1__I,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A4: A,A2: set @ A,B2: A > B > $o,B3: B] :
% 0.30/0.46 ( ( member @ A @ A4 @ A2 )
% 0.30/0.46 => ( ( B2 @ A4 @ B3 )
% 0.30/0.46 => ( complete_Sup_Sup @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B2 @ A2 ) @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP1_I
% 0.30/0.46 thf(fact_227_Sup__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A3: set @ ( set @ A )] :
% 0.30/0.46 ( collect @ A
% 0.30/0.46 @ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_set_def
% 0.30/0.46 thf(fact_228_SUP1__E,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,B2: B > A > $o,A2: set @ B,B3: A] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B2 @ A2 ) @ B3 )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A2 )
% 0.30/0.46 => ~ ( B2 @ X3 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP1_E
% 0.30/0.46 thf(fact_229_SUP__Sup__eq2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > B > $o )
% 0.30/0.46 @ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
% 0.30/0.46 @ ^ [I: set @ ( product_prod @ A @ B ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ I )
% 0.30/0.46 @ S ) )
% 0.30/0.46 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_Sup_eq2
% 0.30/0.46 thf(fact_230_Sup__SUP__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > $o ) )
% 0.30/0.46 = ( ^ [S5: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_SUP_eq
% 0.30/0.46 thf(fact_231_SUP__UN__eq2,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > B > $o )
% 0.30/0.46 @ ( image @ C @ ( A > B > $o )
% 0.30/0.46 @ ^ [I: C,X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( R2 @ I ) )
% 0.30/0.46 @ S ) )
% 0.30/0.46 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_UN_eq2
% 0.30/0.46 thf(fact_232_SUP__Sup__eq,axiom,
% 0.30/0.46 ! [A: $tType,S: set @ ( set @ A )] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > $o )
% 0.30/0.46 @ ( image @ ( set @ A ) @ ( A > $o )
% 0.30/0.46 @ ^ [I: set @ A,X: A] : ( member @ A @ X @ I )
% 0.30/0.46 @ S ) )
% 0.30/0.46 = ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_Sup_eq
% 0.30/0.46 thf(fact_233_Sup1__E,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ ( A > $o ),A4: A] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A4 )
% 0.30/0.46 => ~ ! [P6: A > $o] :
% 0.30/0.46 ( ( member @ ( A > $o ) @ P6 @ A2 )
% 0.30/0.46 => ~ ( P6 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup1_E
% 0.30/0.46 thf(fact_234_UnionE,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 0.30/0.46 => ~ ! [X6: set @ A] :
% 0.30/0.46 ( ( member @ A @ A2 @ X6 )
% 0.30/0.46 => ~ ( member @ ( set @ A ) @ X6 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UnionE
% 0.30/0.46 thf(fact_235_subrelI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ! [X3: A,Y5: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R2 )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subrelI
% 0.30/0.46 thf(fact_236_pred__subset__eq2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o )
% 0.30/0.46 @ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
% 0.30/0.46 @ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.30/0.46
% 0.30/0.46 % pred_subset_eq2
% 0.30/0.46 thf(fact_237_pred__equals__eq2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R ) )
% 0.30/0.46 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S ) ) )
% 0.30/0.46 = ( R = S ) ) ).
% 0.30/0.46
% 0.30/0.46 % pred_equals_eq2
% 0.30/0.46 thf(fact_238_ssubst__Pair__rhs,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R2: A,S3: B,R: set @ ( product_prod @ A @ B ),S6: B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R )
% 0.30/0.46 => ( ( S6 = S3 )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S6 ) @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ssubst_Pair_rhs
% 0.30/0.46 thf(fact_239_top__empty__eq2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( top_top @ ( A > B > $o ) )
% 0.30/0.46 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_empty_eq2
% 0.30/0.46 thf(fact_240_cSup__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ( condit378418413attice @ A )
% 0.30/0.46 & ( no_bot @ A ) )
% 0.30/0.46 => ! [X5: set @ A,A4: A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ X5 )
% 0.30/0.46 => ( ord_less_eq @ A @ X3 @ A4 ) )
% 0.30/0.46 => ( ! [Y5: A] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ X5 )
% 0.30/0.46 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ Y5 ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ A @ X5 )
% 0.30/0.46 = A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cSup_eq
% 0.30/0.46 thf(fact_241_UN__constant__eq,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A4: A,A2: set @ A,F: A > ( set @ B ),C2: set @ B] :
% 0.30/0.46 ( ( member @ A @ A4 @ A2 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( ( F @ X3 )
% 0.30/0.46 = C2 ) )
% 0.30/0.46 => ( ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F @ A2 ) )
% 0.30/0.46 = C2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UN_constant_eq
% 0.30/0.46 thf(fact_242_SUP__eq__top__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comple1035589618norder @ A )
% 0.30/0.46 => ! [F: B > A,A2: set @ B] :
% 0.30/0.46 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 0.30/0.46 = ( top_top @ A ) )
% 0.30/0.46 = ( ! [X: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 0.30/0.46 => ? [Y4: B] :
% 0.30/0.46 ( ( member @ B @ Y4 @ A2 )
% 0.30/0.46 & ( ord_less @ A @ X @ ( F @ Y4 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_eq_top_iff
% 0.30/0.46 thf(fact_243_psubsetI,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ( A2 != B2 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetI
% 0.30/0.46 thf(fact_244_Sup2__I,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R2: A > B > $o,A2: set @ ( A > B > $o ),A4: A,B3: B] :
% 0.30/0.46 ( ( member @ ( A > B > $o ) @ R2 @ A2 )
% 0.30/0.46 => ( ( R2 @ A4 @ B3 )
% 0.30/0.46 => ( complete_Sup_Sup @ ( A > B > $o ) @ A2 @ A4 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup2_I
% 0.30/0.46 thf(fact_245_top2I,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X2: A,Y3: B] : ( top_top @ ( A > B > $o ) @ X2 @ Y3 ) ).
% 0.30/0.46
% 0.30/0.46 % top2I
% 0.30/0.46 thf(fact_246_Sup__eq__top__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple1035589618norder @ A )
% 0.30/0.46 => ! [A2: set @ A] :
% 0.30/0.46 ( ( ( complete_Sup_Sup @ A @ A2 )
% 0.30/0.46 = ( top_top @ A ) )
% 0.30/0.46 = ( ! [X: A] :
% 0.30/0.46 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 0.30/0.46 => ? [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ A2 )
% 0.30/0.46 & ( ord_less @ A @ X @ Y4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_eq_top_iff
% 0.30/0.46 thf(fact_247_SUP2__I,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,A4: A,A2: set @ A,B2: A > B > C > $o,B3: B,C2: C] :
% 0.30/0.46 ( ( member @ A @ A4 @ A2 )
% 0.30/0.46 => ( ( B2 @ A4 @ B3 @ C2 )
% 0.30/0.46 => ( complete_Sup_Sup @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B2 @ A2 ) @ B3 @ C2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP2_I
% 0.30/0.46 thf(fact_248_SUP2__E,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,B2: C > A > B > $o,A2: set @ C,B3: A,C2: B] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B2 @ A2 ) @ B3 @ C2 )
% 0.30/0.46 => ~ ! [X3: C] :
% 0.30/0.46 ( ( member @ C @ X3 @ A2 )
% 0.30/0.46 => ~ ( B2 @ X3 @ B3 @ C2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP2_E
% 0.30/0.46 thf(fact_249_Sup2__E,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ ( A > B > $o ),A4: A,B3: B] :
% 0.30/0.46 ( ( complete_Sup_Sup @ ( A > B > $o ) @ A2 @ A4 @ B3 )
% 0.30/0.46 => ~ ! [R4: A > B > $o] :
% 0.30/0.46 ( ( member @ ( A > B > $o ) @ R4 @ A2 )
% 0.30/0.46 => ~ ( R4 @ A4 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup2_E
% 0.30/0.46 thf(fact_250_Sup__bool__def,axiom,
% 0.30/0.46 ( ( complete_Sup_Sup @ $o )
% 0.30/0.46 = ( member @ $o @ $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_bool_def
% 0.30/0.46 thf(fact_251_less__Sup__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple1035589618norder @ A )
% 0.30/0.46 => ! [A4: A,S: set @ A] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
% 0.30/0.46 = ( ? [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ S )
% 0.30/0.46 & ( ord_less @ A @ A4 @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_Sup_iff
% 0.30/0.46 thf(fact_252_complete__interval,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( condit1037483654norder @ A )
% 0.30/0.46 => ! [A4: A,B3: A,P2: A > $o] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.30/0.46 => ( ( P2 @ A4 )
% 0.30/0.46 => ( ~ ( P2 @ B3 )
% 0.30/0.46 => ? [C5: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ C5 )
% 0.30/0.46 & ( ord_less_eq @ A @ C5 @ B3 )
% 0.30/0.46 & ! [X4: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ A4 @ X4 )
% 0.30/0.46 & ( ord_less @ A @ X4 @ C5 ) )
% 0.30/0.46 => ( P2 @ X4 ) )
% 0.30/0.46 & ! [D3: A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ A4 @ X3 )
% 0.30/0.46 & ( ord_less @ A @ X3 @ D3 ) )
% 0.30/0.46 => ( P2 @ X3 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ D3 @ C5 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % complete_interval
% 0.30/0.46 thf(fact_253_le__Sup__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple1035589618norder @ A )
% 0.30/0.46 => ! [X2: A,A2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) )
% 0.30/0.46 = ( ! [Y4: A] :
% 0.30/0.46 ( ( ord_less @ A @ Y4 @ X2 )
% 0.30/0.46 => ? [X: A] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 & ( ord_less @ A @ Y4 @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_Sup_iff
% 0.30/0.46 thf(fact_254_less__SUP__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comple1035589618norder @ A )
% 0.30/0.46 => ! [A4: A,F: B > A,A2: set @ B] :
% 0.30/0.46 ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) )
% 0.30/0.46 = ( ? [X: B] :
% 0.30/0.46 ( ( member @ B @ X @ A2 )
% 0.30/0.46 & ( ord_less @ A @ A4 @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_SUP_iff
% 0.30/0.46 thf(fact_255_SUP__lessD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [F: B > A,A2: set @ B,Y3: A,I2: B] :
% 0.30/0.46 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ Y3 )
% 0.30/0.46 => ( ( member @ B @ I2 @ A2 )
% 0.30/0.46 => ( ord_less @ A @ ( F @ I2 ) @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SUP_lessD
% 0.30/0.46
% 0.30/0.46 % Type constructors (32)
% 0.30/0.46 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A8 )
% 0.30/0.46 => ( condit378418413attice @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A8 )
% 0.30/0.46 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( complete_Sup @ A8 )
% 0.30/0.46 => ( complete_Sup @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( order_top @ A8 )
% 0.30/0.46 => ( order_top @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( preorder @ A8 )
% 0.30/0.46 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( order @ A8 )
% 0.30/0.46 => ( order @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( top @ A8 )
% 0.30/0.46 => ( top @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( ord @ A8 )
% 0.30/0.46 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.30/0.46 condit1037483654norder @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
% 0.30/0.46 condit378418413attice @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Complete__Lattices_OSup_2,axiom,
% 0.30/0.46 complete_Sup @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
% 0.30/0.46 preorder @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
% 0.30/0.46 order @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
% 0.30/0.46 ord @ nat ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
% 0.30/0.46 ! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_7,axiom,
% 0.30/0.46 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Complete__Lattices_OSup_8,axiom,
% 0.30/0.46 ! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_9,axiom,
% 0.30/0.46 ! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
% 0.30/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder_11,axiom,
% 0.30/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Otop_12,axiom,
% 0.30/0.46 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
% 0.30/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_14,axiom,
% 0.30/0.46 condit378418413attice @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_15,axiom,
% 0.30/0.46 comple187826305attice @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Complete__Lattices_OSup_16,axiom,
% 0.30/0.46 complete_Sup @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_17,axiom,
% 0.30/0.46 order_top @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_18,axiom,
% 0.30/0.46 preorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_19,axiom,
% 0.30/0.46 linorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
% 0.30/0.46 order @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Otop_21,axiom,
% 0.30/0.46 top @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oord_22,axiom,
% 0.30/0.46 ord @ $o ).
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 ( ( ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.46 ( ( paraco876059933e_eval @ I @ p )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) )
% 0.30/0.46 != ( ~ ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ paraco415392788lle_tv ) @ ( image @ ( list @ char ) @ paraco415392788lle_tv @ I @ ( top_top @ ( set @ ( list @ char ) ) ) )
% 0.30/0.46 @ ( paraco506338565domain
% 0.30/0.46 @ ( collect @ nat
% 0.30/0.46 @ ^ [N: nat] : $true ) ) )
% 0.30/0.46 => ( ( paraco876059933e_eval @ I @ p )
% 0.30/0.46 = ( paraco2040174112le_Det @ $true ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.79/1.55 ---- Embedded in THF ---
% 2.79/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.79/1.55 %%% Generated on Tue Feb 24 08:27:28 EST 2026
% 2.79/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.79/1.55 %%% Logic specification used:
% 2.79/1.55 %%% thf(spec, logic, $$dhol).
% 2.79/1.55
% 2.79/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.SMTCspLT4x/DTF2DTF25569.p
% 2.79/1.55 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv, type, paraco415392788lle_tv: $tType).
% 2.79/1.55 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm, type, paraco414474393lle_fm: $tType).
% 2.79/1.55 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.79/1.55 thf(ty_t_String_Ochar, type, char: $tType).
% 2.79/1.55 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.79/1.55 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.79/1.55 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.79/1.55 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Enum_Ofinite__lattice, type, finite_lattice: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Complete__Lattices_OSup, type, complete_Sup: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple1035589618norder: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice, type, condit378418413attice: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.79/1.55 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 2.79/1.55 thf(sy_c_Finite__Set_OFpow, type, finite_Fpow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.79/1.55 thf(sy_c_Fun_Oswap, type, swap: (!> [A:$tType,B:$tType]: ((A > (A > ((A > B) > (A > B))))))).
% 2.79/1.55 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.79/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.79/1.55 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.79/1.55 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.79/1.55 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Odomain, type, paraco506338565domain: ((set @ nat) > (set @ paraco415392788lle_tv))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval, type, paraco876059933e_eval: (((list @ char) > paraco415392788lle_tv) > (paraco414474393lle_fm > paraco415392788lle_tv))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H, type, paraco2100061555le_Con: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql, type, paraco2084319816le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H, type, paraco1628874225le_Eql: (paraco414474393lle_fm > (paraco414474393lle_fm > paraco414474393lle_fm))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H, type, paraco329115265le_Neg: (paraco414474393lle_fm > paraco414474393lle_fm)).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro, type, paraco27778325le_Pro: ((list @ char) > paraco414474393lle_fm)).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet, type, paraco2040174112le_Det: ($o > paraco415392788lle_tv)).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv, type, paraco152590079rec_tv: (!> [A:$tType]: ((($o > A) > ((nat > A) > (paraco415392788lle_tv > A)))))).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid, type, paraco769098683_valid: (paraco414474393lle_fm > $o)).
% 2.79/1.55 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in, type, paraco2086025920lid_in: ((set @ nat) > (paraco414474393lle_fm > $o))).
% 2.79/1.55 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.79/1.55 thf(sy_c_Relation_OPowp, type, powp: (!> [A:$tType]: (((A > $o) > ((set @ A) > $o))))).
% 2.79/1.55 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.79/1.55 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.79/1.55 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.79/1.55 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.79/1.55 thf(sy_v_p, type, p: paraco414474393lle_fm).
% 2.79/1.55 thf(fact_0_universal__domain, axiom, ((paraco506338565domain @ ((collect @ nat) @ (^ [N:nat]: ($true)))) = ((collect @ paraco415392788lle_tv) @ (^ [X:paraco415392788lle_tv]: ($true))))).
% 2.79/1.55 thf(fact_1_tv_Oinject_I1_J, axiom, (! [X1:$o,Y1:$o]: ((((paraco2040174112le_Det @ X1) = (paraco2040174112le_Det @ Y1)) = (X1 = Y1))))).
% 2.79/1.55 thf(fact_2_valid__in__def, axiom, (paraco2086025920lid_in = (^ [U:(set @ nat),P:paraco414474393lle_fm]: ((! [I:((list @ char) > paraco415392788lle_tv)]: (((((ord_less_eq @ (set @ paraco415392788lle_tv)) @ ((((image @ (list @ char)) @ paraco415392788lle_tv) @ I) @ (top_top @ (set @ (list @ char))))) @ (paraco506338565domain @ U)) => (((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true))))))))).
% 2.79/1.55 thf(fact_3_image__ident, axiom, (! [A:$tType,Y:(set @ A)]: ((((((image @ A) @ A) @ (^ [X:A]: (X))) @ Y) = Y)))).
% 2.79/1.55 thf(fact_4_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.79/1.55 thf(fact_5_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.79/1.55 thf(fact_6_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X:D]: ((top_top @ C)))))))).
% 2.79/1.55 thf(fact_7_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.79/1.55 thf(fact_8_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.79/1.55 thf(fact_9_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X2:B,A2:(set @ B)]: (((B3 = (F @ X2)) => ((((member @ B) @ X2) @ A2) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2))))))).
% 2.79/1.55 thf(fact_10_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.79/1.55 thf(fact_11_rangeE, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))) => (~ ((! [X3:B]: ((B3 != (F @ X3)))))))))).
% 2.79/1.55 thf(fact_12_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),G:(B > C)]: ((((((image @ B) @ A) @ (^ [X:B]: ((F @ (G @ X))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F) @ ((((image @ B) @ C) @ G) @ (top_top @ (set @ B)))))))).
% 2.79/1.55 thf(fact_13_image__Collect__subsetI, axiom, (! [A:$tType,B:$tType,P2:(A > $o),F:(A > B),B2:(set @ B)]: (((! [X3:A]: (((P2 @ X3) => (((member @ B) @ (F @ X3)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((collect @ A) @ P2))) @ B2))))).
% 2.79/1.55 thf(fact_14_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.79/1.55 thf(fact_15_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B4:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) @ (^ [X:A]: ((((member @ A) @ X) @ B4)))))))))).
% 2.79/1.55 thf(fact_16_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ A4) @ B3) => (A4 = B3))))))))).
% 2.79/1.55 thf(fact_17_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (((ord_less_eq @ A) @ A5) @ B5))))))))).
% 2.79/1.55 thf(fact_18_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 2.79/1.55 thf(fact_19_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P2:(A > (A > $o)),A4:A,B3:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P2 @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P2 @ B6) @ A6) => ((P2 @ A6) @ B6)))) => ((P2 @ A4) @ B3))))))))).
% 2.79/1.55 thf(fact_20_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.79/1.55 thf(fact_21_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 2.79/1.55 thf(fact_22_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ A4) => (A4 = B3))))))))).
% 2.79/1.55 thf(fact_23_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.79/1.55 thf(fact_24_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C2:A]: (((A4 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.79/1.55 thf(fact_25_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (((ord_less_eq @ A) @ B5) @ A5))))))))).
% 2.79/1.55 thf(fact_26_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X2:A]: (((((ord_less_eq @ A) @ Y3) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y3) = (X2 = Y3))))))))).
% 2.79/1.55 thf(fact_27_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X2)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y3))))))))))))))))).
% 2.79/1.55 thf(fact_28_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.79/1.55 thf(fact_29_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X2)))))))).
% 2.79/1.55 thf(fact_30_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y3:A]: (((X2 = Y3) => (((ord_less_eq @ A) @ X2) @ Y3)))))))).
% 2.79/1.55 thf(fact_31_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X2)))))))).
% 2.79/1.55 thf(fact_32_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X2) => (X2 = Y3))))))))).
% 2.79/1.55 thf(fact_33_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X))))))))).
% 2.79/1.55 thf(fact_34_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A4) @ B3) => (((F @ B3) = C2) => ((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C2)))))))))).
% 2.79/1.55 thf(fact_35_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F:(B > A),B3:B,C2:B]: (((A4 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less_eq @ B) @ X3) @ Y5) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.79/1.55 thf(fact_36_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A4:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C2)))))))))).
% 2.79/1.55 thf(fact_37_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less_eq @ B) @ X3) @ Y5) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.79/1.55 thf(fact_38_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 2.79/1.55 thf(fact_39_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.79/1.55 thf(fact_40_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.79/1.55 thf(fact_41_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.79/1.55 thf(fact_42_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X2:A,A2:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X2) @ A2) => ((B3 = (F @ X2)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 2.79/1.55 thf(fact_43_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A2)) => (P2 @ X3)))) => (! [X4:B]: (((((member @ B) @ X4) @ A2) => (P2 @ (F @ X4))))))))).
% 2.79/1.55 thf(fact_44_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N2:(set @ A),F:(A > B),G:(A > B)]: (((M = N2) => ((! [X3:A]: (((((member @ A) @ X3) @ N2) => ((F @ X3) = (G @ X3))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N2))))))).
% 2.79/1.55 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P2:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P2)) = (P2 @ A4))))).
% 2.79/1.55 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = A2)))).
% 2.79/1.55 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) = (Q @ X3)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.79/1.55 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.79/1.55 thf(fact_49_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A2)) & (P2 @ X4)))) => (? [X3:B]: (((((member @ B) @ X3) @ A2) & (P2 @ (F @ X3))))))))).
% 2.79/1.55 thf(fact_50_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [X:B]: (((((member @ B) @ X) @ A2) & (Z2 = (F @ X))))))))).
% 2.79/1.55 thf(fact_51_imageI, axiom, (! [B:$tType,A:$tType,X2:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X2) @ A2) => (((member @ B) @ (F @ X2)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.79/1.55 thf(fact_52_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P2 @ X) => (Q @ X)))))))).
% 2.79/1.55 thf(fact_53_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z:(set @ A)]: ((Y2 = Z))) = (^ [A3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) & (((ord_less_eq @ (set @ A)) @ B4) @ A3)))))))).
% 2.79/1.55 thf(fact_54_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ A2) @ C3)))))).
% 2.79/1.55 thf(fact_55_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)))))).
% 2.79/1.55 thf(fact_56_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.79/1.55 thf(fact_57_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B4:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A3) => (((member @ A) @ T) @ B4)))))))))).
% 2.79/1.55 thf(fact_58_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.79/1.55 thf(fact_59_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.79/1.55 thf(fact_60_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B4:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B4)))))))))).
% 2.79/1.55 thf(fact_61_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.79/1.55 thf(fact_62_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.79/1.55 thf(fact_63_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))))).
% 2.79/1.55 thf(fact_64_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.79/1.55 thf(fact_65_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.79/1.55 thf(fact_66_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A2)) & (P2 @ X))))) = ((((image @ B) @ A) @ F) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A2) & (P2 @ (F @ X))))))))))).
% 2.79/1.55 thf(fact_67_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),G:(C > B),A2:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G) @ A2)) = ((((image @ C) @ A) @ (^ [X:C]: ((F @ (G @ X))))) @ A2))))).
% 2.79/1.55 thf(fact_68_imageE, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [X3:B]: (((B3 = (F @ X3)) => (~ ((((member @ B) @ X3) @ A2)))))))))))).
% 2.79/1.55 thf(fact_69_Collect__restrict, axiom, (! [A:$tType,X5:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ X5) & (P2 @ X)))))) @ X5)))).
% 2.79/1.55 thf(fact_70_prop__restrict, axiom, (! [A:$tType,X2:A,Z3:(set @ A),X5:(set @ A),P2:(A > $o)]: (((((member @ A) @ X2) @ Z3) => ((((ord_less_eq @ (set @ A)) @ Z3) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ X5) & (P2 @ X)))))) => (P2 @ X2)))))).
% 2.79/1.55 thf(fact_71_Collect__subset, axiom, (! [A:$tType,A2:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A2) & (P2 @ X)))))) @ A2)))).
% 2.79/1.55 thf(fact_72_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X:A]: ($true))))))).
% 2.79/1.55 thf(fact_73_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A4) => (A4 = (top_top @ A))))))))).
% 2.79/1.55 thf(fact_74_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A4:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A4) = (A4 = (top_top @ A))))))))).
% 2.79/1.55 thf(fact_75_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ (top_top @ A)))))))).
% 2.79/1.55 thf(fact_76_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A2) & (B2 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 2.79/1.55 thf(fact_77_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) = (! [X:B]: (((((member @ B) @ X) @ A2) => (((member @ A) @ (F @ X)) @ B2)))))))).
% 2.79/1.55 thf(fact_78_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C4:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C4) @ A2) => (B2 != ((((image @ B) @ A) @ F) @ C4))))))))))).
% 2.79/1.55 thf(fact_79_image__subsetI, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ B) @ (F @ X3)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B2))))).
% 2.79/1.55 thf(fact_80_image__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)))))).
% 2.79/1.55 thf(fact_81_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X2:B]: (((B3 = (F @ X2)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.79/1.55 thf(fact_82_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X2:B]: ((((member @ A) @ (F @ X2)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.79/1.55 thf(fact_83_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.79/1.55 thf(fact_84_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X:B]: ((Y4 = (F @ X)))))))))).
% 2.79/1.55 thf(fact_85_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X3:A]: (((G @ (F @ X3)) = X3))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.79/1.55 thf(fact_86_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y3 != (F @ X3)))))))))).
% 2.79/1.55 thf(fact_87_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y3 = (F @ X3)))))))).
% 2.79/1.55 thf(fact_88_all__subset__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:((set @ A) > $o)]: (((! [B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B4) @ ((((image @ B) @ A) @ F) @ A2)) => (P2 @ B4)))) = (! [B4:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B4) @ A2) => (P2 @ ((((image @ B) @ A) @ F) @ B4))))))))).
% 2.79/1.55 thf(fact_89_valid__def, axiom, (paraco769098683_valid = (^ [P:paraco414474393lle_fm]: ((! [I:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I) @ P) = (paraco2040174112le_Det @ $true)))))))).
% 2.79/1.55 thf(fact_90_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ R)))) @ (^ [X:A]: ((((member @ A) @ X) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 2.79/1.55 thf(fact_91_subset__Collect__iff, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((ord_less_eq @ (set @ A)) @ B2) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A2) & (P2 @ X)))))) = (! [X:A]: (((((member @ A) @ X) @ B2) => (P2 @ X))))))))).
% 2.79/1.55 thf(fact_92_subset__CollectI, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A),Q:(A > $o),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((! [X3:A]: (((((member @ A) @ X3) @ B2) => ((Q @ X3) => (P2 @ X3))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ B2) & (Q @ X)))))) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A2) & (P2 @ X))))))))))).
% 2.79/1.55 thf(fact_93_conj__subset__def, axiom, (! [A:$tType,A2:(set @ A),P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A2) @ ((collect @ A) @ (^ [X:A]: (((P2 @ X) & (Q @ X)))))) = ((((ord_less_eq @ (set @ A)) @ A2) @ ((collect @ A) @ P2)) & (((ord_less_eq @ (set @ A)) @ A2) @ ((collect @ A) @ Q))))))).
% 2.79/1.55 thf(fact_94_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A2:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X:A]: (X))) @ A2)) = (Sup @ A2))))).
% 2.79/1.55 thf(fact_95_predicate1I, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (((ord_less_eq @ (A > $o)) @ P2) @ Q))))).
% 2.79/1.55 thf(fact_96_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 2.79/1.55 thf(fact_97_predicate1D, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o),X2:A]: (((((ord_less_eq @ (A > $o)) @ P2) @ Q) => ((P2 @ X2) => (Q @ X2)))))).
% 2.79/1.55 thf(fact_98_rev__predicate1D, axiom, (! [A:$tType,P2:(A > $o),X2:A,Q:(A > $o)]: (((P2 @ X2) => ((((ord_less_eq @ (A > $o)) @ P2) @ Q) => (Q @ X2)))))).
% 2.79/1.55 thf(fact_99_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C3:(B > A),D2:(B > A),Inf:((set @ A) > A)]: (((A2 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C3 @ X3) = (D2 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C3) @ A2)) = (Inf @ ((((image @ B) @ A) @ D2) @ B2)))))))).
% 2.79/1.55 thf(fact_100_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C3:(B > A),D2:(B > A),Sup:((set @ A) > A)]: (((A2 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C3 @ X3) = (D2 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C3) @ A2)) = (Sup @ ((((image @ B) @ A) @ D2) @ B2)))))))).
% 2.79/1.55 thf(fact_101_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A))))))))).
% 2.79/1.55 thf(fact_102_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A2:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X:A]: (X))) @ A2)) = (Inf @ A2))))).
% 2.79/1.55 thf(fact_103_top__conj_I2_J, axiom, (! [A:$tType,P2:$o,X2:A]: (((P2 & ((top_top @ (A > $o)) @ X2)) = P2)))).
% 2.79/1.55 thf(fact_104_top__conj_I1_J, axiom, (! [A:$tType,X2:A,P2:$o]: (((((top_top @ (A > $o)) @ X2) & P2) = P2)))).
% 2.79/1.55 thf(fact_105_eval_Osimps_I2_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I2) @ paraco251304083_Truth) = (paraco2040174112le_Det @ $true))))).
% 2.79/1.55 thf(fact_106_image__Fpow__mono, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((finite_Fpow @ B) @ A2))) @ ((finite_Fpow @ A) @ B2)))))).
% 2.79/1.55 thf(fact_107_eval_Osimps_I5_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P3:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P3) = ((paraco876059933e_eval @ I2) @ Q2)) => (((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P3) @ Q2)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P3) != ((paraco876059933e_eval @ I2) @ Q2)) => (((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P3) @ Q2)) = (paraco2040174112le_Det @ $false))))))).
% 2.79/1.55 thf(fact_108_fm_Oinject_I4_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm,Y51:paraco414474393lle_fm,Y52:paraco414474393lle_fm]: (((((paraco2084319816le_Eql @ X51) @ X52) = ((paraco2084319816le_Eql @ Y51) @ Y52)) = ((X51 = Y51) & (X52 = Y52)))))).
% 2.79/1.55 thf(fact_109_fm_Odistinct_I15_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.79/1.55 thf(fact_110_Fpow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A2)) @ ((finite_Fpow @ A) @ B2)))))).
% 2.79/1.55 thf(fact_111_eval__equality__simplify, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P3:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((((paraco876059933e_eval @ I2) @ ((paraco2084319816le_Eql @ P3) @ Q2)) = (paraco2040174112le_Det @ (((paraco876059933e_eval @ I2) @ P3) = ((paraco876059933e_eval @ I2) @ Q2))))))).
% 2.79/1.55 thf(fact_112_image__Pow__mono, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((pow @ B) @ A2))) @ ((pow @ A) @ B2)))))).
% 2.79/1.55 thf(fact_113_eval_Osimps_I4_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P3:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P3) = ((paraco876059933e_eval @ I2) @ Q2)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ P3))) & ((((paraco876059933e_eval @ I2) @ P3) != ((paraco876059933e_eval @ I2) @ Q2)) => (((((paraco876059933e_eval @ I2) @ P3) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ Q2))) & ((((paraco876059933e_eval @ I2) @ P3) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ Q2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ P3))) & ((((paraco876059933e_eval @ I2) @ Q2) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco2100061555le_Con @ P3) @ Q2)) = (paraco2040174112le_Det @ $false))))))))))).
% 2.79/1.55 thf(fact_114_Powp__mono, axiom, (! [A:$tType,A2:(A > $o),B2:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ A2) @ B2) => (((ord_less_eq @ ((set @ A) > $o)) @ ((powp @ A) @ A2)) @ ((powp @ A) @ B2)))))).
% 2.79/1.55 thf(fact_115_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P2:(A > $o),X2:A]: (((P2 @ X2) => ((! [Y5:A]: (((P2 @ Y5) => (((ord_less_eq @ A) @ Y5) @ X2)))) => (((order_Greatest @ A) @ P2) = X2))))))))).
% 2.79/1.55 thf(fact_116_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P2:(A > $o),X2:A,Q:(A > $o)]: (((P2 @ X2) => ((! [Y5:A]: (((P2 @ Y5) => (((ord_less_eq @ A) @ Y5) @ X2)))) => ((! [X3:A]: (((P2 @ X3) => ((! [Y6:A]: (((P2 @ Y6) => (((ord_less_eq @ A) @ Y6) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P2))))))))))).
% 2.79/1.55 thf(fact_117_fm_Oinject_I3_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,Y41:paraco414474393lle_fm,Y42:paraco414474393lle_fm]: (((((paraco2100061555le_Con @ X41) @ X42) = ((paraco2100061555le_Con @ Y41) @ Y42)) = ((X41 = Y41) & (X42 = Y42)))))).
% 2.79/1.55 thf(fact_118_PowI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)))))).
% 2.79/1.55 thf(fact_119_Pow__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.79/1.55 thf(fact_120_Pow__UNIV, axiom, (! [A:$tType]: ((((pow @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 2.79/1.55 thf(fact_121_Pow__top, axiom, (! [A:$tType,A2:(set @ A)]: ((((member @ (set @ A)) @ A2) @ ((pow @ A) @ A2))))).
% 2.79/1.55 thf(fact_122_Powp__Pow__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((powp @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = (^ [X:(set @ A)]: ((((member @ (set @ A)) @ X) @ ((pow @ A) @ A2)))))))).
% 2.79/1.55 thf(fact_123_PowD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.79/1.55 thf(fact_124_Cantors__paradox, axiom, (! [A:$tType,A2:(set @ A)]: ((~ ((? [F3:(A > (set @ A))]: ((((((image @ A) @ (set @ A)) @ F3) @ A2) = ((pow @ A) @ A2))))))))).
% 2.79/1.55 thf(fact_125_Pow__def, axiom, (! [A:$tType]: (((pow @ A) = (^ [A3:(set @ A)]: (((collect @ (set @ A)) @ (^ [B4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B4) @ A3)))))))))).
% 2.79/1.55 thf(fact_126_fm_Odistinct_I25_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: ((((paraco2100061555le_Con @ X41) @ X42) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.79/1.55 thf(fact_127_fm_Odistinct_I13_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.79/1.55 thf(fact_128_Pow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow @ A) @ A2)) @ ((pow @ A) @ B2)))))).
% 2.79/1.55 thf(fact_129_image__Pow__surj, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((((image @ B) @ A) @ F) @ A2) = B2) => (((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((pow @ B) @ A2)) = ((pow @ A) @ B2)))))).
% 2.79/1.55 thf(fact_130_conjunction, axiom, (! [P3:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: (((paraco769098683_valid @ ((paraco2100061555le_Con @ P3) @ Q2)) = ((paraco769098683_valid @ P3) & (paraco769098683_valid @ Q2)))))).
% 2.79/1.55 thf(fact_131_Fpow__subset__Pow, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A2)) @ ((pow @ A) @ A2))))).
% 2.79/1.55 thf(fact_132_tv_Osimps_I7_J, axiom, (! [A:$tType,F1:($o > A),F22:(nat > A),X1:$o]: ((((((paraco152590079rec_tv @ A) @ F1) @ F22) @ (paraco2040174112le_Det @ X1)) = (F1 @ X1))))).
% 2.79/1.55 thf(fact_133_strict__mono__inv, axiom, (! [A:$tType,B:$tType]: ((((linorder @ B) & (linorder @ A)) => (! [F:(A > B),G:(B > A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((! [X3:A]: (((G @ (F @ X3)) = X3))) => (((order_strict_mono @ B) @ A) @ G)))))))))).
% 2.79/1.55 thf(fact_134_eval__negation, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P3:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P3) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P3)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P3) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I2) @ P3) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P3)) = (paraco2040174112le_Det @ $false))) & ((((paraco876059933e_eval @ I2) @ P3) != (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P3)) = ((paraco876059933e_eval @ I2) @ P3))))))))).
% 2.79/1.55 thf(fact_135_surj__swap__iff, axiom, (! [B:$tType,A:$tType,A4:B,B3:B,F:(B > A)]: (((((((image @ B) @ A) @ (((((swap @ B) @ A) @ A4) @ B3) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.79/1.55 thf(fact_136_swap__nilpotent, axiom, (! [B:$tType,A:$tType,A4:A,B3:A,F:(A > B)]: (((((((swap @ A) @ B) @ A4) @ B3) @ (((((swap @ A) @ B) @ A4) @ B3) @ F)) = F)))).
% 2.79/1.55 thf(fact_137_swap__self, axiom, (! [B:$tType,A:$tType,A4:A,F:(A > B)]: (((((((swap @ A) @ B) @ A4) @ A4) @ F) = F)))).
% 2.79/1.55 thf(fact_138_swap__apply_I1_J, axiom, (! [A:$tType,B:$tType,A4:B,B3:B,F:(B > A)]: ((((((((swap @ B) @ A) @ A4) @ B3) @ F) @ A4) = (F @ B3))))).
% 2.79/1.55 thf(fact_139_swap__apply_I2_J, axiom, (! [A:$tType,B:$tType,A4:B,B3:B,F:(B > A)]: ((((((((swap @ B) @ A) @ A4) @ B3) @ F) @ B3) = (F @ A4))))).
% 2.79/1.55 thf(fact_140_swap__apply_I3_J, axiom, (! [A:$tType,B:$tType,C2:B,A4:B,B3:B,F:(B > A)]: (((C2 != A4) => ((C2 != B3) => (((((((swap @ B) @ A) @ A4) @ B3) @ F) @ C2) = (F @ C2))))))).
% 2.79/1.55 thf(fact_141_fm_Oinject_I2_J, axiom, (! [X32:paraco414474393lle_fm,Y32:paraco414474393lle_fm]: ((((paraco329115265le_Neg @ X32) = (paraco329115265le_Neg @ Y32)) = (X32 = Y32))))).
% 2.79/1.55 thf(fact_142_swap__image__eq, axiom, (! [B:$tType,A:$tType,A4:A,A2:(set @ A),B3:A,F:(A > B)]: (((((member @ A) @ A4) @ A2) => ((((member @ A) @ B3) @ A2) => (((((image @ A) @ B) @ (((((swap @ A) @ B) @ A4) @ B3) @ F)) @ A2) = ((((image @ A) @ B) @ F) @ A2))))))).
% 2.79/1.55 thf(fact_143_swap__commute, axiom, (! [B:$tType,A:$tType]: ((((swap @ A) @ B) = (^ [A5:A,B5:A]: (((((swap @ A) @ B) @ B5) @ A5))))))).
% 2.79/1.55 thf(fact_144_swap__triple, axiom, (! [B:$tType,A:$tType,A4:A,C2:A,B3:A,F:(A > B)]: (((A4 != C2) => ((B3 != C2) => ((((((swap @ A) @ B) @ A4) @ B3) @ (((((swap @ A) @ B) @ B3) @ C2) @ (((((swap @ A) @ B) @ A4) @ B3) @ F))) = (((((swap @ A) @ B) @ A4) @ C2) @ F))))))).
% 2.79/1.55 thf(fact_145_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X2:A,Y3:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X2) = (F @ Y3)) = (X2 = Y3))))))))).
% 2.79/1.55 thf(fact_146_double__negation, axiom, (paraco876059933e_eval = (^ [I:((list @ char) > paraco415392788lle_tv),P:paraco414474393lle_fm]: (((paraco876059933e_eval @ I) @ (paraco329115265le_Neg @ (paraco329115265le_Neg @ P))))))).
% 2.79/1.55 thf(fact_147_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X2:A,Y3:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y3)) = (((ord_less_eq @ A) @ X2) @ Y3))))))))).
% 2.79/1.55 thf(fact_148_fm_Odistinct_I21_J, axiom, (! [X32:paraco414474393lle_fm,X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X32) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.79/1.55 thf(fact_149_fm_Odistinct_I19_J, axiom, (! [X32:paraco414474393lle_fm,X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X32) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.79/1.55 thf(fact_150_fm_Odistinct_I11_J, axiom, (! [X32:paraco414474393lle_fm]: ((paraco251304083_Truth != (paraco329115265le_Neg @ X32))))).
% 2.79/1.55 thf(fact_151_surj__imp__surj__swap, axiom, (! [B:$tType,A:$tType,F:(B > A),A4:B,B3:B]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ (((((swap @ B) @ A) @ A4) @ B3) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.79/1.55 thf(fact_152_eval__equality, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),P3:paraco414474393lle_fm,Q2:paraco414474393lle_fm]: ((((((paraco876059933e_eval @ I2) @ P3) = ((paraco876059933e_eval @ I2) @ Q2)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P3) @ Q2)) = (paraco2040174112le_Det @ $true))) & ((((paraco876059933e_eval @ I2) @ P3) != ((paraco876059933e_eval @ I2) @ Q2)) => (((((paraco876059933e_eval @ I2) @ P3) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ Q2))) & ((((paraco876059933e_eval @ I2) @ P3) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ Q2) = (paraco2040174112le_Det @ $true)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ P3))) & ((((paraco876059933e_eval @ I2) @ Q2) != (paraco2040174112le_Det @ $true)) => (((((paraco876059933e_eval @ I2) @ P3) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ Q2)))) & ((((paraco876059933e_eval @ I2) @ P3) != (paraco2040174112le_Det @ $false)) => (((((paraco876059933e_eval @ I2) @ Q2) = (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P3) @ Q2)) = ((paraco876059933e_eval @ I2) @ (paraco329115265le_Neg @ P3)))) & ((((paraco876059933e_eval @ I2) @ Q2) != (paraco2040174112le_Det @ $false)) => (((paraco876059933e_eval @ I2) @ ((paraco1628874225le_Eql @ P3) @ Q2)) = (paraco2040174112le_Det @ $false))))))))))))))).
% 2.79/1.55 thf(fact_153_fm_Oexhaust, axiom, (! [Y3:paraco414474393lle_fm]: (((! [X12:(list @ char)]: ((Y3 != (paraco27778325le_Pro @ X12)))) => ((Y3 != paraco251304083_Truth) => ((! [X33:paraco414474393lle_fm]: ((Y3 != (paraco329115265le_Neg @ X33)))) => ((! [X412:paraco414474393lle_fm,X422:paraco414474393lle_fm]: ((Y3 != ((paraco2100061555le_Con @ X412) @ X422)))) => ((! [X512:paraco414474393lle_fm,X522:paraco414474393lle_fm]: ((Y3 != ((paraco2084319816le_Eql @ X512) @ X522)))) => (~ ((! [X61:paraco414474393lle_fm,X62:paraco414474393lle_fm]: ((Y3 != ((paraco1628874225le_Eql @ X61) @ X62)))))))))))))).
% 2.79/1.55 thf(fact_154_fm_Oinduct, axiom, (! [P2:(paraco414474393lle_fm > $o),Fm:paraco414474393lle_fm]: (((! [X3:(list @ char)]: ((P2 @ (paraco27778325le_Pro @ X3)))) => ((P2 @ paraco251304083_Truth) => ((! [X3:paraco414474393lle_fm]: (((P2 @ X3) => (P2 @ (paraco329115265le_Neg @ X3))))) => ((! [X1a:paraco414474393lle_fm,X22:paraco414474393lle_fm]: (((P2 @ X1a) => ((P2 @ X22) => (P2 @ ((paraco2100061555le_Con @ X1a) @ X22)))))) => ((! [X1a:paraco414474393lle_fm,X22:paraco414474393lle_fm]: (((P2 @ X1a) => ((P2 @ X22) => (P2 @ ((paraco2084319816le_Eql @ X1a) @ X22)))))) => ((! [X1a:paraco414474393lle_fm,X22:paraco414474393lle_fm]: (((P2 @ X1a) => ((P2 @ X22) => (P2 @ ((paraco1628874225le_Eql @ X1a) @ X22)))))) => (P2 @ Fm)))))))))).
% 2.79/1.55 thf(fact_155_SUP__subset__mono, axiom, (! [A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ B),B2:(set @ B),F:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ A2) @ B2) => ((! [X3:B]: (((((member @ B) @ X3) @ A2) => (((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B2))))))))))).
% 2.79/1.55 thf(fact_156_UN__ball__bex__simps_I4_J, axiom, (! [F4:$tType,E:$tType,B2:(E > (set @ F4)),A2:(set @ E),P2:(F4 > $o)]: (((? [X:F4]: (((((member @ F4) @ X) @ ((complete_Sup_Sup @ (set @ F4)) @ ((((image @ E) @ (set @ F4)) @ B2) @ A2))) & (P2 @ X)))) = (? [X:E]: (((((member @ E) @ X) @ A2) & (? [Y4:F4]: (((((member @ F4) @ Y4) @ (B2 @ X)) & (P2 @ Y4))))))))))).
% 2.79/1.55 thf(fact_157_UN__ball__bex__simps_I2_J, axiom, (! [C:$tType,B:$tType,B2:(B > (set @ C)),A2:(set @ B),P2:(C > $o)]: (((! [X:C]: (((((member @ C) @ X) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image @ B) @ (set @ C)) @ B2) @ A2))) => (P2 @ X)))) = (! [X:B]: (((((member @ B) @ X) @ A2) => (! [Y4:C]: (((((member @ C) @ Y4) @ (B2 @ X)) => (P2 @ Y4))))))))))).
% 2.79/1.55 thf(fact_158_bex__UN, axiom, (! [A:$tType,B:$tType,B2:(B > (set @ A)),A2:(set @ B),P2:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))) & (P2 @ X)))) = (? [X:B]: (((((member @ B) @ X) @ A2) & (? [Y4:A]: (((((member @ A) @ Y4) @ (B2 @ X)) & (P2 @ Y4))))))))))).
% 2.79/1.55 thf(fact_159_ball__UN, axiom, (! [A:$tType,B:$tType,B2:(B > (set @ A)),A2:(set @ B),P2:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))) => (P2 @ X)))) = (! [X:B]: (((((member @ B) @ X) @ A2) => (! [Y4:A]: (((((member @ A) @ Y4) @ (B2 @ X)) => (P2 @ Y4))))))))))).
% 2.79/1.55 thf(fact_160_Union__Pow__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((complete_Sup_Sup @ (set @ A)) @ ((pow @ A) @ A2)) = A2)))).
% 2.79/1.55 thf(fact_161_fm_Oinject_I5_J, axiom, (! [X612:paraco414474393lle_fm,X622:paraco414474393lle_fm,Y61:paraco414474393lle_fm,Y62:paraco414474393lle_fm]: (((((paraco1628874225le_Eql @ X612) @ X622) = ((paraco1628874225le_Eql @ Y61) @ Y62)) = ((X612 = Y61) & (X622 = Y62)))))).
% 2.79/1.55 thf(fact_162_fm_Oinject_I1_J, axiom, (! [X1:(list @ char),Y1:(list @ char)]: ((((paraco27778325le_Pro @ X1) = (paraco27778325le_Pro @ Y1)) = (X1 = Y1))))).
% 2.79/1.55 thf(fact_163_Sup__apply, axiom, (! [B:$tType,A:$tType]: (((complete_Sup @ B) => ((complete_Sup_Sup @ (A > B)) = (^ [A3:(set @ (A > B)),X:A]: (((complete_Sup_Sup @ B) @ ((((image @ (A > B)) @ B) @ (^ [F2:(A > B)]: ((F2 @ X)))) @ A3))))))))).
% 2.79/1.55 thf(fact_164_UN__iff, axiom, (! [A:$tType,B:$tType,B3:A,B2:(B > (set @ A)),A2:(set @ B)]: (((((member @ A) @ B3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))) = (? [X:B]: (((((member @ B) @ X) @ A2) & (((member @ A) @ B3) @ (B2 @ X))))))))).
% 2.79/1.55 thf(fact_165_UN__I, axiom, (! [B:$tType,A:$tType,A4:A,A2:(set @ A),B3:B,B2:(A > (set @ B))]: (((((member @ A) @ A4) @ A2) => ((((member @ B) @ B3) @ (B2 @ A4)) => (((member @ B) @ B3) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B2) @ A2)))))))).
% 2.79/1.55 thf(fact_166_SUP__apply, axiom, (! [A:$tType,B:$tType,C:$tType]: (((complete_Sup @ A) => (! [F:(C > (B > A)),A2:(set @ C),X2:B]: (((((complete_Sup_Sup @ (B > A)) @ ((((image @ C) @ (B > A)) @ F) @ A2)) @ X2) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ (^ [Y4:C]: (((F @ Y4) @ X2)))) @ A2))))))))).
% 2.79/1.55 thf(fact_167_SUP__identity__eq, axiom, (! [A:$tType]: (((complete_Sup @ A) => (! [A2:(set @ A)]: ((((complete_Sup_Sup @ A) @ ((((image @ A) @ A) @ (^ [X:A]: (X))) @ A2)) = ((complete_Sup_Sup @ A) @ A2)))))))).
% 2.79/1.55 thf(fact_168_Sup__UNIV, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (((complete_Sup_Sup @ A) @ (top_top @ (set @ A))) = (top_top @ A)))))).
% 2.79/1.55 thf(fact_169_SUP__cong, axiom, (! [A:$tType,B:$tType]: (((complete_Sup @ A) => (! [A2:(set @ B),B2:(set @ B),C3:(B > A),D2:(B > A)]: (((A2 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C3 @ X3) = (D2 @ X3))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ C3) @ A2)) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ D2) @ B2))))))))))).
% 2.79/1.55 thf(fact_170_SUP__UNION, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comple187826305attice @ A) => (! [F:(B > A),G:(C > (set @ B)),A2:(set @ C)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C) @ (set @ B)) @ G) @ A2)))) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ (^ [Y4:C]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ (G @ Y4)))))) @ A2))))))))).
% 2.79/1.55 thf(fact_171_Sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((complete_Sup @ B) => ((complete_Sup_Sup @ (A > B)) = (^ [A3:(set @ (A > B)),X:A]: (((complete_Sup_Sup @ B) @ ((((image @ (A > B)) @ B) @ (^ [F2:(A > B)]: ((F2 @ X)))) @ A3))))))))).
% 2.79/1.55 thf(fact_172_SUP__UN__eq, axiom, (! [B:$tType,A:$tType,R2:(B > (set @ A)),S:(set @ B)]: ((((complete_Sup_Sup @ (A > $o)) @ ((((image @ B) @ (A > $o)) @ (^ [I:B,X:A]: ((((member @ A) @ X) @ (R2 @ I))))) @ S)) = (^ [X:A]: ((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ R2) @ S))))))))).
% 2.79/1.55 thf(fact_173_SUP__commute, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comple187826305attice @ A) => (! [F:(B > (C > A)),B2:(set @ C),A2:(set @ B)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [I:B]: (((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ (F @ I)) @ B2))))) @ A2)) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ (^ [J:C]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [I:B]: (((F @ I) @ J)))) @ A2))))) @ B2))))))))).
% 2.79/1.56 thf(fact_174_fm_Odistinct_I9_J, axiom, (! [X1:(list @ char),X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.79/1.56 thf(fact_175_Union__least, axiom, (! [A:$tType,A2:(set @ (set @ A)),C3:(set @ A)]: (((! [X6:(set @ A)]: (((((member @ (set @ A)) @ X6) @ A2) => (((ord_less_eq @ (set @ A)) @ X6) @ C3)))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ C3))))).
% 2.79/1.56 thf(fact_176_Union__upper, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ (set @ A))]: (((((member @ (set @ A)) @ B2) @ A2) => (((ord_less_eq @ (set @ A)) @ B2) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 2.79/1.56 thf(fact_177_Union__subsetI, axiom, (! [A:$tType,A2:(set @ (set @ A)),B2:(set @ (set @ A))]: (((! [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ A2) => (? [Y6:(set @ A)]: (((((member @ (set @ A)) @ Y6) @ B2) & (((ord_less_eq @ (set @ A)) @ X3) @ Y6))))))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ ((complete_Sup_Sup @ (set @ A)) @ B2)))))).
% 2.79/1.56 thf(fact_178_Sup__upper2, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [U2:A,A2:(set @ A),V:A]: (((((member @ A) @ U2) @ A2) => ((((ord_less_eq @ A) @ V) @ U2) => (((ord_less_eq @ A) @ V) @ ((complete_Sup_Sup @ A) @ A2)))))))))).
% 2.79/1.56 thf(fact_179_Sup__le__iff, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),B3:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ B3) = (! [X:A]: (((((member @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ B3))))))))))).
% 2.79/1.56 thf(fact_180_Sup__upper, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [X2:A,A2:(set @ A)]: (((((member @ A) @ X2) @ A2) => (((ord_less_eq @ A) @ X2) @ ((complete_Sup_Sup @ A) @ A2))))))))).
% 2.79/1.56 thf(fact_181_Sup__least, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),Z2:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ Z2)))))))).
% 2.79/1.56 thf(fact_182_Sup__mono, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),B2:(set @ A)]: (((! [A6:A]: (((((member @ A) @ A6) @ A2) => (? [X4:A]: (((((member @ A) @ X4) @ B2) & (((ord_less_eq @ A) @ A6) @ X4))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))).
% 2.79/1.56 thf(fact_183_Sup__eqI, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),X2:A]: (((! [Y5:A]: (((((member @ A) @ Y5) @ A2) => (((ord_less_eq @ A) @ Y5) @ X2)))) => ((! [Y5:A]: (((! [Z4:A]: (((((member @ A) @ Z4) @ A2) => (((ord_less_eq @ A) @ Z4) @ Y5)))) => (((ord_less_eq @ A) @ X2) @ Y5)))) => (((complete_Sup_Sup @ A) @ A2) = X2))))))))).
% 2.79/1.56 thf(fact_184_UN__UN__flatten, axiom, (! [A:$tType,B:$tType,C:$tType,C3:(B > (set @ A)),B2:(C > (set @ B)),A2:(set @ C)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ C3) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C) @ (set @ B)) @ B2) @ A2)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C) @ (set @ A)) @ (^ [Y4:C]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ C3) @ (B2 @ Y4)))))) @ A2)))))).
% 2.79/1.56 thf(fact_185_UN__E, axiom, (! [A:$tType,B:$tType,B3:A,B2:(B > (set @ A)),A2:(set @ B)]: (((((member @ A) @ B3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))) => (~ ((! [X3:B]: (((((member @ B) @ X3) @ A2) => (~ ((((member @ A) @ B3) @ (B2 @ X3))))))))))))).
% 2.79/1.56 thf(fact_186_UN__extend__simps_I9_J, axiom, (! [S2:$tType,R3:$tType,Q3:$tType,C3:(R3 > (set @ S2)),B2:(Q3 > (set @ R3)),A2:(set @ Q3)]: ((((complete_Sup_Sup @ (set @ S2)) @ ((((image @ Q3) @ (set @ S2)) @ (^ [X:Q3]: (((complete_Sup_Sup @ (set @ S2)) @ ((((image @ R3) @ (set @ S2)) @ C3) @ (B2 @ X)))))) @ A2)) = ((complete_Sup_Sup @ (set @ S2)) @ ((((image @ R3) @ (set @ S2)) @ C3) @ ((complete_Sup_Sup @ (set @ R3)) @ ((((image @ Q3) @ (set @ R3)) @ B2) @ A2)))))))).
% 2.79/1.56 thf(fact_187_UN__Pow__subset, axiom, (! [A:$tType,B:$tType,B2:(B > (set @ A)),A2:(set @ B)]: ((((ord_less_eq @ (set @ (set @ A))) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ ((((image @ B) @ (set @ (set @ A))) @ (^ [X:B]: (((pow @ A) @ (B2 @ X))))) @ A2))) @ ((pow @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B2) @ A2))))))).
% 2.79/1.56 thf(fact_188_SUP__eq, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ B),B2:(set @ C),F:(B > A),G:(C > A)]: (((! [I3:B]: (((((member @ B) @ I3) @ A2) => (? [X4:C]: (((((member @ C) @ X4) @ B2) & (((ord_less_eq @ A) @ (F @ I3)) @ (G @ X4)))))))) => ((! [J2:C]: (((((member @ C) @ J2) @ B2) => (? [X4:B]: (((((member @ B) @ X4) @ A2) & (((ord_less_eq @ A) @ (G @ J2)) @ (F @ X4)))))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2)) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B2))))))))))).
% 2.79/1.56 thf(fact_189_Sup__subset__mono, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))).
% 2.79/1.56 thf(fact_190_eval_Osimps_I1_J, axiom, (! [I2:((list @ char) > paraco415392788lle_tv),S3:(list @ char)]: ((((paraco876059933e_eval @ I2) @ (paraco27778325le_Pro @ S3)) = (I2 @ S3))))).
% 2.79/1.56 thf(fact_191_fm_Odistinct_I23_J, axiom, (! [X32:paraco414474393lle_fm,X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: (((paraco329115265le_Neg @ X32) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.79/1.56 thf(fact_192_fm_Odistinct_I3_J, axiom, (! [X1:(list @ char),X32:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != (paraco329115265le_Neg @ X32))))).
% 2.79/1.56 thf(fact_193_fm_Odistinct_I27_J, axiom, (! [X41:paraco414474393lle_fm,X42:paraco414474393lle_fm,X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((((paraco2100061555le_Con @ X41) @ X42) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.79/1.56 thf(fact_194_fm_Odistinct_I29_J, axiom, (! [X51:paraco414474393lle_fm,X52:paraco414474393lle_fm,X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((((paraco2084319816le_Eql @ X51) @ X52) != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.79/1.56 thf(fact_195_Union__mono, axiom, (! [A:$tType,A2:(set @ (set @ A)),B2:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ ((complete_Sup_Sup @ (set @ A)) @ B2)))))).
% 2.79/1.56 thf(fact_196_fm_Odistinct_I5_J, axiom, (! [X1:(list @ char),X41:paraco414474393lle_fm,X42:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2100061555le_Con @ X41) @ X42))))).
% 2.79/1.56 thf(fact_197_fm_Odistinct_I7_J, axiom, (! [X1:(list @ char),X51:paraco414474393lle_fm,X52:paraco414474393lle_fm]: (((paraco27778325le_Pro @ X1) != ((paraco2084319816le_Eql @ X51) @ X52))))).
% 2.79/1.56 thf(fact_198_fm_Odistinct_I17_J, axiom, (! [X612:paraco414474393lle_fm,X622:paraco414474393lle_fm]: ((paraco251304083_Truth != ((paraco1628874225le_Eql @ X612) @ X622))))).
% 2.79/1.56 thf(fact_199_SUP__eqI, axiom, (! [B:$tType,A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ B),F:(B > A),X2:A]: (((! [I3:B]: (((((member @ B) @ I3) @ A2) => (((ord_less_eq @ A) @ (F @ I3)) @ X2)))) => ((! [Y5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A2) => (((ord_less_eq @ A) @ (F @ I4)) @ Y5)))) => (((ord_less_eq @ A) @ X2) @ Y5)))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2)) = X2))))))))).
% 2.79/1.56 thf(fact_200_SUP__mono, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ B),B2:(set @ C),F:(B > A),G:(C > A)]: (((! [N3:B]: (((((member @ B) @ N3) @ A2) => (? [X4:C]: (((((member @ C) @ X4) @ B2) & (((ord_less_eq @ A) @ (F @ N3)) @ (G @ X4)))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B2)))))))))).
% 2.79/1.56 thf(fact_201_SUP__least, axiom, (! [B:$tType,A:$tType]: (((comple187826305attice @ A) => (! [A2:(set @ B),F:(B > A),U2:A]: (((! [I3:B]: (((((member @ B) @ I3) @ A2) => (((ord_less_eq @ A) @ (F @ I3)) @ U2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ U2)))))))).
% 2.79/1.56 thf(fact_202_SUP__mono_H, axiom, (! [A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [F:(B > A),G:(B > A),A2:(set @ B)]: (((! [X3:B]: ((((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ A2)))))))))).
% 2.79/1.56 thf(fact_203_SUP__upper, axiom, (! [A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [I2:B,A2:(set @ B),F:(B > A)]: (((((member @ B) @ I2) @ A2) => (((ord_less_eq @ A) @ (F @ I2)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2)))))))))).
% 2.79/1.56 thf(fact_204_SUP__le__iff, axiom, (! [A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [F:(B > A),A2:(set @ B),U2:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ U2) = (! [X:B]: (((((member @ B) @ X) @ A2) => (((ord_less_eq @ A) @ (F @ X)) @ U2))))))))))).
% 2.79/1.56 thf(fact_205_SUP__upper2, axiom, (! [A:$tType,B:$tType]: (((comple187826305attice @ A) => (! [I2:B,A2:(set @ B),U2:A,F:(B > A)]: (((((member @ B) @ I2) @ A2) => ((((ord_less_eq @ A) @ U2) @ (F @ I2)) => (((ord_less_eq @ A) @ U2) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))))))))))).
% 2.79/1.56 thf(fact_206_fm_Odistinct_I1_J, axiom, (! [X1:(list @ char)]: (((paraco27778325le_Pro @ X1) != paraco251304083_Truth)))).
% 2.79/1.56 thf(fact_207_subset__Pow__Union, axiom, (! [A:$tType,A2:(set @ (set @ A))]: ((((ord_less_eq @ (set @ (set @ A))) @ A2) @ ((pow @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 2.79/1.56 thf(fact_208_UN__extend__simps_I10_J, axiom, (! [V2:$tType,U3:$tType,T2:$tType,B2:(U3 > (set @ V2)),F:(T2 > U3),A2:(set @ T2)]: ((((complete_Sup_Sup @ (set @ V2)) @ ((((image @ T2) @ (set @ V2)) @ (^ [A5:T2]: ((B2 @ (F @ A5))))) @ A2)) = ((complete_Sup_Sup @ (set @ V2)) @ ((((image @ U3) @ (set @ V2)) @ B2) @ ((((image @ T2) @ U3) @ F) @ A2))))))).
% 2.79/1.56 thf(fact_209_image__UN, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),B2:(C > (set @ B)),A2:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C) @ (set @ B)) @ B2) @ A2))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C) @ (set @ A)) @ (^ [X:C]: (((((image @ B) @ A) @ F) @ (B2 @ X))))) @ A2)))))).
% 2.79/1.56 thf(fact_210_UN__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),F:(A > (set @ B)),G:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ (set @ B)) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ (set @ B)) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ F) @ A2))) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ G) @ B2)))))))).
% 2.79/1.56 thf(fact_211_UN__least, axiom, (! [A:$tType,B:$tType,A2:(set @ A),B2:(A > (set @ B)),C3:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ (set @ B)) @ (B2 @ X3)) @ C3)))) => (((ord_less_eq @ (set @ B)) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B2) @ A2))) @ C3))))).
% 2.79/1.56 thf(fact_212_UN__upper, axiom, (! [B:$tType,A:$tType,A4:A,A2:(set @ A),B2:(A > (set @ B))]: (((((member @ A) @ A4) @ A2) => (((ord_less_eq @ (set @ B)) @ (B2 @ A4)) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B2) @ A2))))))).
% 2.79/1.56 thf(fact_213_UN__image__subset, axiom, (! [C:$tType,A:$tType,B:$tType,F:(B > (set @ A)),G:(C > (set @ B)),X2:C,X5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F) @ (G @ X2)))) @ X5) = (((ord_less_eq @ (set @ B)) @ (G @ X2)) @ ((collect @ B) @ (^ [X:B]: ((((ord_less_eq @ (set @ A)) @ (F @ X)) @ X5))))))))).
% 2.79/1.56 thf(fact_214_UN__subset__iff, axiom, (! [A:$tType,B:$tType,A2:(B > (set @ A)),I5:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A2) @ I5))) @ B2) = (! [X:B]: (((((member @ B) @ X) @ I5) => (((ord_less_eq @ (set @ A)) @ (A2 @ X)) @ B2)))))))).
% 2.79/1.56 thf(fact_215_Union__UNIV, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = (top_top @ (set @ A)))))).
% 2.79/1.56 thf(fact_216_image__Union, axiom, (! [A:$tType,B:$tType,F:(B > A),S:(set @ (set @ B))]: ((((((image @ B) @ A) @ F) @ ((complete_Sup_Sup @ (set @ B)) @ S)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ S)))))).
% 2.79/1.56 thf(fact_217_UN__extend__simps_I8_J, axiom, (! [P4:$tType,O:$tType,B2:(O > (set @ P4)),A2:(set @ (set @ O))]: ((((complete_Sup_Sup @ (set @ P4)) @ ((((image @ (set @ O)) @ (set @ P4)) @ (^ [Y4:(set @ O)]: (((complete_Sup_Sup @ (set @ P4)) @ ((((image @ O) @ (set @ P4)) @ B2) @ Y4))))) @ A2)) = ((complete_Sup_Sup @ (set @ P4)) @ ((((image @ O) @ (set @ P4)) @ B2) @ ((complete_Sup_Sup @ (set @ O)) @ A2))))))).
% 2.79/1.56 thf(fact_218_top__finite__def, axiom, (! [A:$tType]: (((finite_lattice @ A) => ((top_top @ A) = ((complete_Sup_Sup @ A) @ (top_top @ (set @ A)))))))).
% 2.79/1.56 thf(fact_219_eval_Ocases, axiom, (! [X2:((product_prod @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm)]: (((! [I3:((list @ char) > paraco415392788lle_tv),S4:(list @ char)]: ((X2 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ (paraco27778325le_Pro @ S4))))) => ((! [I3:((list @ char) > paraco415392788lle_tv)]: ((X2 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ paraco251304083_Truth)))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P5:paraco414474393lle_fm]: ((X2 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ (paraco329115265le_Neg @ P5))))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P5:paraco414474393lle_fm,Q4:paraco414474393lle_fm]: ((X2 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco2100061555le_Con @ P5) @ Q4))))) => ((! [I3:((list @ char) > paraco415392788lle_tv),P5:paraco414474393lle_fm,Q4:paraco414474393lle_fm]: ((X2 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco2084319816le_Eql @ P5) @ Q4))))) => (~ ((! [I3:((list @ char) > paraco415392788lle_tv),P5:paraco414474393lle_fm,Q4:paraco414474393lle_fm]: ((X2 != ((((product_Pair @ ((list @ char) > paraco415392788lle_tv)) @ paraco414474393lle_fm) @ I3) @ ((paraco1628874225le_Eql @ P5) @ Q4))))))))))))))).
% 2.79/1.56 thf(fact_220_cSup__eq__maximum, axiom, (! [A:$tType]: (((condit378418413attice @ A) => (! [Z2:A,X5:(set @ A)]: (((((member @ A) @ Z2) @ X5) => ((! [X3:A]: (((((member @ A) @ X3) @ X5) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((complete_Sup_Sup @ A) @ X5) = Z2))))))))).
% 2.79/1.56 thf(fact_221_UN__ball__bex__simps_I3_J, axiom, (! [D:$tType,A2:(set @ (set @ D)),P2:(D > $o)]: (((? [X:D]: (((((member @ D) @ X) @ ((complete_Sup_Sup @ (set @ D)) @ A2)) & (P2 @ X)))) = (? [X:(set @ D)]: (((((member @ (set @ D)) @ X) @ A2) & (? [Y4:D]: (((((member @ D) @ Y4) @ X) & (P2 @ Y4))))))))))).
% 2.79/1.56 thf(fact_222_UN__ball__bex__simps_I1_J, axiom, (! [A:$tType,A2:(set @ (set @ A)),P2:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) => (P2 @ X)))) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A2) => (! [Y4:A]: (((((member @ A) @ Y4) @ X) => (P2 @ Y4))))))))))).
% 2.79/1.56 thf(fact_223_UnionI, axiom, (! [A:$tType,X5:(set @ A),C3:(set @ (set @ A)),A2:A]: (((((member @ (set @ A)) @ X5) @ C3) => ((((member @ A) @ A2) @ X5) => (((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3))))))).
% 2.79/1.56 thf(fact_224_Union__iff, axiom, (! [A:$tType,A2:A,C3:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3)) = (? [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ C3) & (((member @ A) @ A2) @ X)))))))).
% 2.79/1.56 thf(fact_225_Sup1__I, axiom, (! [A:$tType,P2:(A > $o),A2:(set @ (A > $o)),A4:A]: (((((member @ (A > $o)) @ P2) @ A2) => ((P2 @ A4) => (((complete_Sup_Sup @ (A > $o)) @ A2) @ A4)))))).
% 2.79/1.56 thf(fact_226_SUP1__I, axiom, (! [A:$tType,B:$tType,A4:A,A2:(set @ A),B2:(A > (B > $o)),B3:B]: (((((member @ A) @ A4) @ A2) => (((B2 @ A4) @ B3) => (((complete_Sup_Sup @ (B > $o)) @ ((((image @ A) @ (B > $o)) @ B2) @ A2)) @ B3)))))).
% 2.79/1.56 thf(fact_227_Sup__set__def, axiom, (! [A:$tType]: (((complete_Sup_Sup @ (set @ A)) = (^ [A3:(set @ (set @ A))]: (((collect @ A) @ (^ [X:A]: (((complete_Sup_Sup @ $o) @ ((((image @ (set @ A)) @ $o) @ ((member @ A) @ X)) @ A3))))))))))).
% 2.79/1.56 thf(fact_228_SUP1__E, axiom, (! [B:$tType,A:$tType,B2:(B > (A > $o)),A2:(set @ B),B3:A]: (((((complete_Sup_Sup @ (A > $o)) @ ((((image @ B) @ (A > $o)) @ B2) @ A2)) @ B3) => (~ ((! [X3:B]: (((((member @ B) @ X3) @ A2) => (~ (((B2 @ X3) @ B3)))))))))))).
% 2.79/1.56 thf(fact_229_SUP__Sup__eq2, axiom, (! [B:$tType,A:$tType,S:(set @ (set @ ((product_prod @ A) @ B)))]: ((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image @ (set @ ((product_prod @ A) @ B))) @ (A > (B > $o))) @ (^ [I:(set @ ((product_prod @ A) @ B)),X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ I)))) @ S)) = (^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ S)))))))).
% 2.79/1.56 thf(fact_230_Sup__SUP__eq, axiom, (! [A:$tType]: (((complete_Sup_Sup @ (A > $o)) = (^ [S5:(set @ (A > $o)),X:A]: ((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (A > $o)) @ (set @ A)) @ (collect @ A)) @ S5))))))))).
% 2.79/1.56 thf(fact_231_SUP__UN__eq2, axiom, (! [B:$tType,C:$tType,A:$tType,R2:(C > (set @ ((product_prod @ A) @ B))),S:(set @ C)]: ((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image @ C) @ (A > (B > $o))) @ (^ [I:C,X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ (R2 @ I))))) @ S)) = (^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ C) @ (set @ ((product_prod @ A) @ B))) @ R2) @ S))))))))).
% 2.79/1.56 thf(fact_232_SUP__Sup__eq, axiom, (! [A:$tType,S:(set @ (set @ A))]: ((((complete_Sup_Sup @ (A > $o)) @ ((((image @ (set @ A)) @ (A > $o)) @ (^ [I:(set @ A),X:A]: ((((member @ A) @ X) @ I)))) @ S)) = (^ [X:A]: ((((member @ A) @ X) @ ((complete_Sup_Sup @ (set @ A)) @ S)))))))).
% 2.79/1.56 thf(fact_233_Sup1__E, axiom, (! [A:$tType,A2:(set @ (A > $o)),A4:A]: (((((complete_Sup_Sup @ (A > $o)) @ A2) @ A4) => (~ ((! [P6:(A > $o)]: (((((member @ (A > $o)) @ P6) @ A2) => (~ ((P6 @ A4)))))))))))).
% 2.79/1.56 thf(fact_234_UnionE, axiom, (! [A:$tType,A2:A,C3:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3)) => (~ ((! [X6:(set @ A)]: (((((member @ A) @ A2) @ X6) => (~ ((((member @ (set @ A)) @ X6) @ C3)))))))))))).
% 2.79/1.56 thf(fact_235_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y5)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y5)) @ S3)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S3))))).
% 2.79/1.56 thf(fact_236_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 > $o))) @ (^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ R)))) @ (^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ S)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 2.79/1.56 thf(fact_237_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ R))) = (^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ S)))) = (R = S))))).
% 2.79/1.56 thf(fact_238_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S3:B,R:(set @ ((product_prod @ A) @ B)),S6:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S3)) @ R) => ((S6 = S3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S6)) @ R)))))).
% 2.79/1.56 thf(fact_239_top__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((top_top @ (A > (B > $o))) = (^ [X:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ (top_top @ (set @ ((product_prod @ A) @ B)))))))))).
% 2.79/1.56 thf(fact_240_cSup__eq, axiom, (! [A:$tType]: ((((condit378418413attice @ A) & (no_bot @ A)) => (! [X5:(set @ A),A4:A]: (((! [X3:A]: (((((member @ A) @ X3) @ X5) => (((ord_less_eq @ A) @ X3) @ A4)))) => ((! [Y5:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X5) => (((ord_less_eq @ A) @ X4) @ Y5)))) => (((ord_less_eq @ A) @ A4) @ Y5)))) => (((complete_Sup_Sup @ A) @ X5) = A4))))))))).
% 2.79/1.56 thf(fact_241_UN__constant__eq, axiom, (! [A:$tType,B:$tType,A4:A,A2:(set @ A),F:(A > (set @ B)),C2:(set @ B)]: (((((member @ A) @ A4) @ A2) => ((! [X3:A]: (((((member @ A) @ X3) @ A2) => ((F @ X3) = C2)))) => (((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ F) @ A2)) = C2)))))).
% 2.79/1.56 thf(fact_242_SUP__eq__top__iff, axiom, (! [B:$tType,A:$tType]: (((comple1035589618norder @ A) => (! [F:(B > A),A2:(set @ B)]: (((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2)) = (top_top @ A)) = (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (? [Y4:B]: (((((member @ B) @ Y4) @ A2) & (((ord_less @ A) @ X) @ (F @ Y4))))))))))))))).
% 2.79/1.56 thf(fact_243_psubsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ (set @ A)) @ A2) @ B2)))))).
% 2.79/1.56 thf(fact_244_Sup2__I, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o)),A2:(set @ (A > (B > $o))),A4:A,B3:B]: (((((member @ (A > (B > $o))) @ R2) @ A2) => (((R2 @ A4) @ B3) => ((((complete_Sup_Sup @ (A > (B > $o))) @ A2) @ A4) @ B3)))))).
% 2.79/1.56 thf(fact_245_top2I, axiom, (! [A:$tType,B:$tType,X2:A,Y3:B]: ((((top_top @ (A > (B > $o))) @ X2) @ Y3)))).
% 2.79/1.56 thf(fact_246_Sup__eq__top__iff, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (! [A2:(set @ A)]: (((((complete_Sup_Sup @ A) @ A2) = (top_top @ A)) = (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (? [Y4:A]: (((((member @ A) @ Y4) @ A2) & (((ord_less @ A) @ X) @ Y4)))))))))))))).
% 2.79/1.56 thf(fact_247_SUP2__I, axiom, (! [B:$tType,A:$tType,C:$tType,A4:A,A2:(set @ A),B2:(A > (B > (C > $o))),B3:B,C2:C]: (((((member @ A) @ A4) @ A2) => ((((B2 @ A4) @ B3) @ C2) => ((((complete_Sup_Sup @ (B > (C > $o))) @ ((((image @ A) @ (B > (C > $o))) @ B2) @ A2)) @ B3) @ C2)))))).
% 2.79/1.56 thf(fact_248_SUP2__E, axiom, (! [A:$tType,C:$tType,B:$tType,B2:(C > (A > (B > $o))),A2:(set @ C),B3:A,C2:B]: ((((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image @ C) @ (A > (B > $o))) @ B2) @ A2)) @ B3) @ C2) => (~ ((! [X3:C]: (((((member @ C) @ X3) @ A2) => (~ ((((B2 @ X3) @ B3) @ C2)))))))))))).
% 2.79/1.56 thf(fact_249_Sup2__E, axiom, (! [A:$tType,B:$tType,A2:(set @ (A > (B > $o))),A4:A,B3:B]: ((((((complete_Sup_Sup @ (A > (B > $o))) @ A2) @ A4) @ B3) => (~ ((! [R4:(A > (B > $o))]: (((((member @ (A > (B > $o))) @ R4) @ A2) => (~ (((R4 @ A4) @ B3)))))))))))).
% 2.79/1.56 thf(fact_250_Sup__bool__def, axiom, ((complete_Sup_Sup @ $o) = ((member @ $o) @ $true))).
% 2.79/1.56 thf(fact_251_less__Sup__iff, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (! [A4:A,S:(set @ A)]: (((((ord_less @ A) @ A4) @ ((complete_Sup_Sup @ A) @ S)) = (? [X:A]: (((((member @ A) @ X) @ S) & (((ord_less @ A) @ A4) @ X))))))))))).
% 2.79/1.56 thf(fact_252_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A4:A,B3:A,P2:(A > $o)]: (((((ord_less @ A) @ A4) @ B3) => ((P2 @ A4) => ((~ ((P2 @ B3))) => (? [C5:A]: (((((ord_less_eq @ A) @ A4) @ C5) & ((((ord_less_eq @ A) @ C5) @ B3) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A4) @ X4) & (((ord_less @ A) @ X4) @ C5)) => (P2 @ X4)))) & (! [D3:A]: (((! [X3:A]: ((((((ord_less_eq @ A) @ A4) @ X3) & (((ord_less @ A) @ X3) @ D3)) => (P2 @ X3)))) => (((ord_less_eq @ A) @ D3) @ C5)))))))))))))))))).
% 2.79/1.56 thf(fact_253_le__Sup__iff, axiom, (! [A:$tType]: (((comple1035589618norder @ A) => (! [X2:A,A2:(set @ A)]: (((((ord_less_eq @ A) @ X2) @ ((complete_Sup_Sup @ A) @ A2)) = (! [Y4:A]: (((((ord_less @ A) @ Y4) @ X2) => (? [X:A]: (((((member @ A) @ X) @ A2) & (((ord_less @ A) @ Y4) @ X)))))))))))))).
% 2.79/1.56 thf(fact_254_less__SUP__iff, axiom, (! [A:$tType,B:$tType]: (((comple1035589618norder @ A) => (! [A4:A,F:(B > A),A2:(set @ B)]: (((((ord_less @ A) @ A4) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) = (? [X:B]: (((((member @ B) @ X) @ A2) & (((ord_less @ A) @ A4) @ (F @ X)))))))))))).
% 2.79/1.56 thf(fact_255_SUP__lessD, axiom, (! [B:$tType,A:$tType]: (((comple187826305attice @ A) => (! [F:(B > A),A2:(set @ B),Y3:A,I2:B]: (((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A2))) @ Y3) => ((((member @ B) @ I2) @ A2) => (((ord_less @ A) @ (F @ I2)) @ Y3))))))))).
% 2.79/1.56 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (condit378418413attice @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple187826305attice @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Complete__Lattices_OSup, axiom, (! [A7:$tType,A8:$tType]: (((complete_Sup @ A8) => (complete_Sup @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A7:$tType,A8:$tType]: (((order_top @ A8) => (order_top @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: (((top @ A8) => (top @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.79/1.56 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ nat)).
% 2.79/1.56 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1, axiom, (condit378418413attice @ nat)).
% 2.79/1.56 thf(tcon_Nat_Onat___Complete__Lattices_OSup_2, axiom, (complete_Sup @ nat)).
% 2.79/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_3, axiom, (preorder @ nat)).
% 2.79/1.56 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.79/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_4, axiom, (order @ nat)).
% 2.79/1.56 thf(tcon_Nat_Onat___Orderings_Oord_5, axiom, (ord @ nat)).
% 2.79/1.56 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6, axiom, (! [A7:$tType]: ((condit378418413attice @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_7, axiom, (! [A7:$tType]: ((comple187826305attice @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Complete__Lattices_OSup_8, axiom, (! [A7:$tType]: ((complete_Sup @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Orderings_Oorder__top_9, axiom, (! [A7:$tType]: ((order_top @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_10, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Orderings_Oorder_11, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Orderings_Otop_12, axiom, (! [A7:$tType]: ((top @ (set @ A7))))).
% 2.79/1.56 thf(tcon_Set_Oset___Orderings_Oord_13, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.79/1.56 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_14, axiom, (condit378418413attice @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_15, axiom, (comple187826305attice @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Complete__Lattices_OSup_16, axiom, (complete_Sup @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Orderings_Oorder__top_17, axiom, (order_top @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_18, axiom, (preorder @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder_19, axiom, (linorder @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_20, axiom, (order @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Orderings_Otop_21, axiom, (top @ $o)).
% 2.79/1.56 thf(tcon_HOL_Obool___Orderings_Oord_22, axiom, (ord @ $o)).
% 2.79/1.56 thf(conj_0, conjecture, ((! [I:((list @ char) > paraco415392788lle_tv)]: ((((paraco876059933e_eval @ I) @ p) = (paraco2040174112le_Det @ $true)))) != (~ ((! [I:((list @ char) > paraco415392788lle_tv)]: (((((ord_less_eq @ (set @ paraco415392788lle_tv)) @ ((((image @ (list @ char)) @ paraco415392788lle_tv) @ I) @ (top_top @ (set @ (list @ char))))) @ (paraco506338565domain @ ((collect @ nat) @ (^ [N:nat]: ($true))))) => (((paraco876059933e_eval @ I) @ p) = (paraco2040174112le_Det @ $true))))))))).
% 2.79/1.56 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.SMTCspLT4x/DTF2DTF25569.p
% 2.79/1.56 ------------------------
% 2.93/1.61 ---- Cleaned THF ---
% 2.93/1.61 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
% 2.93/1.61 paraco415392788lle_tv: $tType ).
% 2.93/1.61
% 2.93/1.61 thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
% 2.93/1.61 paraco414474393lle_fm: $tType ).
% 2.93/1.61
% 2.93/1.61 thf(ty_t_Product__Type_Oprod,type,
% 2.93/1.61 product_prod: $tType > $tType > $tType ).
% 2.93/1.61
% 2.93/1.61 thf(ty_t_String_Ochar,type,
% 2.93/1.61 char: $tType ).
% 2.93/1.61
% 2.93/1.61 thf(ty_t_List_Olist,type,
% 2.93/1.61 list: $tType > $tType ).
% 2.93/1.61
% 2.93/1.61 thf(ty_t_Set_Oset,type,
% 2.93/1.61 set: $tType > $tType ).
% 2.93/1.61
% 2.93/1.61 thf(ty_t_Nat_Onat,type,
% 2.93/1.61 nat: $tType ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_HOL_Otype,type,
% 2.93/1.61 type:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Oord,type,
% 2.93/1.61 ord:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Otop,type,
% 2.93/1.61 top:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Oorder,type,
% 2.93/1.61 order:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Ono__bot,type,
% 2.93/1.61 no_bot:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Olinorder,type,
% 2.93/1.61 linorder:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Opreorder,type,
% 2.93/1.61 preorder:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Enum_Ofinite__lattice,type,
% 2.93/1.61 finite_lattice:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Orderings_Oorder__top,type,
% 2.93/1.61 order_top:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Complete__Lattices_OSup,type,
% 2.93/1.61 complete_Sup:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.93/1.61 comple187826305attice:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 2.93/1.61 comple1035589618norder:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 2.93/1.61 condit378418413attice:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.93/1.61 condit1037483654norder:
% 2.93/1.61 !>[A: $tType] : $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 2.93/1.61 complete_Sup_Sup:
% 2.93/1.61 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Finite__Set_OFpow,type,
% 2.93/1.61 finite_Fpow:
% 2.93/1.61 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Fun_Oswap,type,
% 2.93/1.61 swap:
% 2.93/1.61 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.93/1.61 ord_less:
% 2.93/1.61 !>[A: $tType] : ( A > A > $o ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.93/1.61 ord_less_eq:
% 2.93/1.61 !>[A: $tType] : ( A > A > $o ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.93/1.61 order_Greatest:
% 2.93/1.61 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.93/1.61 order_strict_mono:
% 2.93/1.61 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.93/1.61 top_top:
% 2.93/1.61 !>[A: $tType] : A ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Odomain,type,
% 2.93/1.61 paraco506338565domain: ( set @ nat ) > ( set @ paraco415392788lle_tv ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
% 2.93/1.61 paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
% 2.93/1.61 paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
% 2.93/1.61 paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
% 2.93/1.61 paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
% 2.93/1.61 paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
% 2.93/1.61 paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
% 2.93/1.61 paraco251304083_Truth: paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
% 2.93/1.61 paraco2040174112le_Det: $o > paraco415392788lle_tv ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
% 2.93/1.61 paraco152590079rec_tv:
% 2.93/1.61 !>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
% 2.93/1.61 paraco769098683_valid: paraco414474393lle_fm > $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
% 2.93/1.61 paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Product__Type_OPair,type,
% 2.93/1.61 product_Pair:
% 2.93/1.61 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Relation_OPowp,type,
% 2.93/1.61 powp:
% 2.93/1.61 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Set_OCollect,type,
% 2.93/1.61 collect:
% 2.93/1.61 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Set_OPow,type,
% 2.93/1.61 pow:
% 2.93/1.61 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_Set_Oimage,type,
% 2.93/1.61 image:
% 2.93/1.61 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_c_member,type,
% 2.93/1.61 member:
% 2.93/1.61 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.93/1.61
% 2.93/1.61 thf(sy_v_p,type,
% 2.93/1.61 p: paraco414474393lle_fm ).
% 2.93/1.61
% 2.93/1.61 thf(fact_0_universal__domain,axiom,
% 2.93/1.61 ( ( paraco506338565domain
% 2.93/1.61 @ ( collect @ nat
% 2.93/1.61 @ ^ [N: nat] : $true ) )
% 2.93/1.61 = ( collect @ paraco415392788lle_tv
% 2.93/1.61 @ ^ [X: paraco415392788lle_tv] : $true ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_1_tv_Oinject_I1_J,axiom,
% 2.93/1.61 ! [X1: $o,Y1: $o] :
% 2.93/1.61 ( ( ( paraco2040174112le_Det @ X1 )
% 2.93/1.61 = ( paraco2040174112le_Det @ Y1 ) )
% 2.93/1.61 = ( X1 = Y1 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_2_valid__in__def,axiom,
% 2.93/1.61 ( paraco2086025920lid_in
% 2.93/1.61 = ( ^ [U: set @ nat,P: paraco414474393lle_fm] :
% 2.93/1.61 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ paraco415392788lle_tv ) @ ( image @ ( list @ char ) @ paraco415392788lle_tv @ I @ ( top_top @ ( set @ ( list @ char ) ) ) ) @ ( paraco506338565domain @ U ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I @ P )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_3_image__ident,axiom,
% 2.93/1.61 ! [A: $tType,Y: set @ A] :
% 2.93/1.61 ( ( image @ A @ A
% 2.93/1.61 @ ^ [X: A] : X
% 2.93/1.61 @ Y )
% 2.93/1.61 = Y ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_4_UNIV__I,axiom,
% 2.93/1.61 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_5_iso__tuple__UNIV__I,axiom,
% 2.93/1.61 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_6_top__apply,axiom,
% 2.93/1.61 ! [C: $tType,D: $tType] :
% 2.93/1.61 ( ( top @ C )
% 2.93/1.61 => ( ( top_top @ ( D > C ) )
% 2.93/1.61 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_7_subsetI,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( member @ A @ X3 @ B2 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_8_subset__antisym,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.93/1.61 => ( A2 = B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_9_image__eqI,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B,A2: set @ B] :
% 2.93/1.61 ( ( B3
% 2.93/1.61 = ( F @ X2 ) )
% 2.93/1.61 => ( ( member @ B @ X2 @ A2 )
% 2.93/1.61 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_10_order__refl,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( preorder @ A )
% 2.93/1.61 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_11_rangeE,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B3: A,F: B > A] :
% 2.93/1.61 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 2.93/1.61 => ~ ! [X3: B] :
% 2.93/1.61 ( B3
% 2.93/1.61 != ( F @ X3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_12_range__composition,axiom,
% 2.93/1.61 ! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
% 2.93/1.61 ( ( image @ B @ A
% 2.93/1.61 @ ^ [X: B] : ( F @ ( G @ X ) )
% 2.93/1.61 @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_13_image__Collect__subsetI,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,P2: A > $o,F: A > B,B2: set @ B] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( P2 @ X3 )
% 2.93/1.61 => ( member @ B @ ( F @ X3 ) @ B2 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_14_top__set__def,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( top_top @ ( set @ A ) )
% 2.93/1.61 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_15_less__eq__set__def,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.61 = ( ^ [A3: set @ A,B4: set @ A] :
% 2.93/1.61 ( ord_less_eq @ ( A > $o )
% 2.93/1.61 @ ^ [X: A] : ( member @ A @ X @ A3 )
% 2.93/1.61 @ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_16_dual__order_Oantisym,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [B3: A,A4: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.93/1.61 => ( A4 = B3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_17_dual__order_Oeq__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.93/1.61 = ( ^ [A5: A,B5: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.93/1.61 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_18_dual__order_Otrans,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [B3: A,A4: A,C2: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.93/1.61 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_19_linorder__wlog,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( linorder @ A )
% 2.93/1.61 => ! [P2: A > A > $o,A4: A,B3: A] :
% 2.93/1.61 ( ! [A6: A,B6: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.93/1.61 => ( P2 @ A6 @ B6 ) )
% 2.93/1.61 => ( ! [A6: A,B6: A] :
% 2.93/1.61 ( ( P2 @ B6 @ A6 )
% 2.93/1.61 => ( P2 @ A6 @ B6 ) )
% 2.93/1.61 => ( P2 @ A4 @ B3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_20_dual__order_Orefl,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_21_order__trans,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( preorder @ A )
% 2.93/1.61 => ! [X2: A,Y3: A,Z2: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.93/1.61 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_22_order__class_Oorder_Oantisym,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [A4: A,B3: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.93/1.61 => ( A4 = B3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_23_ord__le__eq__trans,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ord @ A )
% 2.93/1.61 => ! [A4: A,B3: A,C2: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.93/1.61 => ( ( B3 = C2 )
% 2.93/1.61 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_24_ord__eq__le__trans,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ord @ A )
% 2.93/1.61 => ! [A4: A,B3: A,C2: A] :
% 2.93/1.61 ( ( A4 = B3 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.93/1.61 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_25_order__class_Oorder_Oeq__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.93/1.61 = ( ^ [A5: A,B5: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.93/1.61 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_26_antisym__conv,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [Y3: A,X2: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ Y3 @ X2 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.93/1.61 = ( X2 = Y3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_27_le__cases3,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( linorder @ A )
% 2.93/1.61 => ! [X2: A,Y3: A,Z2: A] :
% 2.93/1.61 ( ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.93/1.61 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.93/1.61 => ( ( ( ord_less_eq @ A @ Y3 @ X2 )
% 2.93/1.61 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.93/1.61 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.93/1.61 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.93/1.61 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.93/1.61 => ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
% 2.93/1.61 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.93/1.61 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 2.93/1.61 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.93/1.61 => ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_28_order_Otrans,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [A4: A,B3: A,C2: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.93/1.61 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_29_le__cases,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( linorder @ A )
% 2.93/1.61 => ! [X2: A,Y3: A] :
% 2.93/1.61 ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
% 2.93/1.61 => ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_30_eq__refl,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( preorder @ A )
% 2.93/1.61 => ! [X2: A,Y3: A] :
% 2.93/1.61 ( ( X2 = Y3 )
% 2.93/1.61 => ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_31_linear,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( linorder @ A )
% 2.93/1.61 => ! [X2: A,Y3: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.93/1.61 | ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_32_antisym,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [X2: A,Y3: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ Y3 @ X2 )
% 2.93/1.61 => ( X2 = Y3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_33_eq__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.93/1.61 = ( ^ [X: A,Y4: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.93/1.61 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_34_ord__le__eq__subst,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( ( ord @ B )
% 2.93/1.61 & ( ord @ A ) )
% 2.93/1.61 => ! [A4: A,B3: A,F: A > B,C2: B] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.93/1.61 => ( ( ( F @ B3 )
% 2.93/1.61 = C2 )
% 2.93/1.61 => ( ! [X3: A,Y5: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 2.93/1.61 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 2.93/1.61 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_35_ord__eq__le__subst,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( ( ord @ B )
% 2.93/1.61 & ( ord @ A ) )
% 2.93/1.61 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 2.93/1.61 ( ( A4
% 2.93/1.61 = ( F @ B3 ) )
% 2.93/1.61 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.93/1.61 => ( ! [X3: B,Y5: B] :
% 2.93/1.61 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 2.93/1.61 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_36_order__subst2,axiom,
% 2.93/1.61 ! [A: $tType,C: $tType] :
% 2.93/1.61 ( ( ( order @ C )
% 2.93/1.61 & ( order @ A ) )
% 2.93/1.61 => ! [A4: A,B3: A,F: A > C,C2: C] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.93/1.61 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 2.93/1.61 => ( ! [X3: A,Y5: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 2.93/1.61 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 2.93/1.61 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_37_order__subst1,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( ( order @ B )
% 2.93/1.61 & ( order @ A ) )
% 2.93/1.61 => ! [A4: A,F: B > A,B3: B,C2: B] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 2.93/1.61 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.93/1.61 => ( ! [X3: B,Y5: B] :
% 2.93/1.61 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 2.93/1.61 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_38_le__fun__def,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( ord @ B )
% 2.93/1.61 => ( ( ord_less_eq @ ( A > B ) )
% 2.93/1.61 = ( ^ [F2: A > B,G2: A > B] :
% 2.93/1.61 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_39_le__funI,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( ord @ B )
% 2.93/1.61 => ! [F: A > B,G: A > B] :
% 2.93/1.61 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.93/1.61 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_40_le__funE,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( ord @ B )
% 2.93/1.61 => ! [F: A > B,G: A > B,X2: A] :
% 2.93/1.61 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.93/1.61 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_41_le__funD,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( ord @ B )
% 2.93/1.61 => ! [F: A > B,G: A > B,X2: A] :
% 2.93/1.61 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.93/1.61 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_42_rev__image__eqI,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,X2: A,A2: set @ A,B3: B,F: A > B] :
% 2.93/1.61 ( ( member @ A @ X2 @ A2 )
% 2.93/1.61 => ( ( B3
% 2.93/1.61 = ( F @ X2 ) )
% 2.93/1.61 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_43_ball__imageD,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 => ( P2 @ X3 ) )
% 2.93/1.61 => ! [X4: B] :
% 2.93/1.61 ( ( member @ B @ X4 @ A2 )
% 2.93/1.61 => ( P2 @ ( F @ X4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_44_image__cong,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
% 2.93/1.61 ( ( M = N2 )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ N2 )
% 2.93/1.61 => ( ( F @ X3 )
% 2.93/1.61 = ( G @ X3 ) ) )
% 2.93/1.61 => ( ( image @ A @ B @ F @ M )
% 2.93/1.61 = ( image @ A @ B @ G @ N2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_45_mem__Collect__eq,axiom,
% 2.93/1.61 ! [A: $tType,A4: A,P2: A > $o] :
% 2.93/1.61 ( ( member @ A @ A4 @ ( collect @ A @ P2 ) )
% 2.93/1.61 = ( P2 @ A4 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_46_Collect__mem__eq,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] :
% 2.93/1.61 ( ( collect @ A
% 2.93/1.61 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 2.93/1.61 = A2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_47_Collect__cong,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( P2 @ X3 )
% 2.93/1.61 = ( Q @ X3 ) )
% 2.93/1.61 => ( ( collect @ A @ P2 )
% 2.93/1.61 = ( collect @ A @ Q ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_48_ext,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( F @ X3 )
% 2.93/1.61 = ( G @ X3 ) )
% 2.93/1.61 => ( F = G ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_49_bex__imageD,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 2.93/1.61 ( ? [X4: A] :
% 2.93/1.61 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 & ( P2 @ X4 ) )
% 2.93/1.61 => ? [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ A2 )
% 2.93/1.61 & ( P2 @ ( F @ X3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_50_image__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
% 2.93/1.61 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 = ( ? [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 & ( Z2
% 2.93/1.61 = ( F @ X ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_51_imageI,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,X2: A,A2: set @ A,F: A > B] :
% 2.93/1.61 ( ( member @ A @ X2 @ A2 )
% 2.93/1.61 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_52_Collect__mono__iff,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 2.93/1.61 = ( ! [X: A] :
% 2.93/1.61 ( ( P2 @ X )
% 2.93/1.61 => ( Q @ X ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_53_set__eq__subset,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 2.93/1.61 = ( ^ [A3: set @ A,B4: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.93/1.61 & ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_54_subset__trans,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_55_Collect__mono,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( P2 @ X3 )
% 2.93/1.61 => ( Q @ X3 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_56_subset__refl,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_57_subset__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.61 = ( ^ [A3: set @ A,B4: set @ A] :
% 2.93/1.61 ! [T: A] :
% 2.93/1.61 ( ( member @ A @ T @ A3 )
% 2.93/1.61 => ( member @ A @ T @ B4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_58_equalityD2,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( A2 = B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_59_equalityD1,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( A2 = B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_60_subset__eq,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) )
% 2.93/1.61 = ( ^ [A3: set @ A,B4: set @ A] :
% 2.93/1.61 ! [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ A3 )
% 2.93/1.61 => ( member @ A @ X @ B4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_61_equalityE,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( A2 = B2 )
% 2.93/1.61 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_62_subsetD,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ( member @ A @ C2 @ A2 )
% 2.93/1.61 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_63_in__mono,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ( member @ A @ X2 @ A2 )
% 2.93/1.61 => ( member @ A @ X2 @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_64_UNIV__witness,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_65_UNIV__eq__I,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] :
% 2.93/1.61 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( ( top_top @ ( set @ A ) )
% 2.93/1.61 = A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_66_Compr__image__eq,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 2.93/1.61 ( ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 & ( P2 @ X ) ) )
% 2.93/1.61 = ( image @ B @ A @ F
% 2.93/1.61 @ ( collect @ B
% 2.93/1.61 @ ^ [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 & ( P2 @ ( F @ X ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_67_image__image,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A2: set @ C] :
% 2.93/1.61 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A2 ) )
% 2.93/1.61 = ( image @ C @ A
% 2.93/1.61 @ ^ [X: C] : ( F @ ( G @ X ) )
% 2.93/1.61 @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_68_imageE,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
% 2.93/1.61 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 => ~ ! [X3: B] :
% 2.93/1.61 ( ( B3
% 2.93/1.61 = ( F @ X3 ) )
% 2.93/1.61 => ~ ( member @ B @ X3 @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_69_Collect__restrict,axiom,
% 2.93/1.61 ! [A: $tType,X5: set @ A,P2: A > $o] :
% 2.93/1.61 ( ord_less_eq @ ( set @ A )
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ X5 )
% 2.93/1.61 & ( P2 @ X ) ) )
% 2.93/1.61 @ X5 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_70_prop__restrict,axiom,
% 2.93/1.61 ! [A: $tType,X2: A,Z3: set @ A,X5: set @ A,P2: A > $o] :
% 2.93/1.61 ( ( member @ A @ X2 @ Z3 )
% 2.93/1.61 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ X5 )
% 2.93/1.61 & ( P2 @ X ) ) ) )
% 2.93/1.61 => ( P2 @ X2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_71_Collect__subset,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,P2: A > $o] :
% 2.93/1.61 ( ord_less_eq @ ( set @ A )
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ A2 )
% 2.93/1.61 & ( P2 @ X ) ) )
% 2.93/1.61 @ A2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_72_UNIV__def,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( top_top @ ( set @ A ) )
% 2.93/1.61 = ( collect @ A
% 2.93/1.61 @ ^ [X: A] : $true ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_73_top_Oextremum__uniqueI,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order_top @ A )
% 2.93/1.61 => ! [A4: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 2.93/1.61 => ( A4
% 2.93/1.61 = ( top_top @ A ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_74_top_Oextremum__unique,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order_top @ A )
% 2.93/1.61 => ! [A4: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 2.93/1.61 = ( A4
% 2.93/1.61 = ( top_top @ A ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_75_top__greatest,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order_top @ A )
% 2.93/1.61 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_76_subset__image__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 = ( ? [AA: set @ B] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 2.93/1.61 & ( B2
% 2.93/1.61 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_77_image__subset__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.93/1.61 = ( ! [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 => ( member @ A @ ( F @ X ) @ B2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_78_subset__imageE,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 => ~ ! [C4: set @ B] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ B ) @ C4 @ A2 )
% 2.93/1.61 => ( B2
% 2.93/1.61 != ( image @ B @ A @ F @ C4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_79_image__subsetI,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( member @ B @ ( F @ X3 ) @ B2 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_80_image__mono,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_81_range__eqI,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B3: A,F: B > A,X2: B] :
% 2.93/1.61 ( ( B3
% 2.93/1.61 = ( F @ X2 ) )
% 2.93/1.61 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_82_rangeI,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_83_subset__UNIV,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_84_surj__def,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,F: B > A] :
% 2.93/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) )
% 2.93/1.61 = ( ! [Y4: A] :
% 2.93/1.61 ? [X: B] :
% 2.93/1.61 ( Y4
% 2.93/1.61 = ( F @ X ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_85_surjI,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( G @ ( F @ X3 ) )
% 2.93/1.61 = X3 )
% 2.93/1.61 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_86_surjE,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 2.93/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) )
% 2.93/1.61 => ~ ! [X3: B] :
% 2.93/1.61 ( Y3
% 2.93/1.61 != ( F @ X3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_87_surjD,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 2.93/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) )
% 2.93/1.61 => ? [X3: B] :
% 2.93/1.61 ( Y3
% 2.93/1.61 = ( F @ X3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_88_all__subset__image,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: ( set @ A ) > $o] :
% 2.93/1.61 ( ( ! [B4: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 => ( P2 @ B4 ) ) )
% 2.93/1.61 = ( ! [B4: set @ B] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ B ) @ B4 @ A2 )
% 2.93/1.61 => ( P2 @ ( image @ B @ A @ F @ B4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_89_valid__def,axiom,
% 2.93/1.61 ( paraco769098683_valid
% 2.93/1.61 = ( ^ [P: paraco414474393lle_fm] :
% 2.93/1.61 ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.93/1.61 ( ( paraco876059933e_eval @ I @ P )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_90_pred__subset__eq,axiom,
% 2.93/1.61 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( A > $o )
% 2.93/1.61 @ ^ [X: A] : ( member @ A @ X @ R )
% 2.93/1.61 @ ^ [X: A] : ( member @ A @ X @ S ) )
% 2.93/1.61 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_91_subset__Collect__iff,axiom,
% 2.93/1.61 ! [A: $tType,B2: set @ A,A2: set @ A,P2: A > $o] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.93/1.61 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ A2 )
% 2.93/1.61 & ( P2 @ X ) ) ) )
% 2.93/1.61 = ( ! [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ B2 )
% 2.93/1.61 => ( P2 @ X ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_92_subset__CollectI,axiom,
% 2.93/1.61 ! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P2: A > $o] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ B2 )
% 2.93/1.61 => ( ( Q @ X3 )
% 2.93/1.61 => ( P2 @ X3 ) ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A )
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ B2 )
% 2.93/1.61 & ( Q @ X ) ) )
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ A2 )
% 2.93/1.61 & ( P2 @ X ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_93_conj__subset__def,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,P2: A > $o,Q: A > $o] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2
% 2.93/1.61 @ ( collect @ A
% 2.93/1.61 @ ^ [X: A] :
% 2.93/1.61 ( ( P2 @ X )
% 2.93/1.61 & ( Q @ X ) ) ) )
% 2.93/1.61 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P2 ) )
% 2.93/1.61 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_94_Sup_OSUP__identity__eq,axiom,
% 2.93/1.61 ! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
% 2.93/1.61 ( ( Sup
% 2.93/1.61 @ ( image @ A @ A
% 2.93/1.61 @ ^ [X: A] : X
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( Sup @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_95_predicate1I,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( P2 @ X3 )
% 2.93/1.61 => ( Q @ X3 ) )
% 2.93/1.61 => ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_96_top1I,axiom,
% 2.93/1.61 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_97_predicate1D,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,Q: A > $o,X2: A] :
% 2.93/1.61 ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 2.93/1.61 => ( ( P2 @ X2 )
% 2.93/1.61 => ( Q @ X2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_98_rev__predicate1D,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,X2: A,Q: A > $o] :
% 2.93/1.61 ( ( P2 @ X2 )
% 2.93/1.61 => ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 2.93/1.61 => ( Q @ X2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_99_Inf_OINF__cong,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D2: B > A,Inf: ( set @ A ) > A] :
% 2.93/1.61 ( ( A2 = B2 )
% 2.93/1.61 => ( ! [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ B2 )
% 2.93/1.61 => ( ( C3 @ X3 )
% 2.93/1.61 = ( D2 @ X3 ) ) )
% 2.93/1.61 => ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
% 2.93/1.61 = ( Inf @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_100_Sup_OSUP__cong,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D2: B > A,Sup: ( set @ A ) > A] :
% 2.93/1.61 ( ( A2 = B2 )
% 2.93/1.61 => ( ! [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ B2 )
% 2.93/1.61 => ( ( C3 @ X3 )
% 2.93/1.61 = ( D2 @ X3 ) ) )
% 2.93/1.61 => ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
% 2.93/1.61 = ( Sup @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_101_top__empty__eq,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( top_top @ ( A > $o ) )
% 2.93/1.61 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_102_Inf_OINF__identity__eq,axiom,
% 2.93/1.61 ! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
% 2.93/1.61 ( ( Inf
% 2.93/1.61 @ ( image @ A @ A
% 2.93/1.61 @ ^ [X: A] : X
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( Inf @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_103_top__conj_I2_J,axiom,
% 2.93/1.61 ! [A: $tType,P2: $o,X2: A] :
% 2.93/1.61 ( ( P2
% 2.93/1.61 & ( top_top @ ( A > $o ) @ X2 ) )
% 2.93/1.61 = P2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_104_top__conj_I1_J,axiom,
% 2.93/1.61 ! [A: $tType,X2: A,P2: $o] :
% 2.93/1.61 ( ( ( top_top @ ( A > $o ) @ X2 )
% 2.93/1.61 & P2 )
% 2.93/1.61 = P2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_105_eval_Osimps_I2_J,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv] :
% 2.93/1.61 ( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_106_image__Fpow__mono,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A2 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_107_eval_Osimps_I5_J,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ Q2 ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco876059933e_eval @ I2 @ Q2 ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_108_fm_Oinject_I4_J,axiom,
% 2.93/1.61 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.93/1.61 = ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
% 2.93/1.61 = ( ( X51 = Y51 )
% 2.93/1.61 & ( X52 = Y52 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_109_fm_Odistinct_I15_J,axiom,
% 2.93/1.61 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.93/1.61 ( paraco251304083_Truth
% 2.93/1.61 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_110_Fpow__mono,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_111_eval__equality__simplify,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco2040174112le_Det
% 2.93/1.61 @ ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ Q2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_112_image__Pow__mono,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_113_eval_Osimps_I4_J,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ Q2 ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ P3 ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco876059933e_eval @ I2 @ Q2 ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ Q2 ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ P3 ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_114_Powp__mono,axiom,
% 2.93/1.61 ! [A: $tType,A2: A > $o,B2: A > $o] :
% 2.93/1.61 ( ( ord_less_eq @ ( A > $o ) @ A2 @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A2 ) @ ( powp @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_115_Greatest__equality,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [P2: A > $o,X2: A] :
% 2.93/1.61 ( ( P2 @ X2 )
% 2.93/1.61 => ( ! [Y5: A] :
% 2.93/1.61 ( ( P2 @ Y5 )
% 2.93/1.61 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 2.93/1.61 => ( ( order_Greatest @ A @ P2 )
% 2.93/1.61 = X2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_116_GreatestI2__order,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( order @ A )
% 2.93/1.61 => ! [P2: A > $o,X2: A,Q: A > $o] :
% 2.93/1.61 ( ( P2 @ X2 )
% 2.93/1.61 => ( ! [Y5: A] :
% 2.93/1.61 ( ( P2 @ Y5 )
% 2.93/1.61 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( P2 @ X3 )
% 2.93/1.61 => ( ! [Y6: A] :
% 2.93/1.61 ( ( P2 @ Y6 )
% 2.93/1.61 => ( ord_less_eq @ A @ Y6 @ X3 ) )
% 2.93/1.61 => ( Q @ X3 ) ) )
% 2.93/1.61 => ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_117_fm_Oinject_I3_J,axiom,
% 2.93/1.61 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.93/1.61 = ( paraco2100061555le_Con @ Y41 @ Y42 ) )
% 2.93/1.61 = ( ( X41 = Y41 )
% 2.93/1.61 & ( X42 = Y42 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_118_PowI,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_119_Pow__iff,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.93/1.61 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_120_Pow__UNIV,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 2.93/1.61 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_121_Pow__top,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_122_Powp__Pow__eq,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] :
% 2.93/1.61 ( ( powp @ A
% 2.93/1.61 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 2.93/1.61 = ( ^ [X: set @ A] : ( member @ ( set @ A ) @ X @ ( pow @ A @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_123_PowD,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_124_Cantors__paradox,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] :
% 2.93/1.61 ~ ? [F3: A > ( set @ A )] :
% 2.93/1.61 ( ( image @ A @ ( set @ A ) @ F3 @ A2 )
% 2.93/1.61 = ( pow @ A @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_125_Pow__def,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( pow @ A )
% 2.93/1.61 = ( ^ [A3: set @ A] :
% 2.93/1.61 ( collect @ ( set @ A )
% 2.93/1.61 @ ^ [B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_126_fm_Odistinct_I25_J,axiom,
% 2.93/1.61 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.93/1.61 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_127_fm_Odistinct_I13_J,axiom,
% 2.93/1.61 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.93/1.61 ( paraco251304083_Truth
% 2.93/1.61 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_128_Pow__mono,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_129_image__Pow__surj,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.93/1.61 ( ( ( image @ B @ A @ F @ A2 )
% 2.93/1.61 = B2 )
% 2.93/1.61 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
% 2.93/1.61 = ( pow @ A @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_130_conjunction,axiom,
% 2.93/1.61 ! [P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
% 2.93/1.61 = ( ( paraco769098683_valid @ P3 )
% 2.93/1.61 & ( paraco769098683_valid @ Q2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_131_Fpow__subset__Pow,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( pow @ A @ A2 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_132_tv_Osimps_I7_J,axiom,
% 2.93/1.61 ! [A: $tType,F1: $o > A,F22: nat > A,X1: $o] :
% 2.93/1.61 ( ( paraco152590079rec_tv @ A @ F1 @ F22 @ ( paraco2040174112le_Det @ X1 ) )
% 2.93/1.61 = ( F1 @ X1 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_133_strict__mono__inv,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( ( linorder @ B )
% 2.93/1.61 & ( linorder @ A ) )
% 2.93/1.61 => ! [F: A > B,G: B > A] :
% 2.93/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.93/1.61 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.93/1.61 = ( top_top @ ( set @ B ) ) )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( G @ ( F @ X3 ) )
% 2.93/1.61 = X3 )
% 2.93/1.61 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_134_eval__negation,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $false ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ P3 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_135_surj__swap__iff,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: B,B3: B,F: B > A] :
% 2.93/1.61 ( ( ( image @ B @ A @ ( swap @ B @ A @ A4 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) )
% 2.93/1.61 = ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_136_swap__nilpotent,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: A,B3: A,F: A > B] :
% 2.93/1.61 ( ( swap @ A @ B @ A4 @ B3 @ ( swap @ A @ B @ A4 @ B3 @ F ) )
% 2.93/1.61 = F ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_137_swap__self,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: A,F: A > B] :
% 2.93/1.61 ( ( swap @ A @ B @ A4 @ A4 @ F )
% 2.93/1.61 = F ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_138_swap__apply_I1_J,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A4: B,B3: B,F: B > A] :
% 2.93/1.61 ( ( swap @ B @ A @ A4 @ B3 @ F @ A4 )
% 2.93/1.61 = ( F @ B3 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_139_swap__apply_I2_J,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A4: B,B3: B,F: B > A] :
% 2.93/1.61 ( ( swap @ B @ A @ A4 @ B3 @ F @ B3 )
% 2.93/1.61 = ( F @ A4 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_140_swap__apply_I3_J,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C2: B,A4: B,B3: B,F: B > A] :
% 2.93/1.61 ( ( C2 != A4 )
% 2.93/1.61 => ( ( C2 != B3 )
% 2.93/1.61 => ( ( swap @ B @ A @ A4 @ B3 @ F @ C2 )
% 2.93/1.61 = ( F @ C2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_141_fm_Oinject_I2_J,axiom,
% 2.93/1.61 ! [X32: paraco414474393lle_fm,Y32: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( paraco329115265le_Neg @ X32 )
% 2.93/1.61 = ( paraco329115265le_Neg @ Y32 ) )
% 2.93/1.61 = ( X32 = Y32 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_142_swap__image__eq,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: A,F: A > B] :
% 2.93/1.61 ( ( member @ A @ A4 @ A2 )
% 2.93/1.61 => ( ( member @ A @ B3 @ A2 )
% 2.93/1.61 => ( ( image @ A @ B @ ( swap @ A @ B @ A4 @ B3 @ F ) @ A2 )
% 2.93/1.61 = ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_143_swap__commute,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( swap @ A @ B )
% 2.93/1.61 = ( ^ [A5: A,B5: A] : ( swap @ A @ B @ B5 @ A5 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_144_swap__triple,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: A,C2: A,B3: A,F: A > B] :
% 2.93/1.61 ( ( A4 != C2 )
% 2.93/1.61 => ( ( B3 != C2 )
% 2.93/1.61 => ( ( swap @ A @ B @ A4 @ B3 @ ( swap @ A @ B @ B3 @ C2 @ ( swap @ A @ B @ A4 @ B3 @ F ) ) )
% 2.93/1.61 = ( swap @ A @ B @ A4 @ C2 @ F ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_145_strict__mono__eq,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( ( linorder @ A )
% 2.93/1.61 & ( order @ B ) )
% 2.93/1.61 => ! [F: A > B,X2: A,Y3: A] :
% 2.93/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.93/1.61 => ( ( ( F @ X2 )
% 2.93/1.61 = ( F @ Y3 ) )
% 2.93/1.61 = ( X2 = Y3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_146_double__negation,axiom,
% 2.93/1.61 ( paraco876059933e_eval
% 2.93/1.61 = ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_147_strict__mono__less__eq,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( ( linorder @ A )
% 2.93/1.61 & ( order @ B ) )
% 2.93/1.61 => ! [F: A > B,X2: A,Y3: A] :
% 2.93/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.93/1.61 => ( ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.93/1.61 = ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_148_fm_Odistinct_I21_J,axiom,
% 2.93/1.61 ! [X32: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco329115265le_Neg @ X32 )
% 2.93/1.61 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_149_fm_Odistinct_I19_J,axiom,
% 2.93/1.61 ! [X32: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco329115265le_Neg @ X32 )
% 2.93/1.61 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_150_fm_Odistinct_I11_J,axiom,
% 2.93/1.61 ! [X32: paraco414474393lle_fm] :
% 2.93/1.61 ( paraco251304083_Truth
% 2.93/1.61 != ( paraco329115265le_Neg @ X32 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_151_surj__imp__surj__swap,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,F: B > A,A4: B,B3: B] :
% 2.93/1.61 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) )
% 2.93/1.61 => ( ( image @ B @ A @ ( swap @ B @ A @ A4 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_152_eval__equality,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ Q2 ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco876059933e_eval @ I2 @ Q2 ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ Q2 ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ P3 ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $true ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q2 ) ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ P3 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $false ) )
% 2.93/1.61 => ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) ) ) )
% 2.93/1.61 & ( ( ( paraco876059933e_eval @ I2 @ Q2 )
% 2.93/1.61 != ( paraco2040174112le_Det @ $false ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
% 2.93/1.61 = ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_153_fm_Oexhaust,axiom,
% 2.93/1.61 ! [Y3: paraco414474393lle_fm] :
% 2.93/1.61 ( ! [X12: list @ char] :
% 2.93/1.61 ( Y3
% 2.93/1.61 != ( paraco27778325le_Pro @ X12 ) )
% 2.93/1.61 => ( ( Y3 != paraco251304083_Truth )
% 2.93/1.61 => ( ! [X33: paraco414474393lle_fm] :
% 2.93/1.61 ( Y3
% 2.93/1.61 != ( paraco329115265le_Neg @ X33 ) )
% 2.93/1.61 => ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
% 2.93/1.61 ( Y3
% 2.93/1.61 != ( paraco2100061555le_Con @ X412 @ X422 ) )
% 2.93/1.61 => ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
% 2.93/1.61 ( Y3
% 2.93/1.61 != ( paraco2084319816le_Eql @ X512 @ X522 ) )
% 2.93/1.61 => ~ ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
% 2.93/1.61 ( Y3
% 2.93/1.61 != ( paraco1628874225le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_154_fm_Oinduct,axiom,
% 2.93/1.61 ! [P2: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
% 2.93/1.61 ( ! [X3: list @ char] : ( P2 @ ( paraco27778325le_Pro @ X3 ) )
% 2.93/1.61 => ( ( P2 @ paraco251304083_Truth )
% 2.93/1.61 => ( ! [X3: paraco414474393lle_fm] :
% 2.93/1.61 ( ( P2 @ X3 )
% 2.93/1.61 => ( P2 @ ( paraco329115265le_Neg @ X3 ) ) )
% 2.93/1.61 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.93/1.61 ( ( P2 @ X1a )
% 2.93/1.61 => ( ( P2 @ X22 )
% 2.93/1.61 => ( P2 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
% 2.93/1.61 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.93/1.61 ( ( P2 @ X1a )
% 2.93/1.61 => ( ( P2 @ X22 )
% 2.93/1.61 => ( P2 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
% 2.93/1.61 => ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
% 2.93/1.61 ( ( P2 @ X1a )
% 2.93/1.61 => ( ( P2 @ X22 )
% 2.93/1.61 => ( P2 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
% 2.93/1.61 => ( P2 @ Fm ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_155_SUP__subset__mono,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ B,B2: set @ B,F: B > A,G: B > A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
% 2.93/1.61 => ( ! [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_156_UN__ball__bex__simps_I4_J,axiom,
% 2.93/1.61 ! [F4: $tType,E: $tType,B2: E > ( set @ F4 ),A2: set @ E,P2: F4 > $o] :
% 2.93/1.61 ( ( ? [X: F4] :
% 2.93/1.61 ( ( member @ F4 @ X @ ( complete_Sup_Sup @ ( set @ F4 ) @ ( image @ E @ ( set @ F4 ) @ B2 @ A2 ) ) )
% 2.93/1.61 & ( P2 @ X ) ) )
% 2.93/1.61 = ( ? [X: E] :
% 2.93/1.61 ( ( member @ E @ X @ A2 )
% 2.93/1.61 & ? [Y4: F4] :
% 2.93/1.61 ( ( member @ F4 @ Y4 @ ( B2 @ X ) )
% 2.93/1.61 & ( P2 @ Y4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_157_UN__ball__bex__simps_I2_J,axiom,
% 2.93/1.61 ! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P2: C > $o] :
% 2.93/1.61 ( ( ! [X: C] :
% 2.93/1.61 ( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
% 2.93/1.61 => ( P2 @ X ) ) )
% 2.93/1.61 = ( ! [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 => ! [Y4: C] :
% 2.93/1.61 ( ( member @ C @ Y4 @ ( B2 @ X ) )
% 2.93/1.61 => ( P2 @ Y4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_158_bex__UN,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
% 2.93/1.61 ( ( ? [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 2.93/1.61 & ( P2 @ X ) ) )
% 2.93/1.61 = ( ? [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 & ? [Y4: A] :
% 2.93/1.61 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 2.93/1.61 & ( P2 @ Y4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_159_ball__UN,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
% 2.93/1.61 ( ( ! [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 2.93/1.61 => ( P2 @ X ) ) )
% 2.93/1.61 = ( ! [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 => ! [Y4: A] :
% 2.93/1.61 ( ( member @ A @ Y4 @ ( B2 @ X ) )
% 2.93/1.61 => ( P2 @ Y4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_160_Union__Pow__eq,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
% 2.93/1.61 = A2 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_161_fm_Oinject_I5_J,axiom,
% 2.93/1.61 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
% 2.93/1.61 ( ( ( paraco1628874225le_Eql @ X612 @ X622 )
% 2.93/1.61 = ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
% 2.93/1.61 = ( ( X612 = Y61 )
% 2.93/1.61 & ( X622 = Y62 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_162_fm_Oinject_I1_J,axiom,
% 2.93/1.61 ! [X1: list @ char,Y1: list @ char] :
% 2.93/1.61 ( ( ( paraco27778325le_Pro @ X1 )
% 2.93/1.61 = ( paraco27778325le_Pro @ Y1 ) )
% 2.93/1.61 = ( X1 = Y1 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_163_Sup__apply,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( complete_Sup @ B )
% 2.93/1.61 => ( ( complete_Sup_Sup @ ( A > B ) )
% 2.93/1.61 = ( ^ [A3: set @ ( A > B ),X: A] :
% 2.93/1.61 ( complete_Sup_Sup @ B
% 2.93/1.61 @ ( image @ ( A > B ) @ B
% 2.93/1.61 @ ^ [F2: A > B] : ( F2 @ X )
% 2.93/1.61 @ A3 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_164_UN__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
% 2.93/1.61 ( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 2.93/1.61 = ( ? [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 & ( member @ A @ B3 @ ( B2 @ X ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_165_UN__I,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
% 2.93/1.61 ( ( member @ A @ A4 @ A2 )
% 2.93/1.61 => ( ( member @ B @ B3 @ ( B2 @ A4 ) )
% 2.93/1.61 => ( member @ B @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_166_SUP__apply,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C: $tType] :
% 2.93/1.61 ( ( complete_Sup @ A )
% 2.93/1.61 => ! [F: C > B > A,A2: set @ C,X2: B] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F @ A2 ) @ X2 )
% 2.93/1.61 = ( complete_Sup_Sup @ A
% 2.93/1.61 @ ( image @ C @ A
% 2.93/1.61 @ ^ [Y4: C] : ( F @ Y4 @ X2 )
% 2.93/1.61 @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_167_SUP__identity__eq,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( complete_Sup @ A )
% 2.93/1.61 => ! [A2: set @ A] :
% 2.93/1.61 ( ( complete_Sup_Sup @ A
% 2.93/1.61 @ ( image @ A @ A
% 2.93/1.61 @ ^ [X: A] : X
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ A @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_168_Sup__UNIV,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 2.93/1.61 = ( top_top @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_169_SUP__cong,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( complete_Sup @ A )
% 2.93/1.61 => ! [A2: set @ B,B2: set @ B,C3: B > A,D2: B > A] :
% 2.93/1.61 ( ( A2 = B2 )
% 2.93/1.61 => ( ! [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ B2 )
% 2.93/1.61 => ( ( C3 @ X3 )
% 2.93/1.61 = ( D2 @ X3 ) ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C3 @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_170_SUP__UNION,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [F: B > A,G: C > ( set @ B ),A2: set @ C] :
% 2.93/1.61 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G @ A2 ) ) ) )
% 2.93/1.61 = ( complete_Sup_Sup @ A
% 2.93/1.61 @ ( image @ C @ A
% 2.93/1.61 @ ^ [Y4: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G @ Y4 ) ) )
% 2.93/1.61 @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_171_Sup__fun__def,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( complete_Sup @ B )
% 2.93/1.61 => ( ( complete_Sup_Sup @ ( A > B ) )
% 2.93/1.61 = ( ^ [A3: set @ ( A > B ),X: A] :
% 2.93/1.61 ( complete_Sup_Sup @ B
% 2.93/1.61 @ ( image @ ( A > B ) @ B
% 2.93/1.61 @ ^ [F2: A > B] : ( F2 @ X )
% 2.93/1.61 @ A3 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_172_SUP__UN__eq,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > $o )
% 2.93/1.61 @ ( image @ B @ ( A > $o )
% 2.93/1.61 @ ^ [I: B,X: A] : ( member @ A @ X @ ( R2 @ I ) )
% 2.93/1.61 @ S ) )
% 2.93/1.61 = ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_173_SUP__commute,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [F: B > C > A,B2: set @ C,A2: set @ B] :
% 2.93/1.61 ( ( complete_Sup_Sup @ A
% 2.93/1.61 @ ( image @ B @ A
% 2.93/1.61 @ ^ [I: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F @ I ) @ B2 ) )
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ A
% 2.93/1.61 @ ( image @ C @ A
% 2.93/1.61 @ ^ [J: C] :
% 2.93/1.61 ( complete_Sup_Sup @ A
% 2.93/1.61 @ ( image @ B @ A
% 2.93/1.61 @ ^ [I: B] : ( F @ I @ J )
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 @ B2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_174_fm_Odistinct_I9_J,axiom,
% 2.93/1.61 ! [X1: list @ char,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco27778325le_Pro @ X1 )
% 2.93/1.61 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_175_Union__least,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
% 2.93/1.61 ( ! [X6: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ X6 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ X6 @ C3 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_176_Union__upper,axiom,
% 2.93/1.61 ! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ B2 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_177_Union__subsetI,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 2.93/1.61 ( ! [X3: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ X3 @ A2 )
% 2.93/1.61 => ? [Y6: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ Y6 @ B2 )
% 2.93/1.61 & ( ord_less_eq @ ( set @ A ) @ X3 @ Y6 ) ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_178_Sup__upper2,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [U2: A,A2: set @ A,V: A] :
% 2.93/1.61 ( ( member @ A @ U2 @ A2 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ V @ U2 )
% 2.93/1.61 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_179_Sup__le__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ A,B3: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
% 2.93/1.61 = ( ! [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_180_Sup__upper,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [X2: A,A2: set @ A] :
% 2.93/1.61 ( ( member @ A @ X2 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_181_Sup__least,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ A,Z2: A] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_182_Sup__mono,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ! [A6: A] :
% 2.93/1.61 ( ( member @ A @ A6 @ A2 )
% 2.93/1.61 => ? [X4: A] :
% 2.93/1.61 ( ( member @ A @ X4 @ B2 )
% 2.93/1.61 & ( ord_less_eq @ A @ A6 @ X4 ) ) )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_183_Sup__eqI,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ A,X2: A] :
% 2.93/1.61 ( ! [Y5: A] :
% 2.93/1.61 ( ( member @ A @ Y5 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 2.93/1.61 => ( ! [Y5: A] :
% 2.93/1.61 ( ! [Z4: A] :
% 2.93/1.61 ( ( member @ A @ Z4 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ Z4 @ Y5 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ A2 )
% 2.93/1.61 = X2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_184_UN__UN__flatten,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C: $tType,C3: B > ( set @ A ),B2: C > ( set @ B ),A2: set @ C] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) ) )
% 2.93/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.93/1.61 @ ( image @ C @ ( set @ A )
% 2.93/1.61 @ ^ [Y4: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( B2 @ Y4 ) ) )
% 2.93/1.61 @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_185_UN__E,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
% 2.93/1.61 ( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
% 2.93/1.61 => ~ ! [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ A2 )
% 2.93/1.61 => ~ ( member @ A @ B3 @ ( B2 @ X3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_186_UN__extend__simps_I9_J,axiom,
% 2.93/1.61 ! [S2: $tType,R3: $tType,Q3: $tType,C3: R3 > ( set @ S2 ),B2: Q3 > ( set @ R3 ),A2: set @ Q3] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ S2 )
% 2.93/1.61 @ ( image @ Q3 @ ( set @ S2 )
% 2.93/1.61 @ ^ [X: Q3] : ( complete_Sup_Sup @ ( set @ S2 ) @ ( image @ R3 @ ( set @ S2 ) @ C3 @ ( B2 @ X ) ) )
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ ( set @ S2 ) @ ( image @ R3 @ ( set @ S2 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R3 ) @ ( image @ Q3 @ ( set @ R3 ) @ B2 @ A2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_187_UN__Pow__subset,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
% 2.93/1.61 ( ord_less_eq @ ( set @ ( set @ A ) )
% 2.93/1.61 @ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
% 2.93/1.61 @ ( image @ B @ ( set @ ( set @ A ) )
% 2.93/1.61 @ ^ [X: B] : ( pow @ A @ ( B2 @ X ) )
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_188_SUP__eq,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,C: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
% 2.93/1.61 ( ! [I3: B] :
% 2.93/1.61 ( ( member @ B @ I3 @ A2 )
% 2.93/1.61 => ? [X4: C] :
% 2.93/1.61 ( ( member @ C @ X4 @ B2 )
% 2.93/1.61 & ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
% 2.93/1.61 => ( ! [J2: C] :
% 2.93/1.61 ( ( member @ C @ J2 @ B2 )
% 2.93/1.61 => ? [X4: B] :
% 2.93/1.61 ( ( member @ B @ X4 @ A2 )
% 2.93/1.61 & ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_189_Sup__subset__mono,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_190_eval_Osimps_I1_J,axiom,
% 2.93/1.61 ! [I2: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
% 2.93/1.61 ( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S3 ) )
% 2.93/1.61 = ( I2 @ S3 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_191_fm_Odistinct_I23_J,axiom,
% 2.93/1.61 ! [X32: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco329115265le_Neg @ X32 )
% 2.93/1.61 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_192_fm_Odistinct_I3_J,axiom,
% 2.93/1.61 ! [X1: list @ char,X32: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco27778325le_Pro @ X1 )
% 2.93/1.61 != ( paraco329115265le_Neg @ X32 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_193_fm_Odistinct_I27_J,axiom,
% 2.93/1.61 ! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco2100061555le_Con @ X41 @ X42 )
% 2.93/1.61 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_194_fm_Odistinct_I29_J,axiom,
% 2.93/1.61 ! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco2084319816le_Eql @ X51 @ X52 )
% 2.93/1.61 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_195_Union__mono,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_196_fm_Odistinct_I5_J,axiom,
% 2.93/1.61 ! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco27778325le_Pro @ X1 )
% 2.93/1.61 != ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_197_fm_Odistinct_I7_J,axiom,
% 2.93/1.61 ! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
% 2.93/1.61 ( ( paraco27778325le_Pro @ X1 )
% 2.93/1.61 != ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_198_fm_Odistinct_I17_J,axiom,
% 2.93/1.61 ! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
% 2.93/1.61 ( paraco251304083_Truth
% 2.93/1.61 != ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_199_SUP__eqI,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ B,F: B > A,X2: A] :
% 2.93/1.61 ( ! [I3: B] :
% 2.93/1.61 ( ( member @ B @ I3 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ I3 ) @ X2 ) )
% 2.93/1.61 => ( ! [Y5: A] :
% 2.93/1.61 ( ! [I4: B] :
% 2.93/1.61 ( ( member @ B @ I4 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ I4 ) @ Y5 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 = X2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_200_SUP__mono,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,C: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
% 2.93/1.61 ( ! [N3: B] :
% 2.93/1.61 ( ( member @ B @ N3 @ A2 )
% 2.93/1.61 => ? [X4: C] :
% 2.93/1.61 ( ( member @ C @ X4 @ B2 )
% 2.93/1.61 & ( ord_less_eq @ A @ ( F @ N3 ) @ ( G @ X4 ) ) ) )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_201_SUP__least,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [A2: set @ B,F: B > A,U2: A] :
% 2.93/1.61 ( ! [I3: B] :
% 2.93/1.61 ( ( member @ B @ I3 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ I3 ) @ U2 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_202_SUP__mono_H,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [F: B > A,G: B > A,A2: set @ B] :
% 2.93/1.61 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_203_SUP__upper,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [I2: B,A2: set @ B,F: B > A] :
% 2.93/1.61 ( ( member @ B @ I2 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ I2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_204_SUP__le__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [F: B > A,A2: set @ B,U2: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U2 )
% 2.93/1.61 = ( ! [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 => ( ord_less_eq @ A @ ( F @ X ) @ U2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_205_SUP__upper2,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [I2: B,A2: set @ B,U2: A,F: B > A] :
% 2.93/1.61 ( ( member @ B @ I2 @ A2 )
% 2.93/1.61 => ( ( ord_less_eq @ A @ U2 @ ( F @ I2 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_206_fm_Odistinct_I1_J,axiom,
% 2.93/1.61 ! [X1: list @ char] :
% 2.93/1.61 ( ( paraco27778325le_Pro @ X1 )
% 2.93/1.61 != paraco251304083_Truth ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_207_subset__Pow__Union,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_208_UN__extend__simps_I10_J,axiom,
% 2.93/1.61 ! [V2: $tType,U3: $tType,T2: $tType,B2: U3 > ( set @ V2 ),F: T2 > U3,A2: set @ T2] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ V2 )
% 2.93/1.61 @ ( image @ T2 @ ( set @ V2 )
% 2.93/1.61 @ ^ [A5: T2] : ( B2 @ ( F @ A5 ) )
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ ( set @ V2 ) @ ( image @ U3 @ ( set @ V2 ) @ B2 @ ( image @ T2 @ U3 @ F @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_209_image__UN,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,C: $tType,F: B > A,B2: C > ( set @ B ),A2: set @ C] :
% 2.93/1.61 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) )
% 2.93/1.61 = ( complete_Sup_Sup @ ( set @ A )
% 2.93/1.61 @ ( image @ C @ ( set @ A )
% 2.93/1.61 @ ^ [X: C] : ( image @ B @ A @ F @ ( B2 @ X ) )
% 2.93/1.61 @ A2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_210_UN__mono,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F @ A2 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B2 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_211_UN__least,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A2: set @ A,B2: A > ( set @ B ),C3: set @ B] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ C3 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) @ C3 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_212_UN__upper,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,A4: A,A2: set @ A,B2: A > ( set @ B )] :
% 2.93/1.61 ( ( member @ A @ A4 @ A2 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ B ) @ ( B2 @ A4 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_213_UN__image__subset,axiom,
% 2.93/1.61 ! [C: $tType,A: $tType,B: $tType,F: B > ( set @ A ),G: C > ( set @ B ),X2: C,X5: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ ( G @ X2 ) ) ) @ X5 )
% 2.93/1.61 = ( ord_less_eq @ ( set @ B ) @ ( G @ X2 )
% 2.93/1.61 @ ( collect @ B
% 2.93/1.61 @ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F @ X ) @ X5 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_214_UN__subset__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A2: B > ( set @ A ),I5: set @ B,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A2 @ I5 ) ) @ B2 )
% 2.93/1.61 = ( ! [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ I5 )
% 2.93/1.61 => ( ord_less_eq @ ( set @ A ) @ ( A2 @ X ) @ B2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_215_Union__UNIV,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 2.93/1.61 = ( top_top @ ( set @ A ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_216_image__Union,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,F: B > A,S: set @ ( set @ B )] :
% 2.93/1.61 ( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
% 2.93/1.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_217_UN__extend__simps_I8_J,axiom,
% 2.93/1.61 ! [P4: $tType,O: $tType,B2: O > ( set @ P4 ),A2: set @ ( set @ O )] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ P4 )
% 2.93/1.61 @ ( image @ ( set @ O ) @ ( set @ P4 )
% 2.93/1.61 @ ^ [Y4: set @ O] : ( complete_Sup_Sup @ ( set @ P4 ) @ ( image @ O @ ( set @ P4 ) @ B2 @ Y4 ) )
% 2.93/1.61 @ A2 ) )
% 2.93/1.61 = ( complete_Sup_Sup @ ( set @ P4 ) @ ( image @ O @ ( set @ P4 ) @ B2 @ ( complete_Sup_Sup @ ( set @ O ) @ A2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_218_top__finite__def,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( finite_lattice @ A )
% 2.93/1.61 => ( ( top_top @ A )
% 2.93/1.61 = ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_219_eval_Ocases,axiom,
% 2.93/1.61 ! [X2: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
% 2.93/1.61 ( ! [I3: ( list @ char ) > paraco415392788lle_tv,S4: list @ char] :
% 2.93/1.61 ( X2
% 2.93/1.61 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S4 ) ) )
% 2.93/1.61 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
% 2.93/1.61 ( X2
% 2.93/1.61 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
% 2.93/1.61 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm] :
% 2.93/1.61 ( X2
% 2.93/1.61 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P5 ) ) )
% 2.93/1.61 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
% 2.93/1.61 ( X2
% 2.93/1.61 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P5 @ Q4 ) ) )
% 2.93/1.61 => ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
% 2.93/1.61 ( X2
% 2.93/1.61 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P5 @ Q4 ) ) )
% 2.93/1.61 => ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
% 2.93/1.61 ( X2
% 2.93/1.61 != ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P5 @ Q4 ) ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_220_cSup__eq__maximum,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( condit378418413attice @ A )
% 2.93/1.61 => ! [Z2: A,X5: set @ A] :
% 2.93/1.61 ( ( member @ A @ Z2 @ X5 )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ X5 )
% 2.93/1.61 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ X5 )
% 2.93/1.61 = Z2 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_221_UN__ball__bex__simps_I3_J,axiom,
% 2.93/1.61 ! [D: $tType,A2: set @ ( set @ D ),P2: D > $o] :
% 2.93/1.61 ( ( ? [X: D] :
% 2.93/1.61 ( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
% 2.93/1.61 & ( P2 @ X ) ) )
% 2.93/1.61 = ( ? [X: set @ D] :
% 2.93/1.61 ( ( member @ ( set @ D ) @ X @ A2 )
% 2.93/1.61 & ? [Y4: D] :
% 2.93/1.61 ( ( member @ D @ Y4 @ X )
% 2.93/1.61 & ( P2 @ Y4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_222_UN__ball__bex__simps_I1_J,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ ( set @ A ),P2: A > $o] :
% 2.93/1.61 ( ( ! [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 2.93/1.61 => ( P2 @ X ) ) )
% 2.93/1.61 = ( ! [X: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ X @ A2 )
% 2.93/1.61 => ! [Y4: A] :
% 2.93/1.61 ( ( member @ A @ Y4 @ X )
% 2.93/1.61 => ( P2 @ Y4 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_223_UnionI,axiom,
% 2.93/1.61 ! [A: $tType,X5: set @ A,C3: set @ ( set @ A ),A2: A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ X5 @ C3 )
% 2.93/1.61 => ( ( member @ A @ A2 @ X5 )
% 2.93/1.61 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_224_Union__iff,axiom,
% 2.93/1.61 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 2.93/1.61 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 2.93/1.61 = ( ? [X: set @ A] :
% 2.93/1.61 ( ( member @ ( set @ A ) @ X @ C3 )
% 2.93/1.61 & ( member @ A @ A2 @ X ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_225_Sup1__I,axiom,
% 2.93/1.61 ! [A: $tType,P2: A > $o,A2: set @ ( A > $o ),A4: A] :
% 2.93/1.61 ( ( member @ ( A > $o ) @ P2 @ A2 )
% 2.93/1.61 => ( ( P2 @ A4 )
% 2.93/1.61 => ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A4 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_226_SUP1__I,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A4: A,A2: set @ A,B2: A > B > $o,B3: B] :
% 2.93/1.61 ( ( member @ A @ A4 @ A2 )
% 2.93/1.61 => ( ( B2 @ A4 @ B3 )
% 2.93/1.61 => ( complete_Sup_Sup @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B2 @ A2 ) @ B3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_227_Sup__set__def,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( set @ A ) )
% 2.93/1.61 = ( ^ [A3: set @ ( set @ A )] :
% 2.93/1.61 ( collect @ A
% 2.93/1.61 @ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A3 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_228_SUP1__E,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,B2: B > A > $o,A2: set @ B,B3: A] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B2 @ A2 ) @ B3 )
% 2.93/1.61 => ~ ! [X3: B] :
% 2.93/1.61 ( ( member @ B @ X3 @ A2 )
% 2.93/1.61 => ~ ( B2 @ X3 @ B3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_229_SUP__Sup__eq2,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > B > $o )
% 2.93/1.61 @ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
% 2.93/1.61 @ ^ [I: set @ ( product_prod @ A @ B ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ I )
% 2.93/1.61 @ S ) )
% 2.93/1.61 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_230_Sup__SUP__eq,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > $o ) )
% 2.93/1.61 = ( ^ [S5: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S5 ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_231_SUP__UN__eq2,axiom,
% 2.93/1.61 ! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > B > $o )
% 2.93/1.61 @ ( image @ C @ ( A > B > $o )
% 2.93/1.61 @ ^ [I: C,X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( R2 @ I ) )
% 2.93/1.61 @ S ) )
% 2.93/1.61 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_232_SUP__Sup__eq,axiom,
% 2.93/1.61 ! [A: $tType,S: set @ ( set @ A )] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > $o )
% 2.93/1.61 @ ( image @ ( set @ A ) @ ( A > $o )
% 2.93/1.61 @ ^ [I: set @ A,X: A] : ( member @ A @ X @ I )
% 2.93/1.61 @ S ) )
% 2.93/1.61 = ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_233_Sup1__E,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ ( A > $o ),A4: A] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A4 )
% 2.93/1.61 => ~ ! [P6: A > $o] :
% 2.93/1.61 ( ( member @ ( A > $o ) @ P6 @ A2 )
% 2.93/1.61 => ~ ( P6 @ A4 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_234_UnionE,axiom,
% 2.93/1.61 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 2.93/1.61 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 2.93/1.61 => ~ ! [X6: set @ A] :
% 2.93/1.61 ( ( member @ A @ A2 @ X6 )
% 2.93/1.61 => ~ ( member @ ( set @ A ) @ X6 @ C3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_235_subrelI,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.93/1.61 ( ! [X3: A,Y5: B] :
% 2.93/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R2 )
% 2.93/1.61 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) )
% 2.93/1.61 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_236_pred__subset__eq2,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.93/1.61 ( ( ord_less_eq @ ( A > B > $o )
% 2.93/1.61 @ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
% 2.93/1.61 @ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S ) )
% 2.93/1.61 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_237_pred__equals__eq2,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.93/1.61 ( ( ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R ) )
% 2.93/1.61 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S ) ) )
% 2.93/1.61 = ( R = S ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_238_ssubst__Pair__rhs,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,R2: A,S3: B,R: set @ ( product_prod @ A @ B ),S6: B] :
% 2.93/1.61 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R )
% 2.93/1.61 => ( ( S6 = S3 )
% 2.93/1.61 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S6 ) @ R ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_239_top__empty__eq2,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( top_top @ ( A > B > $o ) )
% 2.93/1.61 = ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_240_cSup__eq,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( ( condit378418413attice @ A )
% 2.93/1.61 & ( no_bot @ A ) )
% 2.93/1.61 => ! [X5: set @ A,A4: A] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ X5 )
% 2.93/1.61 => ( ord_less_eq @ A @ X3 @ A4 ) )
% 2.93/1.61 => ( ! [Y5: A] :
% 2.93/1.61 ( ! [X4: A] :
% 2.93/1.61 ( ( member @ A @ X4 @ X5 )
% 2.93/1.61 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ A4 @ Y5 ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ A @ X5 )
% 2.93/1.61 = A4 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_241_UN__constant__eq,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A4: A,A2: set @ A,F: A > ( set @ B ),C2: set @ B] :
% 2.93/1.61 ( ( member @ A @ A4 @ A2 )
% 2.93/1.61 => ( ! [X3: A] :
% 2.93/1.61 ( ( member @ A @ X3 @ A2 )
% 2.93/1.61 => ( ( F @ X3 )
% 2.93/1.61 = C2 ) )
% 2.93/1.61 => ( ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F @ A2 ) )
% 2.93/1.61 = C2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_242_SUP__eq__top__iff,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( comple1035589618norder @ A )
% 2.93/1.61 => ! [F: B > A,A2: set @ B] :
% 2.93/1.61 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
% 2.93/1.61 = ( top_top @ A ) )
% 2.93/1.61 = ( ! [X: A] :
% 2.93/1.61 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 2.93/1.61 => ? [Y4: B] :
% 2.93/1.61 ( ( member @ B @ Y4 @ A2 )
% 2.93/1.61 & ( ord_less @ A @ X @ ( F @ Y4 ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_243_psubsetI,axiom,
% 2.93/1.61 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.93/1.61 => ( ( A2 != B2 )
% 2.93/1.61 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_244_Sup2__I,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,R2: A > B > $o,A2: set @ ( A > B > $o ),A4: A,B3: B] :
% 2.93/1.61 ( ( member @ ( A > B > $o ) @ R2 @ A2 )
% 2.93/1.61 => ( ( R2 @ A4 @ B3 )
% 2.93/1.61 => ( complete_Sup_Sup @ ( A > B > $o ) @ A2 @ A4 @ B3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_245_top2I,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,X2: A,Y3: B] : ( top_top @ ( A > B > $o ) @ X2 @ Y3 ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_246_Sup__eq__top__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple1035589618norder @ A )
% 2.93/1.61 => ! [A2: set @ A] :
% 2.93/1.61 ( ( ( complete_Sup_Sup @ A @ A2 )
% 2.93/1.61 = ( top_top @ A ) )
% 2.93/1.61 = ( ! [X: A] :
% 2.93/1.61 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 2.93/1.61 => ? [Y4: A] :
% 2.93/1.61 ( ( member @ A @ Y4 @ A2 )
% 2.93/1.61 & ( ord_less @ A @ X @ Y4 ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_247_SUP2__I,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType,C: $tType,A4: A,A2: set @ A,B2: A > B > C > $o,B3: B,C2: C] :
% 2.93/1.61 ( ( member @ A @ A4 @ A2 )
% 2.93/1.61 => ( ( B2 @ A4 @ B3 @ C2 )
% 2.93/1.61 => ( complete_Sup_Sup @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B2 @ A2 ) @ B3 @ C2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_248_SUP2__E,axiom,
% 2.93/1.61 ! [A: $tType,C: $tType,B: $tType,B2: C > A > B > $o,A2: set @ C,B3: A,C2: B] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B2 @ A2 ) @ B3 @ C2 )
% 2.93/1.61 => ~ ! [X3: C] :
% 2.93/1.61 ( ( member @ C @ X3 @ A2 )
% 2.93/1.61 => ~ ( B2 @ X3 @ B3 @ C2 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_249_Sup2__E,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType,A2: set @ ( A > B > $o ),A4: A,B3: B] :
% 2.93/1.61 ( ( complete_Sup_Sup @ ( A > B > $o ) @ A2 @ A4 @ B3 )
% 2.93/1.61 => ~ ! [R4: A > B > $o] :
% 2.93/1.61 ( ( member @ ( A > B > $o ) @ R4 @ A2 )
% 2.93/1.61 => ~ ( R4 @ A4 @ B3 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_250_Sup__bool__def,axiom,
% 2.93/1.61 ( ( complete_Sup_Sup @ $o )
% 2.93/1.61 = ( member @ $o @ $true ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_251_less__Sup__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple1035589618norder @ A )
% 2.93/1.61 => ! [A4: A,S: set @ A] :
% 2.93/1.61 ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
% 2.93/1.61 = ( ? [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ S )
% 2.93/1.61 & ( ord_less @ A @ A4 @ X ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_252_complete__interval,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( condit1037483654norder @ A )
% 2.93/1.61 => ! [A4: A,B3: A,P2: A > $o] :
% 2.93/1.61 ( ( ord_less @ A @ A4 @ B3 )
% 2.93/1.61 => ( ( P2 @ A4 )
% 2.93/1.61 => ( ~ ( P2 @ B3 )
% 2.93/1.61 => ? [C5: A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ A4 @ C5 )
% 2.93/1.61 & ( ord_less_eq @ A @ C5 @ B3 )
% 2.93/1.61 & ! [X4: A] :
% 2.93/1.61 ( ( ( ord_less_eq @ A @ A4 @ X4 )
% 2.93/1.61 & ( ord_less @ A @ X4 @ C5 ) )
% 2.93/1.61 => ( P2 @ X4 ) )
% 2.93/1.61 & ! [D3: A] :
% 2.93/1.61 ( ! [X3: A] :
% 2.93/1.61 ( ( ( ord_less_eq @ A @ A4 @ X3 )
% 2.93/1.61 & ( ord_less @ A @ X3 @ D3 ) )
% 2.93/1.61 => ( P2 @ X3 ) )
% 2.93/1.61 => ( ord_less_eq @ A @ D3 @ C5 ) ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_253_le__Sup__iff,axiom,
% 2.93/1.61 ! [A: $tType] :
% 2.93/1.61 ( ( comple1035589618norder @ A )
% 2.93/1.61 => ! [X2: A,A2: set @ A] :
% 2.93/1.61 ( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) )
% 2.93/1.61 = ( ! [Y4: A] :
% 2.93/1.61 ( ( ord_less @ A @ Y4 @ X2 )
% 2.93/1.61 => ? [X: A] :
% 2.93/1.61 ( ( member @ A @ X @ A2 )
% 2.93/1.61 & ( ord_less @ A @ Y4 @ X ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_254_less__SUP__iff,axiom,
% 2.93/1.61 ! [A: $tType,B: $tType] :
% 2.93/1.61 ( ( comple1035589618norder @ A )
% 2.93/1.61 => ! [A4: A,F: B > A,A2: set @ B] :
% 2.93/1.61 ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) )
% 2.93/1.61 = ( ? [X: B] :
% 2.93/1.61 ( ( member @ B @ X @ A2 )
% 2.93/1.61 & ( ord_less @ A @ A4 @ ( F @ X ) ) ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(fact_255_SUP__lessD,axiom,
% 2.93/1.61 ! [B: $tType,A: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A )
% 2.93/1.61 => ! [F: B > A,A2: set @ B,Y3: A,I2: B] :
% 2.93/1.61 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ Y3 )
% 2.93/1.61 => ( ( member @ B @ I2 @ A2 )
% 2.93/1.61 => ( ord_less @ A @ ( F @ I2 ) @ Y3 ) ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A8 )
% 2.93/1.61 => ( condit378418413attice @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( comple187826305attice @ A8 )
% 2.93/1.61 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( complete_Sup @ A8 )
% 2.93/1.61 => ( complete_Sup @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( order_top @ A8 )
% 2.93/1.61 => ( order_top @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( preorder @ A8 )
% 2.93/1.61 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( order @ A8 )
% 2.93/1.61 => ( order @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Orderings_Otop,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( top @ A8 )
% 2.93/1.61 => ( top @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_fun___Orderings_Oord,axiom,
% 2.93/1.61 ! [A7: $tType,A8: $tType] :
% 2.93/1.61 ( ( ord @ A8 )
% 2.93/1.61 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.93/1.61 condit1037483654norder @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
% 2.93/1.61 condit378418413attice @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Complete__Lattices_OSup_2,axiom,
% 2.93/1.61 complete_Sup @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
% 2.93/1.61 preorder @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.93/1.61 linorder @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
% 2.93/1.61 order @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
% 2.93/1.61 ord @ nat ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
% 2.93/1.61 ! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_7,axiom,
% 2.93/1.61 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Complete__Lattices_OSup_8,axiom,
% 2.93/1.61 ! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Orderings_Oorder__top_9,axiom,
% 2.93/1.61 ! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
% 2.93/1.61 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Orderings_Oorder_11,axiom,
% 2.93/1.61 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Orderings_Otop_12,axiom,
% 2.93/1.61 ! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
% 2.93/1.61 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_14,axiom,
% 2.93/1.61 condit378418413attice @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_15,axiom,
% 2.93/1.61 comple187826305attice @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Complete__Lattices_OSup_16,axiom,
% 2.93/1.61 complete_Sup @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Orderings_Oorder__top_17,axiom,
% 2.93/1.61 order_top @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Orderings_Opreorder_18,axiom,
% 2.93/1.61 preorder @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Orderings_Olinorder_19,axiom,
% 2.93/1.61 linorder @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
% 2.93/1.61 order @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Orderings_Otop_21,axiom,
% 2.93/1.61 top @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(tcon_HOL_Obool___Orderings_Oord_22,axiom,
% 2.93/1.61 ord @ $o ).
% 2.93/1.61
% 2.93/1.61 thf(conj_0,conjecture,
% 2.93/1.61 ( ( ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.93/1.61 ( ( paraco876059933e_eval @ I @ p )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) )
% 2.93/1.61 != ( ~ ! [I: ( list @ char ) > paraco415392788lle_tv] :
% 2.93/1.61 ( ( ord_less_eq @ ( set @ paraco415392788lle_tv ) @ ( image @ ( list @ char ) @ paraco415392788lle_tv @ I @ ( top_top @ ( set @ ( list @ char ) ) ) )
% 2.93/1.61 @ ( paraco506338565domain
% 2.93/1.61 @ ( collect @ nat
% 2.93/1.61 @ ^ [N: nat] : $true ) ) )
% 2.93/1.61 => ( ( paraco876059933e_eval @ I @ p )
% 2.93/1.61 = ( paraco2040174112le_Det @ $true ) ) ) ) ) ).
% 2.93/1.61 ------------------------
% 2.93/1.63 % (25669)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_25569 for (2999ds/4Mi)
% 2.93/1.63 % (25671)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/2Mi)
% 2.93/1.64 % (25671)Instruction limit reached!
% 2.93/1.64 % (25671)------------------------------
% 2.93/1.64 % (25671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (25671)Termination reason: Unknown
% 2.93/1.64 % (25671)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (25671)Memory used [KB]: 1407
% 2.93/1.64 % (25671)Time elapsed: 0.002 s
% 2.93/1.64 % (25671)Instructions burned: 2 (million)
% 2.93/1.64 % (25671)------------------------------
% 2.93/1.64 % (25671)------------------------------
% 2.93/1.64 % (25674)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/18Mi)
% 2.93/1.64 % (25669)Instruction limit reached!
% 2.93/1.64 % (25669)------------------------------
% 2.93/1.64 % (25669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (25669)Termination reason: Unknown
% 2.93/1.64 % (25669)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (25669)Memory used [KB]: 1535
% 2.93/1.64 % (25669)Time elapsed: 0.003 s
% 2.93/1.64 % (25669)Instructions burned: 5 (million)
% 2.93/1.64 % (25669)------------------------------
% 2.93/1.64 % (25669)------------------------------
% 2.93/1.64 % (25668)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/183Mi)
% 2.93/1.64 % (25674)Instruction limit reached!
% 2.93/1.64 % (25674)------------------------------
% 2.93/1.64 % (25674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (25674)Termination reason: Unknown
% 2.93/1.64 % (25674)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (25674)Memory used [KB]: 1918
% 2.93/1.64 % (25674)Time elapsed: 0.009 s
% 2.93/1.64 % (25674)Instructions burned: 19 (million)
% 2.93/1.64 % (25674)------------------------------
% 2.93/1.64 % (25674)------------------------------
% 2.93/1.64 % (25670)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_25569 for (2999ds/27Mi)
% 2.93/1.64 % (25673)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_25569 for (2999ds/275Mi)
% 2.93/1.64 % (25672)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_25569 for (2999ds/2Mi)
% 2.93/1.64 % (25672)Instruction limit reached!
% 2.93/1.64 % (25672)------------------------------
% 2.93/1.64 % (25672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (25672)Termination reason: Unknown
% 2.93/1.64 % (25672)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (25672)Memory used [KB]: 1407
% 2.93/1.64 % (25672)Time elapsed: 0.004 s
% 2.93/1.64 % (25672)Instructions burned: 3 (million)
% 2.93/1.64 % (25672)------------------------------
% 2.93/1.64 % (25672)------------------------------
% 2.93/1.65 % (25676)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_25569 for (2999ds/37Mi)
% 2.93/1.65 % (25677)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_25569 for (2999ds/15Mi)
% 2.93/1.65 % (25675)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/3Mi)
% 2.93/1.65 % (25675)Instruction limit reached!
% 2.93/1.65 % (25675)------------------------------
% 2.93/1.65 % (25675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.65 % (25675)Termination reason: Unknown
% 2.93/1.65 % (25675)Termination phase: shuffling
% 2.93/1.65
% 2.93/1.65 % (25675)Memory used [KB]: 1407
% 2.93/1.65 % (25675)Time elapsed: 0.003 s
% 2.93/1.65 % (25675)Instructions burned: 3 (million)
% 2.93/1.65 % (25675)------------------------------
% 2.93/1.65 % (25675)------------------------------
% 2.93/1.66 % (25676)Instruction limit reached!
% 2.93/1.66 % (25676)------------------------------
% 2.93/1.66 % (25676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (25676)Termination reason: Unknown
% 2.93/1.66 % (25676)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (25676)Memory used [KB]: 2174
% 2.93/1.66 % (25676)Time elapsed: 0.012 s
% 2.93/1.66 % (25676)Instructions burned: 38 (million)
% 2.93/1.66 % (25676)------------------------------
% 2.93/1.66 % (25676)------------------------------
% 2.93/1.66 % (25677)Instruction limit reached!
% 2.93/1.66 % (25677)------------------------------
% 2.93/1.66 % (25677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (25677)Termination reason: Unknown
% 2.93/1.66 % (25677)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (25677)Memory used [KB]: 1918
% 2.93/1.66 % (25677)Time elapsed: 0.008 s
% 2.93/1.66 % (25677)Instructions burned: 17 (million)
% 2.93/1.66 % (25677)------------------------------
% 2.93/1.66 % (25677)------------------------------
% 2.93/1.66 % (25670)Instruction limit reached!
% 2.93/1.66 % (25670)------------------------------
% 2.93/1.66 % (25670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (25670)Termination reason: Unknown
% 2.93/1.66 % (25670)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (25670)Memory used [KB]: 2046
% 2.93/1.66 % (25670)Time elapsed: 0.019 s
% 2.93/1.66 % (25670)Instructions burned: 27 (million)
% 2.93/1.66 % (25670)------------------------------
% 2.93/1.66 % (25670)------------------------------
% 2.93/1.66 % (25678)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_25569 for (2999ds/3Mi)
% 2.93/1.66 % (25678)Instruction limit reached!
% 2.93/1.66 % (25678)------------------------------
% 2.93/1.66 % (25678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (25678)Termination reason: Unknown
% 2.93/1.66 % (25678)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (25678)Memory used [KB]: 1407
% 2.93/1.66 % (25678)Time elapsed: 0.004 s
% 2.93/1.66 % (25678)Instructions burned: 3 (million)
% 2.93/1.66 % (25678)------------------------------
% 2.93/1.66 % (25678)------------------------------
% 2.93/1.66 % (25679)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_25569 for (2999ds/1041Mi)
% 2.93/1.67 % (25680)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/7Mi)
% 2.93/1.67 % (25680)Instruction limit reached!
% 2.93/1.67 % (25680)------------------------------
% 2.93/1.67 % (25680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.67 % (25680)Termination reason: Unknown
% 2.93/1.67 % (25680)Termination phase: shuffling
% 2.93/1.67
% 2.93/1.67 % (25680)Memory used [KB]: 1663
% 2.93/1.67 % (25680)Time elapsed: 0.004 s
% 2.93/1.67 % (25680)Instructions burned: 10 (million)
% 2.93/1.67 % (25680)------------------------------
% 2.93/1.67 % (25680)------------------------------
% 2.93/1.67 % (25681)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/16Mi)
% 2.93/1.68 % (25681)Instruction limit reached!
% 2.93/1.68 % (25681)------------------------------
% 2.93/1.68 % (25681)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (25681)Termination reason: Unknown
% 2.93/1.68 % (25681)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (25681)Memory used [KB]: 1791
% 2.93/1.68 % (25681)Time elapsed: 0.008 s
% 2.93/1.68 % (25681)Instructions burned: 18 (million)
% 2.93/1.68 % (25681)------------------------------
% 2.93/1.68 % (25681)------------------------------
% 2.93/1.68 % (25682)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_25569 for (2999ds/3Mi)
% 2.93/1.68 % (25682)Instruction limit reached!
% 2.93/1.68 % (25682)------------------------------
% 2.93/1.68 % (25682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (25682)Termination reason: Unknown
% 2.93/1.68 % (25682)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (25682)Memory used [KB]: 1407
% 2.93/1.68 % (25682)Time elapsed: 0.004 s
% 2.93/1.68 % (25682)Instructions burned: 3 (million)
% 2.93/1.68 % (25682)------------------------------
% 2.93/1.68 % (25682)------------------------------
% 2.93/1.68 % (25684)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/7Mi)
% 2.93/1.68 % (25684)Instruction limit reached!
% 2.93/1.68 % (25684)------------------------------
% 2.93/1.68 % (25684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (25684)Termination reason: Unknown
% 2.93/1.68 % (25684)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (25684)Memory used [KB]: 1535
% 2.93/1.68 % (25684)Time elapsed: 0.004 s
% 2.93/1.68 % (25684)Instructions burned: 9 (million)
% 2.93/1.68 % (25684)------------------------------
% 2.93/1.68 % (25684)------------------------------
% 2.93/1.68 % (25683)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_25569 for (2999ds/3Mi)
% 2.93/1.68 % (25683)Instruction limit reached!
% 2.93/1.68 % (25683)------------------------------
% 2.93/1.68 % (25683)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (25683)Termination reason: Unknown
% 2.93/1.68 % (25683)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (25683)Memory used [KB]: 1407
% 2.93/1.68 % (25683)Time elapsed: 0.004 s
% 2.93/1.68 % (25683)Instructions burned: 3 (million)
% 2.93/1.68 % (25683)------------------------------
% 2.93/1.68 % (25683)------------------------------
% 2.93/1.69 % (25685)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/3Mi)
% 2.93/1.69 % (25685)Instruction limit reached!
% 2.93/1.69 % (25685)------------------------------
% 2.93/1.69 % (25685)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.69 % (25685)Termination reason: Unknown
% 2.93/1.69 % (25685)Termination phase: shuffling
% 2.93/1.69
% 2.93/1.69 % (25685)Memory used [KB]: 1407
% 2.93/1.69 % (25685)Time elapsed: 0.003 s
% 2.93/1.69 % (25685)Instructions burned: 4 (million)
% 2.93/1.69 % (25685)------------------------------
% 2.93/1.69 % (25685)------------------------------
% 2.93/1.69 % (25686)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/4Mi)
% 2.93/1.69 % (25687)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/18Mi)
% 2.93/1.69 % (25686)Instruction limit reached!
% 2.93/1.69 % (25686)------------------------------
% 2.93/1.69 % (25686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.69 % (25686)Termination reason: Unknown
% 2.93/1.69 % (25686)Termination phase: shuffling
% 2.93/1.69
% 2.93/1.69 % (25686)Memory used [KB]: 1535
% 2.93/1.69 % (25686)Time elapsed: 0.003 s
% 2.93/1.69 % (25686)Instructions burned: 4 (million)
% 2.93/1.69 % (25686)------------------------------
% 2.93/1.69 % (25686)------------------------------
% 2.99/1.70 % (25687)Instruction limit reached!
% 2.99/1.70 % (25687)------------------------------
% 2.99/1.70 % (25687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.70 % (25687)Termination reason: Unknown
% 2.99/1.70 % (25687)Termination phase: shuffling
% 2.99/1.70
% 2.99/1.70 % (25687)Memory used [KB]: 1791
% 2.99/1.70 % (25687)Time elapsed: 0.007 s
% 2.99/1.70 % (25687)Instructions burned: 19 (million)
% 2.99/1.70 % (25687)------------------------------
% 2.99/1.70 % (25687)------------------------------
% 2.99/1.70 % (25689)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/6Mi)
% 2.99/1.70 % (25688)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_25569 for (2999ds/710Mi)
% 2.99/1.70 % (25689)Instruction limit reached!
% 2.99/1.70 % (25689)------------------------------
% 2.99/1.70 % (25689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.70 % (25689)Termination reason: Unknown
% 2.99/1.70 % (25689)Termination phase: shuffling
% 2.99/1.70
% 2.99/1.70 % (25689)Memory used [KB]: 1535
% 2.99/1.70 % (25689)Time elapsed: 0.004 s
% 2.99/1.70 % (25689)Instructions burned: 6 (million)
% 2.99/1.70 % (25689)------------------------------
% 2.99/1.70 % (25689)------------------------------
% 2.99/1.70 % (25673)First to succeed.
% 2.99/1.70 % (25690)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_25569 for (2999ds/902Mi)
% 2.99/1.70 % (25673)Refutation found. Thanks to Tanya!
% 2.99/1.70 % SZS status Theorem for DTF2THF_25569
% 2.99/1.70 % SZS output start Proof for DTF2THF_25569
% 2.99/1.70 thf(type_def_5, type, set: ($tType) > $tType).
% 2.99/1.70 thf(type_def_7, type, nat: $tType).
% 2.99/1.70 thf(type_def_8, type, paraco415392788lle_tv: $tType).
% 2.99/1.70 thf(type_def_9, type, char: $tType).
% 2.99/1.70 thf(type_def_10, type, list: ($tType) > $tType).
% 2.99/1.70 thf(type_def_11, type, paraco414474393lle_fm: $tType).
% 2.99/1.70 thf(type_def_12, type, product_prod: ($tType * $tType) > $tType).
% 2.99/1.70 thf(func_def_0, type, paraco415392788lle_tv: $tType).
% 2.99/1.70 thf(func_def_1, type, paraco414474393lle_fm: $tType).
% 2.99/1.70 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 2.99/1.70 thf(func_def_3, type, char: $tType).
% 2.99/1.70 thf(func_def_4, type, list: ($tType) > $tType).
% 2.99/1.70 thf(func_def_5, type, set: ($tType) > $tType).
% 2.99/1.70 thf(func_def_6, type, nat: $tType).
% 2.99/1.70 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_9, type, top: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_11, type, no_bot: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_12, type, linorder: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_13, type, preorder: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_14, type, finite_lattice: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_15, type, order_top: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_16, type, complete_Sup: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_17, type, comple187826305attice: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_18, type, comple1035589618norder: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_19, type, condit378418413attice: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_20, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.99/1.70 thf(func_def_21, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 2.99/1.70 thf(func_def_22, type, finite_Fpow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.99/1.70 thf(func_def_23, type, swap: !>[X0: $tType, X1: $tType]:(X0 > X0 > (X0 > X1) > X0 > X1)).
% 2.99/1.70 thf(func_def_24, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.99/1.70 thf(func_def_25, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.99/1.70 thf(func_def_26, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.99/1.70 thf(func_def_27, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.99/1.70 thf(func_def_28, type, top_top: !>[X0: $tType]:(X0)).
% 2.99/1.70 thf(func_def_29, type, paraco506338565domain: set @ nat > set @ paraco415392788lle_tv).
% 2.99/1.70 thf(func_def_30, type, paraco876059933e_eval: (list @ char > paraco415392788lle_tv) > paraco414474393lle_fm > paraco415392788lle_tv).
% 2.99/1.70 thf(func_def_31, type, paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_32, type, paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_33, type, paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_34, type, paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_35, type, paraco27778325le_Pro: list @ char > paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_36, type, paraco251304083_Truth: paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_37, type, paraco2040174112le_Det: $o > paraco415392788lle_tv).
% 2.99/1.70 thf(func_def_38, type, paraco152590079rec_tv: !>[X0: $tType]:(($o > X0) > (nat > X0) > paraco415392788lle_tv > X0)).
% 2.99/1.70 thf(func_def_39, type, paraco769098683_valid: paraco414474393lle_fm > $o).
% 2.99/1.70 thf(func_def_40, type, paraco2086025920lid_in: set @ nat > paraco414474393lle_fm > $o).
% 2.99/1.70 thf(func_def_41, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.99/1.70 thf(func_def_42, type, powp: !>[X0: $tType]:((X0 > $o) > set @ X0 > $o)).
% 2.99/1.70 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.99/1.70 thf(func_def_44, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.99/1.70 thf(func_def_45, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.99/1.70 thf(func_def_46, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.99/1.70 thf(func_def_47, type, p: paraco414474393lle_fm).
% 2.99/1.70 thf(func_def_59, type, sK0: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X0)).
% 2.99/1.70 thf(func_def_60, type, sK1: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X1)).
% 2.99/1.70 thf(func_def_61, type, sK2: !>[X0: $tType, X1: $tType]:(X1 > set @ X0 > (X0 > X1 > $o) > X0)).
% 2.99/1.70 thf(func_def_62, type, sK3: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_63, type, sK4: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_64, type, sK5: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > (X0 > $o) > X0)).
% 2.99/1.70 thf(func_def_65, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > X0 > X1)).
% 2.99/1.70 thf(func_def_66, type, sK7: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.99/1.70 thf(func_def_67, type, sK8: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.99/1.70 thf(func_def_68, type, sK9: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 2.99/1.70 thf(func_def_69, type, sK10: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > set @ X1) > set @ X0 > X0)).
% 2.99/1.70 thf(func_def_70, type, sK11: list @ char > paraco415392788lle_tv).
% 2.99/1.70 thf(func_def_71, type, sK12: list @ char > paraco415392788lle_tv).
% 2.99/1.70 thf(func_def_72, type, sK13: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.99/1.70 thf(func_def_73, type, sK14: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.99/1.70 thf(func_def_74, type, sK15: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > set @ X0) > X0 > X1)).
% 2.99/1.70 thf(func_def_75, type, sK16: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > set @ X0)).
% 2.99/1.70 thf(func_def_76, type, sK17: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.99/1.70 thf(func_def_77, type, sK18: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.99/1.70 thf(func_def_78, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_79, type, sK20: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_80, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.99/1.70 thf(func_def_81, type, sK22: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 2.99/1.70 thf(func_def_82, type, sK23: !>[X0: $tType]:(set @ set @ X0 > X0 > set @ X0)).
% 2.99/1.70 thf(func_def_83, type, sK24: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_84, type, sK25: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.99/1.70 thf(func_def_85, type, sK26: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_86, type, sK27: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 2.99/1.70 thf(func_def_87, type, sK28: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_88, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_89, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_90, type, sK31: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.99/1.70 thf(func_def_91, type, sK32: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > set @ X0)).
% 2.99/1.70 thf(func_def_92, type, sK33: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.99/1.70 thf(func_def_93, type, sK34: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.99/1.70 thf(func_def_94, type, sK35: !>[X0: $tType]:(set @ X0 > set @ set @ X0 > set @ X0)).
% 2.99/1.70 thf(func_def_95, type, sK36: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > set @ X0) > X0 > X1)).
% 2.99/1.70 thf(func_def_96, type, sK37: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.99/1.70 thf(func_def_97, type, sK38: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_98, type, sK39: !>[X0: $tType]:(set @ set @ X0 > set @ set @ X0 > set @ X0)).
% 2.99/1.70 thf(func_def_99, type, sK40: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.99/1.70 thf(func_def_100, type, sK41: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 2.99/1.70 thf(func_def_101, type, sK42: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 2.99/1.70 thf(func_def_102, type, sK43: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 2.99/1.70 thf(func_def_103, type, sK44: !>[X0: $tType]:(X0)).
% 2.99/1.70 thf(func_def_104, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > X0 > set @ X1 > (X1 > X0 > X2 > $o) > X1)).
% 2.99/1.70 thf(func_def_105, type, sK46: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_106, type, sK47: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 2.99/1.70 thf(func_def_107, type, sK48: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_108, type, sK49: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 2.99/1.70 thf(func_def_109, type, sK50: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_110, type, sK51: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_111, type, sK52: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.99/1.70 thf(func_def_112, type, sK53: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.99/1.70 thf(func_def_113, type, sK54: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.99/1.70 thf(func_def_114, type, sK55: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > set @ X1) > set @ X0 > X0)).
% 2.99/1.70 thf(func_def_115, type, sK56: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_116, type, sK57: !>[X0: $tType]:(set @ X0 > X0)).
% 2.99/1.70 thf(func_def_117, type, sK58: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > set @ X0) > set @ X1 > X1)).
% 2.99/1.70 thf(func_def_118, type, sK59: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 2.99/1.70 thf(func_def_119, type, sK60: !>[X0: $tType, X1: $tType]:(X1 > X0 > set @ (X0 > X1 > $o) > X0 > X1 > $o)).
% 2.99/1.70 thf(func_def_120, type, sK61: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > set @ X0) > X1)).
% 2.99/1.70 thf(func_def_121, type, sK62: !>[X0: $tType]:(X0 > set @ (X0 > $o) > X0 > $o)).
% 2.99/1.70 thf(func_def_122, type, sK63: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.99/1.70 thf(func_def_123, type, sK64: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.99/1.70 thf(func_def_124, type, sK65: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_125, type, sK66: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X1)).
% 2.99/1.70 thf(func_def_126, type, sK67: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > set @ X0) > X0)).
% 2.99/1.70 thf(func_def_127, type, sK68: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.99/1.70 thf(func_def_128, type, sK69: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.99/1.70 thf(func_def_129, type, sK70: !>[X0: $tType]:(set @ set @ X0 > X0 > set @ X0)).
% 2.99/1.70 thf(func_def_131, type, ph72: !>[X0: $tType]:(X0)).
% 2.99/1.70 thf(f1360,plain,(
% 2.99/1.70 $false),
% 2.99/1.70 inference(avatar_sat_refutation,[status(thm)],[f1321,f1330,f1343,f1359])).
% 2.99/1.70 thf(f1359,plain,(
% 2.99/1.70 spl71_2 | ~spl71_3),
% 2.99/1.70 inference(avatar_contradiction_clause,[status(thm)],[f1358])).
% 2.99/1.70 thf(f1358,plain,(
% 2.99/1.70 $false | (spl71_2 | ~spl71_3)),
% 2.99/1.70 inference(subsumption_resolution,[status(thm)],[f1320,f1329])).
% 2.99/1.70 thf(f1329,plain,(
% 2.99/1.70 ( ! [X2 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true))) ) | ~spl71_3),
% 2.99/1.70 inference(avatar_component_clause,[status(thm)],[f1328])).
% 2.99/1.70 thf(f1328,definition,(
% 2.99/1.70 spl71_3 <=> ! [X2 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true))),
% 2.99/1.70 introduced(definition,[new_symbols(naming,[spl71_3])],[avatar_definition])).
% 2.99/1.70 thf(f1320,plain,(
% 2.99/1.70 ((paraco876059933e_eval @ sK12 @ p) != (paraco2040174112le_Det @ $true)) | spl71_2),
% 2.99/1.70 inference(avatar_component_clause,[status(thm)],[f1318])).
% 2.99/1.70 thf(f1318,definition,(
% 2.99/1.70 spl71_2 <=> ((paraco876059933e_eval @ sK12 @ p) = (paraco2040174112le_Det @ $true))),
% 2.99/1.70 introduced(definition,[new_symbols(naming,[spl71_2])],[avatar_definition])).
% 2.99/1.70 thf(f1343,plain,(
% 2.99/1.70 spl71_1 | ~spl71_3),
% 2.99/1.70 inference(avatar_contradiction_clause,[status(thm)],[f1342])).
% 2.99/1.70 thf(f1342,plain,(
% 2.99/1.70 $false | (spl71_1 | ~spl71_3)),
% 2.99/1.70 inference(trivial_inequality_removal,[status(thm)],[f1331])).
% 2.99/1.70 thf(f1331,plain,(
% 2.99/1.70 ((paraco2040174112le_Det @ $true) != (paraco2040174112le_Det @ $true)) | (spl71_1 | ~spl71_3)),
% 2.99/1.70 inference(superposition,[status(thm)],[f1316,f1329])).
% 2.99/1.70 thf(f1316,plain,(
% 2.99/1.70 ((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ sK11 @ p)) | spl71_1),
% 2.99/1.70 inference(avatar_component_clause,[status(thm)],[f1314])).
% 2.99/1.70 thf(f1314,definition,(
% 2.99/1.70 spl71_1 <=> ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ sK11 @ p))),
% 2.99/1.70 introduced(definition,[new_symbols(naming,[spl71_1])],[avatar_definition])).
% 2.99/1.70 thf(f1330,plain,(
% 2.99/1.70 spl71_3 | spl71_3),
% 2.99/1.70 inference(avatar_split_clause,[status(thm)],[f1326,f1328,f1328])).
% 2.99/1.70 thf(f1326,plain,(
% 2.99/1.70 ( ! [X2 : list @ char > paraco415392788lle_tv,X3 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X3 @ p) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true))) )),
% 2.99/1.70 inference(subsumption_resolution,[status(thm)],[f1325,f1118])).
% 2.99/1.70 thf(f1118,plain,(
% 2.99/1.70 ( ! [X0 : $tType,X1 : set @ X0] : (($true = (ord_less_eq @ set @ X0 @ X1 @ top_top @ set @ X0))) )),
% 2.99/1.70 inference(cnf_transformation,[status(thm)],[f371])).
% 2.99/1.70 thf(f371,plain,(
% 2.99/1.70 ! [X0 : $tType,X1 : set @ X0] : ($true = (ord_less_eq @ set @ X0 @ X1 @ top_top @ set @ X0))),
% 2.99/1.70 inference(fool_elimination,[status(thm)],[f370])).
% 2.99/1.70 thf(f370,plain,(
% 2.99/1.70 ! [X0 : $tType,X1 : set @ X0] : (ord_less_eq @ set @ X0 @ X1 @ top_top @ set @ X0)),
% 2.99/1.70 inference(rectify,[status(thm)],[f84])).
% 2.99/1.70 thf(f84,axiom,(
% 2.99/1.70 ! [X0 : $tType,X13 : set @ X0] : (ord_less_eq @ set @ X0 @ X13 @ top_top @ set @ X0)),
% 2.99/1.70 file('/export/starexec/sandbox/tmp/tmp.SMTCspLT4x/DTF2THF_25569.p',fact_83_subset__UNIV)).
% 2.99/1.70 thf(f1325,plain,(
% 2.99/1.70 ( ! [X2 : list @ char > paraco415392788lle_tv,X3 : list @ char > paraco415392788lle_tv] : (($true != (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X3 @ top_top @ set @ list @ char) @ top_top @ set @ paraco415392788lle_tv)) | ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X3 @ p) = (paraco2040174112le_Det @ $true))) )),
% 2.99/1.70 inference(forward_demodulation,[status(thm)],[f1324,f1094])).
% 2.99/1.70 thf(f1094,plain,(
% 2.99/1.70 ( ! [X0 : $tType] : ((top_top @ set @ X0 = (collect @ X0 @ (^[Y0 : X0]: ($true))))) )),
% 2.99/1.70 inference(cnf_transformation,[status(thm)],[f496])).
% 2.99/1.70 thf(f496,plain,(
% 2.99/1.70 ! [X0 : $tType] : (top_top @ set @ X0 = (collect @ X0 @ (^[Y0 : X0]: ($true))))),
% 2.99/1.70 inference(fool_elimination,[status(thm)],[f495])).
% 2.99/1.70 thf(f495,plain,(
% 2.99/1.70 ! [X0 : $tType] : (top_top @ set @ X0 = (collect @ X0 @ (^[X1 : X0] : ($true))))),
% 2.99/1.70 inference(rectify,[status(thm)],[f73])).
% 2.99/1.70 thf(f73,axiom,(
% 2.99/1.70 ! [X0 : $tType] : (top_top @ set @ X0 = (collect @ X0 @ (^[X3 : X0] : ($true))))),
% 2.99/1.70 file('/export/starexec/sandbox/tmp/tmp.SMTCspLT4x/DTF2THF_25569.p',fact_72_UNIV__def)).
% 2.99/1.70 thf(f1324,plain,(
% 2.99/1.70 ( ! [X2 : list @ char > paraco415392788lle_tv,X3 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X3 @ p) = (paraco2040174112le_Det @ $true)) | ((ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X3 @ top_top @ set @ list @ char) @ (collect @ paraco415392788lle_tv @ (^[Y0 : paraco415392788lle_tv]: ($true)))) != $true)) )),
% 2.99/1.70 inference(forward_demodulation,[status(thm)],[f1102,f1130])).
% 2.99/1.70 thf(f1130,plain,(
% 2.99/1.70 ((collect @ paraco415392788lle_tv @ (^[Y0 : paraco415392788lle_tv]: ($true))) = (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))),
% 2.99/1.70 inference(cnf_transformation,[status(thm)],[f337])).
% 2.99/1.70 thf(f337,plain,(
% 2.99/1.70 ((collect @ paraco415392788lle_tv @ (^[Y0 : paraco415392788lle_tv]: ($true))) = (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))),
% 2.99/1.70 inference(fool_elimination,[status(thm)],[f336])).
% 2.99/1.70 thf(f336,plain,(
% 2.99/1.70 ((collect @ paraco415392788lle_tv @ (^[X0 : paraco415392788lle_tv] : ($true))) = (paraco506338565domain @ (collect @ nat @ (^[X1 : nat] : ($true)))))),
% 2.99/1.70 inference(rectify,[status(thm)],[f1])).
% 2.99/1.70 thf(f1,axiom,(
% 2.99/1.70 ((collect @ paraco415392788lle_tv @ (^[X3 : paraco415392788lle_tv] : ($true))) = (paraco506338565domain @ (collect @ nat @ (^[X2 : nat] : ($true)))))),
% 2.99/1.70 file('/export/starexec/sandbox/tmp/tmp.SMTCspLT4x/DTF2THF_25569.p',fact_0_universal__domain)).
% 2.99/1.70 thf(f1102,plain,(
% 2.99/1.70 ( ! [X2 : list @ char > paraco415392788lle_tv,X3 : list @ char > paraco415392788lle_tv] : (((paraco876059933e_eval @ X3 @ p) = (paraco2040174112le_Det @ $true)) | ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true)) | ($true != (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X3 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true))))))) )),
% 2.99/1.70 inference(cnf_transformation,[status(thm)],[f943])).
% 2.99/1.70 thf(f943,plain,(
% 2.99/1.70 ((((ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ sK11 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true))))) = $true) & ((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ sK11 @ p))) | ((paraco876059933e_eval @ sK12 @ p) != (paraco2040174112le_Det @ $true))) & (! [X2 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true)) | ! [X3 : list @ char > paraco415392788lle_tv] : (($true != (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X3 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) | ((paraco876059933e_eval @ X3 @ p) = (paraco2040174112le_Det @ $true))))),
% 2.99/1.70 inference(skolemisation,[status(esa),new_symbols(skolem,[sK11,sK12])],[f940,f942,f941])).
% 2.99/1.70 thf(f941,plain,(
% 2.99/1.70 ? [X0 : list @ char > paraco415392788lle_tv] : (($true = (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) & ((paraco876059933e_eval @ X0 @ p) != (paraco2040174112le_Det @ $true))) => (((ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ sK11 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true))))) = $true) & ((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ sK11 @ p)))),
% 2.99/1.70 introduced(definition,[],[choice_axiom])).
% 2.99/1.70 thf(f942,plain,(
% 2.99/1.70 ? [X1 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X1 @ p) != (paraco2040174112le_Det @ $true)) => ((paraco876059933e_eval @ sK12 @ p) != (paraco2040174112le_Det @ $true))),
% 2.99/1.70 introduced(definition,[],[choice_axiom])).
% 2.99/1.70 thf(f940,plain,(
% 2.99/1.70 (? [X0 : list @ char > paraco415392788lle_tv] : (($true = (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) & ((paraco876059933e_eval @ X0 @ p) != (paraco2040174112le_Det @ $true))) | ? [X1 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X1 @ p) != (paraco2040174112le_Det @ $true))) & (! [X2 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true)) | ! [X3 : list @ char > paraco415392788lle_tv] : (($true != (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X3 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) | ((paraco876059933e_eval @ X3 @ p) = (paraco2040174112le_Det @ $true))))),
% 2.99/1.70 inference(rectify,[status(thm)],[f939])).
% 2.99/1.70 thf(f939,plain,(
% 2.99/1.70 (? [X0 : list @ char > paraco415392788lle_tv] : (($true = (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) & ((paraco876059933e_eval @ X0 @ p) != (paraco2040174112le_Det @ $true))) | ? [X1 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X1 @ p) != (paraco2040174112le_Det @ $true))) & (! [X1 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X1 @ p) = (paraco2040174112le_Det @ $true)) | ! [X0 : list @ char > paraco415392788lle_tv] : (($true != (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) | ((paraco876059933e_eval @ X0 @ p) = (paraco2040174112le_Det @ $true))))),
% 2.99/1.70 inference(nnf_transformation,[status(thm)],[f889])).
% 2.99/1.70 thf(f889,plain,(
% 2.99/1.70 ? [X0 : list @ char > paraco415392788lle_tv] : (($true = (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) & ((paraco876059933e_eval @ X0 @ p) != (paraco2040174112le_Det @ $true))) <=> ! [X1 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X1 @ p) = (paraco2040174112le_Det @ $true))),
% 2.99/1.70 inference(ennf_transformation,[status(thm)],[f789])).
% 2.99/1.70 thf(f789,plain,(
% 2.99/1.70 ~(! [X1 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X1 @ p) = (paraco2040174112le_Det @ $true)) <~> ~! [X0 : list @ char > paraco415392788lle_tv] : (($true = (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) => ((paraco876059933e_eval @ X0 @ p) = (paraco2040174112le_Det @ $true))))),
% 2.99/1.70 inference(rectify,[status(thm)],[f441])).
% 2.99/1.70 thf(f441,plain,(
% 2.99/1.70 ~(~! [X0 : list @ char > paraco415392788lle_tv] : (($true = (ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[Y0 : nat]: ($true)))))) => ((paraco876059933e_eval @ X0 @ p) = (paraco2040174112le_Det @ $true))) <~> ! [X2 : list @ char > paraco415392788lle_tv] : ((paraco876059933e_eval @ X2 @ p) = (paraco2040174112le_Det @ $true)))),
% 2.99/1.70 inference(fool_elimination,[status(thm)],[f440])).
% 2.99/1.70 thf(f440,plain,(
% 2.99/1.70 ~(~! [X0 : list @ char > paraco415392788lle_tv] : ((ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X0 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[X1 : nat] : ($true))))) => ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X0 @ p))) != ! [X2 : list @ char > paraco415392788lle_tv] : ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X2 @ p)))),
% 2.99/1.70 inference(rectify,[status(thm)],[f290])).
% 2.99/1.70 thf(f290,negated_conjecture,(
% 2.99/1.70 ~(~! [X8 : list @ char > paraco415392788lle_tv] : ((ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X8 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[X2 : nat] : ($true))))) => ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X8 @ p))) != ! [X8 : list @ char > paraco415392788lle_tv] : ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X8 @ p)))),
% 2.99/1.70 inference(negated_conjecture,[status(cth)],[f289])).
% 2.99/1.70 thf(f289,conjecture,(
% 2.99/1.70 (~! [X8 : list @ char > paraco415392788lle_tv] : ((ord_less_eq @ set @ paraco415392788lle_tv @ (image @ list @ char @ paraco415392788lle_tv @ X8 @ top_top @ set @ list @ char) @ (paraco506338565domain @ (collect @ nat @ (^[X2 : nat] : ($true))))) => ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X8 @ p))) != ! [X8 : list @ char > paraco415392788lle_tv] : ((paraco2040174112le_Det @ $true) = (paraco876059933e_eval @ X8 @ p)))),
% 2.99/1.70 file('/export/starexec/sandbox/tmp/tmp.SMTCspLT4x/DTF2THF_25569.p',conj_0)).
% 2.99/1.70 thf(f1321,plain,(
% 2.99/1.70 ~spl71_1 | ~spl71_2),
% 2.99/1.70 inference(avatar_split_clause,[status(thm)],[f1103,f1318,f1314])).
% 2.99/1.70 thf(f1103,plain,(
% 2.99/1.70 ((paraco876059933e_eval @ sK12 @ p) != (paraco2040174112le_Det @ $true)) | ((paraco2040174112le_Det @ $true) != (paraco876059933e_eval @ sK11 @ p))),
% 2.99/1.70 inference(cnf_transformation,[status(thm)],[f943])).
% 2.99/1.70 % SZS output end Proof for DTF2THF_25569
% 2.99/1.70 % (25673)------------------------------
% 2.99/1.70 % (25673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.70 % (25673)Termination reason: Refutation
% 2.99/1.70
% 2.99/1.70 % (25673)Memory used [KB]: 7291
% 2.99/1.70 % (25673)Time elapsed: 0.060 s
% 2.99/1.70 % (25673)Instructions burned: 109 (million)
% 2.99/1.70 % (25673)------------------------------
% 2.99/1.70 % (25673)------------------------------
% 2.99/1.70 % (25667)Success in time 0.09 s
%------------------------------------------------------------------------------